2018 Mathematics G9 Q3 Past Paper PDF

2018 LEARNING MODULE Mathematics G9 | Q3 Geometry, Parallelogram and Triangle Similarity  NOTICE TO THE SCHOOLS This learning module (LM) was developed by the Private Education Assistance Committee under the GASTPE Program of the Department of Education. The learnin...

2018 LEARNING MODULE Mathematics G9 | Q3 Geometry, Parallelogram and Triangle Similarity  NOTICE TO THE SCHOOLS This learning module (LM) was developed by the Private Education Assistance Committee under the GASTPE Program of the Department of Education. The learning modules were written by the PEAC Junior High School (JHS) Trainers and were used as exemplars either as a sample for presentation or for workshop purposes in the JHS In- Service Training (INSET) program for teachers in private schools. The LM is designed for online learning and can also be used for blended learning and remote learning modalities. The year indicated on the cover of this LM refers to the year when the LM was used as an exemplar in the JHS INSET and the year it was written or revised. For instance, 2017 means the LM was written in SY 2016-2017 and was used in the 2017 Summer JHS INSET. The quarter indicated on the cover refers to the quarter of the current curriculum guide at the time the LM was written. The most recently revised LMs were in 2018 and 2019. The LM is also designed such that it encourages independent and self-regulated learning among the students and develops their 21st century skills. It is written in such a way that the teacher is communicating directly to the learner. Participants in the JHS INSET are trained how to unpack the standards and competencies from the K-12 curriculum guides to identify desired results and design standards-based assessment and instruction. Hence, the teachers are trained how to write their own standards-based learning plan. The parts or stages of this LM include Explore, Firm Up, Deepen and Transfer. It is possible that some links or online resources in some parts of this LM may no longer be available, thus, teachers are urged to provide alternative learning resources or reading materials they deem fit for their students which are aligned with the standards and competencies. Teachers are encouraged to write their own standards-based learning plan or learning module with respect to attainment of their school’s vision and mission. The learning modules developed by PEAC are aligned with the K to 12 Basic Education Curriculum of the Department of Education. Public school teachers may also download and use the learning modules. Schools, teachers and students may reproduce the LM so long as such reproduction is limited to (i) non-commercial, non-profit educational purposes; and to (ii) personal use or a limited audience under the doctrine of fair use (Section 185, IP Code). They may also share copies of the LM and customize the learning activities as they see fit so long as these are done for non-commercial, non-profit educational purposes and limited to personal use or to a limited audience and fall within the limits of fair use. This document is password-protected to prevent unauthorized processing such as copying and pasting. MATHEMATICS 9 Module 3: Geometry Parallelogram and Triangle Similarity MODULE INTRODUCTION AND FOCUS QUESTION(S): v Developed by the Private Education Assistance Committee 1 under the GASTPE Program of the Department of Education Have you ever wondered how engineers make structural designs? How do artist choose shapes in their art pieces? In this module you will try find answers to these questions. You will acquire knowledge and skills needed to solve problems involving shapes and geometric relationships. You will learn how to name and classify quadrilaterals. You will also verify some of the properties of parallelograms, rectangles, rhombuses, squares, trapezoids and kites and use these properties to differentiate one from the other. It is hoped that you will acquire deep understanding of the lesson to enable you to determine the best way to solve problems involving quadrilaterals and triangle similarity. A quadrilateral can be classified into many different forms, but in this learning module you will focus on the most important family of quadrilaterals—trapezoids, parallelograms and kites, along with their sub-shapes. You will explore uses of quadrilaterals in real life. Make sure to write down insights gained on how understanding of quadrilaterals can be effectively used in real life as you do the various learning tasks in this module. To be able to succeed in this module you need to ensure that you have a good understanding of what polygons are, the relationship been sides and angles as well as concepts of parallel and perpendicular lines. This module contains four lessons. Each lesson includes applications of the shapes to real life situations for better appreciation of the topics. Remember to search for the answer to the following question: What is the best way to solve problems involving quadrilaterals and triangle inequality? MODULE LESSONS AND COVERAGE: In this module, you will examine this question when you take the following lessons: Lesson No. Title You’ll learn to… Lesson 1 Parallelogram Identify quadrilateral that are parallelogram, trapezoid and kite Determines the conditions that guarantee a quadrilateral is a parallelogram. Use properties to find measures of angles, sides and other quantities involving parallelograms Prove theorems on different kinds of parallelograms. Developed by the Private Education Assistance Committee 2 under the GASTPE Program of the Department of Education Lesson No. Title You’ll learn to… Prove the Midline Theorem Lesson 2 Trapezoid and Prove theorems on trapezoids and kites Kites Describe special trapezoid and their properties Demonstrate uses of quadrilaterals in real life. Solve problems involving parallelograms, trapezoids and kites. Lesson 3 Triangle Describes a proportion. Similarity Applies the fundamental theorems of proportionality to solve problems involving proportions Illustrates similarity of figures. Proves the conditions for similarity of triangles a. SAS Similarity Theorem b. SSS Similarity Theorem c. AA Similarity Theorem d. Right Triangle Similarity Theorem Lesson 4 Special Right Illustrates Special Right Triangle Theorem. Triangle Applies the theorems to show that give triangles are similar. Proves the Pythagorean Theorem. Solves problems that involve triangles similarity and right triangles. Developed by the Private Education Assistance Committee 3 under the GASTPE Program of the Department of Education  Concept Map of the Module Here is a simple map of the above lessons you will cover: Quadrilateral Triangle Pythagore s Similarity an SAS Special Kites Right Parallelogra Trapezoi Triangle ms ds SSS Rectangle Isosceles s Trapezoi 30-60- AA 90 Squares Right 45-45-90 Trapezoi Right Triangle Similarit Scalene y Rhombi Trapezoi Midline Theorem Applications Developed by the Private Education Assistance Committee 4 under the GASTPE Program of the Department of Education  Expected Skills To do well in this module, you need to remember and do the following: 1. Carefully read the module and do the activities neatly and accurately. 2. Break tasks into manageable parts. 3. Complete all activities even if you may not be asked to hand these in, but they will help you learn the material. 4. Keep copies of all accomplished activities. These are needed to assess your progress and for grading. 5. If you are having problems, do NOT wait to request help. The longer you wait the bigger the problem becomes! 6. Form study groups if possible. Developed by the Private Education Assistance Committee 5 under the GASTPE Program of the Department of Education PRE-ASSESSMENT Let’s find out how much you already know about this module. Click on the letter that you think best answers the question. Please answer all items. After taking this short test, you will see your score. Take note of the items that you were not able to correctly answer and look for the right answer as you go through this module. 1. Which statement best differentiates squares from the rectangles? A. Squares must have four 90˚ angles, rectangles do not have all 90˚angles. B. Squares have two sets of equal sides, rectangles have only one pair of equal sides. C. Squares have four equal sides. Rectangles have two pairs of equal opposite sides. D. Squares have the diagonals that bisect each other. Rectangles have diagonals that are perpendicular. 2. When comparing a trapezoid and a kite, one similarity is: A. They both have congruent diagonals. B. They both have at least one set of parallel sides. C. They both have four congruent sides. D. They both have four sides 3. Points (2,1), (4,3) and (3,4) are vertices of a quadrilateral. What should be the coordinates of the fourth point to form a parallelogram? A. (1,2) B. (2, 4) C. (1, 4) D. (1,3) 4. Suppose we are told two things about a quadrilateral: first, that it is aparallelogram, and second, that one of its interior angles measures 60°. The measure of the angle adjacent to the 60° angle is A. 60° B. 90° C. 120° D. impossible to know without more information 5. ABCD is a rhombus. How will you prove that its diagonals are perpendicular? Developed by the Private Education Assistance Committee 6 under the GASTPE Program of the Department of Education  A. Show AXB   CXB, Since AC and BD intersect to form congruent adjacent angles, AC  BD. B. Show AXB CXB. Since congruent parts of congruent triangles are congruent, then, AC  BD. C. Show BAD BCD. Since congruent parts of congruent triangles are congruent, then AXB CXB , then AC  BD. D. Show AXB CXD. Since AXB CXD , then AC  BD. 6. A carpenter accurately measure four boards to frame a door: two sides of 8 inches and a top and bottom of 40 inches. What else should a carpenter do to ensure that it will fit a rectangular door? A. Set one side piece at right angle to the floor piece. B. Ensure that the sides parallel. C. Connect the opposite vertices with a pieces of woods of equal length forming diagonals. D. Connect the two sides at their midpoints with a piece of wood. 7. Maria knows the following information about quadrilateral BEST: BT = ES , TS // BE , and T   S. Maria concludes that BEST is an isosceles trapezoid. Why can’t Maria make this conclusion? A. BEST is a rectangle B. BEST is a square C. There is not enough information D. Length of BT is and ES not known. 8. A Mothers’ Club is making a quilt consisting of squares with each side measuring 40 cm. The quilt has five rows and 6 columns and with cord edging. How many meters of cord should the Club buy for the quilt? A. 880 B. 88 C. 9 D. 8 9. If you are to design a room in the attic of a Victorian style house which looks like an isosceles triangle in the front and back view whose ceiling is parallel to the floor, furniture and fixtures are also designed in such a way to maximize the space. The possible things which may happen includes the following; Developed by the Private Education Assistance Committee 7 under the GASTPE Program of the Department of Education  1. The floor area is wider than the ceiling. 2. The ceiling is wider than the floor area. 3. The bed can be attached to the side wall. 4. The built-in cabinets on the side wall are rectangular prisms. A. 1 only B. 2 only C. 1 and 3 only D. 1 and 4 only 10. One liter of a certain paint can cover about 80 square feet. I want to paint a circle with a diameter of 28 ft. How many liters of paint will I buy? A. 7 B. 8 C. 196 D. 616 11. What is the longest stick that can be placed inside a box with inside dimensions of 24 inches, 30inches, and 18 inches? A. 38.4 inches B. 30 C. 30 2 inches D. Can no be determined 12. A triangular plot of land has boundary lines 45 meters, 60 meters, and 70 meters long. The 60 meter boundary line runs north-south. Is there a boundary line for the property that runs due east-west? A. Yes. It’s the 45 meters boundary. B. Yes. It’s the 70 meters boundary. C. No. The plot is not a right triangle. D. Cannot be determined. 13. Which of the following supports statement 3 in the proof? C Figure: b a B m D n A Statements Reasons 1.Draw CD  AB. 1. There is only one and only one line perpendicular to a given line form an external point. 2. CD is the altitude to AB. 2. Definition of Altitude 3. h2 = mn 3.________ a2 = m(m+n) b2 = n(m+n) 4.a2 + b2 = m(m+n) + n(m+n) 4. Addition Property of Equality Developed by the Private Education Assistance Committee 8 under the GASTPE Program of the Department of Education  5.a2 + b2 = (m+n)(m +n) 5. Factoring 6. m + n = c 6.Definition of Betweeness 7. a2 + b2 = (c)(c) 7.Addition Property of Equality a2+b2 = c2 Simplifying A. Geometric Mean Theorem B. SSS Similarity Theorem C. SAS Similarity Theorem D. AA Similarity Theorem 14. A new pipeline is being constructed to re-route the water flow around the exterior of the City Park. The plan showing the old pipeline and the new route is shown below. About how many extra miles will the water flow once the new route is established? A. 24 B. 68 C. 92 D. 160 15. The dimensions of a rectangular doorway are 200 cm by 80 cm. Can a circular mirror with a diameter of 210 cm be carried through the doorway? A. No. Diameter is longer than the length of the door. B. Yes. The diameter is less than 200 cm by 80 cm. C. Yes. Hold it along the diagonal of the door. D. Additional information is needed to answer the question. 16. A baseball diamond is a square. The distance fromthe base to base is 90 ft. How far does the second baseman throw a ball to home plate? A. 6 5 ft B. 180 ft C. 90 2 ft D. 45 5 ft 17. Makee would like to have a new corner cabinet for his room. He is trying to figure out how to design it so that his TV which is 30” high, 34 inches wide and 20 “ deep would fit. He wants the new cabinet to be the same length on each side (along the two walls). How long should each side of the cabinet be? A. 84 inches Developed by the Private Education Assistance Committee 9 under the GASTPE Program of the Department of Education  B. 64 inches C. 42.2 inches D. 37 inches 18. Which triangles must be similar? 20 34 A. Two obtuse triangles B. Two scalene triangles with congruent bases C. Two right triangles D. Two isosceles triangles with congruent 19. Which of the following best describes the triangles at the right? A. Both are similar and congruent B. Similar but not congruent. C. Congruent but not similar D. Neither similar nor congruent. 20. SM ║ EL. Which of the following guarantee that ∆ SME is similar to ∆CDE? A. SAS Similarity B. SSS Similarity C. AA Similarity D. None of the above Did you do well in the pretest? Are there items that you were not sure of your answers? You can go back to those items as you gain new knowledge and skills. Now, proceed to Lesson 1. Take time to list ideas and concepts in the lessons. Developed by the Private Education Assistance Committee 10 under the GASTPE Program of the Department of Education Lesson 1: Quadrilaterals In this lesson you will learn the following: 1. Describe the properties of quadrilaterals. 2. Determine the conditions that guarantee a quadrilateral is a parallelogram. 3. Use properties to find measures of angles, sides and other quantities involving parallelograms 4. Prove theorems on different kinds of parallelograms. 5. Demonstrate uses of quadrilaterals in real life. 6. Solve problems involving parallelograms. What are quadrilaterals? How do we use them? Where do we see them in real life? Quadrilaterals are everywhere. We see them on signs, buildings, work of art, books, computers, floor designs and many more. A polygon with four sides is a quadrilateral. In this lesson you will know more about quadrilaterals. Start by doing Activity 1. ACTIVITY 1a Map It Arrange the boxes to form a concept map that will show the relations between and among the different shapes. Quadrilaterals Parallelograms Trapezoids Squares Rhombi kites Rectangles Isosceles Right Trapezoids Trapezoids Developed by the Private Education Assistance Committee 11 under the GASTPE Program of the Department of Education Your concept map here. Questions to Answer: 1. What is the basis of your arrangement? 2. How can you differentiate a shape from the other? 3. How can we use the properties of quadrilaterals to create designs and solve problems? YOUR ANSWERS In Activity 1, you made a concept map on your initial understanding of the different quadrilaterals. In the next activity you will see actual use of these shapes in real life. Take note of some questions that you might want to ask from your teacher especially on the reason why a shape is used in a particular situation. Developed by the Private Education Assistance Committee 12 under the GASTPE Program of the Department of Education  ACTIVITY 2a Quadrilaterals Everywhere Below are some pictures of quadrilaterals in real life. Can you identify them? Can you give possible reasons why the shape is such? A B C D E F You see why you need to have a good knowledge of quadrilaterals. Identifying these shapes and the understanding the properties will enable you to use appropriately these shapes. Questions to Answer: Can you cite other examples of the uses of the quadrilaterals in real life? In the next activity you will learn how to classify quadrilaterals. Doing so, will help you identify similarities and differences between and among quadrilaterals. At the end of this module you are expected to make a model that will demonstrate the best solution to a problem. Challenge yourself to finish the module in not more than two weeks. Developed by the Private Education Assistance Committee 13 under the GASTPE Program of the Department of Education  ACTIVITY 3a To Be or Not To Be Given a set of quadrilaterals, group them based on their common features by dragging them in the column for examples. Name and describe each group in terms of their identifying characteristics in 5 minutes. Type of Definition Examples Quadrilaterals Questions to Answer: 1. What are quadrilaterals? 2. How are the different types of quadrilaterals named? 3. How are quadrilaterals denoted? 4. Are there other ways of grouping quadrilaterals? How? 5. What are the properties of each quadrilateral? 6. Why is there a need to know the properties of these quadrilaterals? Developed by the Private Education Assistance Committee 14 under the GASTPE Program of the Department of Education  YOUR ANSWERS Now, you have a clear idea of what quadrilaterals are. In the next activity, you will build up your knowledge and skills in parallelograms by mastering their properties. End of EXPLORE: You just have classified quadrilaterals. Let us now strengthen that insight by doing the succeeding activities. What you will be learning in this section will help you perform well in your final performance task which will challenge you to use what you know to create a design that will help you use the materials efficiently or maximize the use of space. How do you know a shape is a quadrilateral? Why are quadrilaterals important? How are they used in real life? What is the best way to solve problems involving quadrilaterals? Discover the answers to these questions as you do the next tasks. Your goal in this section is to have a good understanding of the properties of quadrilaterals. Properties of quadrilaterals will enable you to differentiate one from the other and use these shapes efficiently. Developed by the Private Education Assistance Committee 15 under the GASTPE Program of the Department of Education  ACTIVITY 4a Show and Tell Access http://www.mathsisfun.com/geometry/quadrilaterals-interactive.html. Click on angles and diagonals to answer the questions below. Drag the dots, observe the measures of the angles and the length and positions of the diagonals. Use your discovery to answer the following questions. Questions to Answer: 1. Given a quadrilateral ABCD, when are its side said to be opposite? Consecutive? 2. What angle is opposite A ? B? C?D? 3. Which are consecutive angles? 4. What is the sum of the interior angles of a quadrilateral? How did get this number? 5. Can an interior angle of a quadrilateral measure more than 180˚? Draw a sample of this quadrilateral. Your answers here. Developed by the Private Education Assistance Committee 16 under the GASTPE Program of the Department of Education  ACTIVITY 5a Try This Out Use your answers to the questions above to find the measure of the indicated angles below. Drag the points to show the measures of the four angles. Are your answers correct? E M a) mE ? 70 92 85 H O b) What is x? F A 108 2x + 18 x + 30 R 84 I Now summarize the properties of a quadrilateral by completing the phrase” polygon is a quadrilateral if …” considering the following: A polygon is a quadrilateral if … Sides vertices Interior angles Developed by the Private Education Assistance Committee 17 under the GASTPE Program of the Department of Education  ACTIVITY 6a Check your Understanding Do self-assessment by accessing the website http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SH AP&ID2=AB.MATH.JR.SHAP.SHAP&lesson=html/video_interactives/classificatio ns/classificationsInteractive.html Click on new and answer the items. Click check when you are done. Did you do well in the quiz? Are there questions that you would like to ask? Post it in the Discussion Forum. Before you proceed to the next activity do a self- assessment. This help you identify areas that you are doing well and areas that you need to work harder. ACTIVITY 7a Shaping Up Review 1- thing I loved about quadrilaterals 1- thing that is going around my head at this point … 1. 3-things 2. I learned Developed by the Private Education Assistance Committee 18 under the GASTPE Program of the Department of Education Your answers here. From the previous activity, you tried to classify quadrilaterals into three groups: parallelogram, trapezoid and kite. How can you differentiate one from the other? In what way are they similar? How can knowledge on the properties of these shapes help you determine their best use? 1.2 Parallelograms In this section you will firm up your knowledge on quadrilaterals by identifying special quadrilaterals known as parallelograms. Questions that you need to answer in this section are: When is a quadrilateral called a parallelogram? How can you find measures of angles, sides and other quantities involving parallelograms? How are parallelograms used in real life? Write down your answers to these questions as you do the next activities. Properties of Parallelograms In the next activity you will focus on quadrilaterals which are parallelograms. Here you will put to use your knowledge on parallel lines and their transversals to identify properties of parallelograms. Developed by the Private Education Assistance Committee 19 under the GASTPE Program of the Department of Education  ACTIVITY 8a Draw and Discover Use the diagram to help you follow the instructions below. You are expected to identify properties of quadrilateral formed by parallel lines in terms of the relationships of its sides, angles and diagonals.  A C  D B Questions to Answer: 1. Draw two horizontal lines passing through points A and B and name them 1 and 2. 2. Draw a transversal t 1 passing through point A. 3. Draw a transversal t 2 passing through B parallel to the first transversal. How do you know the transversals are parallel? 4. Label the other intersections C and D. 5. Measure the length of the two opposite sides AC and DB. What can you say about the two opposite sides? Do the same for sides AD and CB. What theorem on parallel lines supports your answer? 6. Now, cut the parallelogram and duplicate by tracing the cut out figure. 7. Using a straight edge connect the two opposite vertices A and B and cut along this line. Developed by the Private Education Assistance Committee 20 under the GASTPE Program of the Department of Education  8. Superimpose the two triangles such that corresponding parts coincides. Does one triangle completely cover the other triangle? Are the corresponding sides congruent? Are the corresponding angles congruent? Are the two triangles congruent? 9. Do the same for the duplicate parallelogram connecting CD and cutting along this diagonal. 10. By superimposing the cut out triangles, can you say that the triangles formed are congruent? What conclusion can you make about the diagonals of the parallelogram? 11. Using the cut out figures show that diagonals of a parallelogram bisect each other. Describe your process. 12. With the help of the same cut out figures answer the question: Are the diagonals of a parallelogram equal? 13. When is a quadrilateral said to be a parallelogram? Summarize your findings by completing the table below: Write your answers here… If a quadrilateral is a parallelogram, EXAMPLE then... SIDES ANGLES Developed by the Private Education Assistance Committee 21 under the GASTPE Program of the Department of Education  DIAGONALS Now check your findings by going back to the interactive site: http://www.mathsisfun.com/geometry/quadrilaterals-interactive.html. Click on parallelograms and angles. Drag the points to verify your findings. Click on diagonals. Drag points and verify your findings. Is there a need to revise your initial findings. Go back to your summary table and finalize your answers. Can you draw other shapes of parallelograms? Now that you know the properties of parallelograms, put it to use by answering the next activity. ACTIVITY 9a Parallelograms Challenge Use the properties of parallelograms to answer the following: 1. Find the measures of the other angles of parallelogram CDEB. B O 14x-2 124 N E 8x+6 2. In the figure below, find m A, m C, m  D, CD, and AD. A  12 B  100 6 D  C 3. Find m  MOE, m  NOE, and m MYO. Developed by the Private Education Assistance Committee 22 under the GASTPE Program of the Department of Education  M O Y E N mMON= 120 Write your answers here… Were you able to answer all the items? Are there questions that you would like to ask? Post it in the Discussion Forum. Now rate your progress in understanding of the lesson based on your performance in Activities 6. I need to shine Leveling up! Good job! Excellent! my star! You have learned in the previous section that parallelograms are special type of quadrilaterals. Are there special types of parallelograms? How do you differentiate one from the other? You will continue to build up your understanding of parallelograms in the succeeding activities. Developed by the Private Education Assistance Committee 23 under the GASTPE Program of the Department of Education END OF FIRM - UP In this section you have learned that parallelograms are quadrilaterals. What make them special quadrilaterals are their unique properties: they have two pairs of opposite sides that are parallel, opposite angles are congruent and consecutive angles are supplementary. In the next section you will deepen your knowledge on parallelograms and study special parallelograms. When are parallelograms said to be special? Now you know that a quadrilateral can be a parallelogram. In the next activity, you will put to test what you know about parallelograms. Doing this activity will deepen your knowledge on parallelograms and help you solve problems involving quadrilateral. After this section answer the question: What is the best way to solve problems involving quadrilaterals? Then compare your answer to your Classifying Parallelograms ACTIVITY 10 What Makes Them Special? Study the special quadrilaterals above. Like markings denote congruent sides or angles and arrows denote parallel sides. When is a parallelogram a rectangle? A square? A rhombus? Summarize your answers in the table below. Special Definition Draw an illustrative example. If Parallelogram ABCD is a … Rectangle Square Developed by the Private Education Assistance Committee 24 under the GASTPE Program of the Department of Education  Rhombus Look at your definitions of the three quadrilaterals. Will you be able to compare and contrast a rectangle from the square? from a rhombus? Are there other ways of representing these parallelograms? The rectangle, rhombus, and square have a few other special properties. First, remember that these figures are all parallelograms; therefore, they possess the same properties as any parallelogram. However, because these figures are special parallelograms, they also have additional properties. What are the properties of special quadrilaterals? How can you use these properties to differentiate one from the other? How can these properties help you identify the best way to solve problems involving quadrilaterals? You will discover these special properties by doing an investigation. 1.2.1 Rectangles In this section you will deepen your knowledge on special parallelogram called rectangle. What makes it special? How is it different from the other types of parallelograms? Discover answers to these questions by doing the next activity. ACTIVITY 11a When Am I a Rectangle? Read and answer the questions below about rectangles. A B A rectangle is a parallelogram with equal angles What is the full meaning of this definition? If the rectangle is to be equiangular, what is D C the measure of each angle? Draw the diagonals. What relationship exist between the diagonals? Developed by the Private Education Assistance Committee 25 under the GASTPE Program of the Department of Education Summarize your answer in the table below. If a parallelogram a rectangle , then... EXAMPLE SIDES ANGLES DIAGONALS Verify your answer by accessing the same website. This time, click rectangles and angles. Click on diagonals. Observe. Do the measures of the interior angles change when you increase or decrease the length of the rectangle? Now let us go back to our previous activity about quadrilateral let us see if you have now a better understanding. Lesson 1.1 Quadrilaterals What are quadrilaterals? How do we use them? Where do we see them in real life? Quadrilaterals are everywhere. We see them on signs, buildings, work of art, books, computers, floor designs and many more. A polygon with four sides is a quadrilateral. In this lesson you will know more about quadrilaterals. Start by doing the next Activity. ACTIVITY 12a Map It Arrange the boxes to form a concept map that will show the relations between and among the different shapes. Quadrilaterals Parallelograms Trapezoids Squares Rhombi kites Rectangles Isosceles Right Trapezoids Trapezoids Developed by the Private Education Assistance Committee 26 under the GASTPE Program of the Department of Education Your concept map here. Questions to Answer: 1. What is the basis of your arrangement? 2. How can you differentiate a shape from the other? 3. How can we use the properties of quadrilaterals to create designs and solve problems? Write your answers here…. Proving that the Diagonals of Rectangles are Congruent ACTIVITY 13a Are we Congruent? Given: Rectangle FLAG. F L Prove: FA  LG G A Developed by the Private Education Assistance Committee 27 under the GASTPE Program of the Department of Education Do this by supplying the reasons for each of the given statement. We know that Because… FL = GA LFGFGA FG = FG GFL  GAL GL = FA What did you realize about the diagonals of a rectangle? Write it down. Diagonals of rectangles are ______________________________________. Now probe deeper into the properties of rectangles. You have discovered that diagonals of rectangles are congruent. Now prove that the diagonals bisect each other. Proving Properties of Rectangles ACTIVITY 14a Mutually Bisecting Diagonals? ABCD is a rectangle. Prove the diagonals of a rectangle bisect each other. A B O D C Developed by the Private Education Assistance Committee 28 under the GASTPE Program of the Department of Education  We know that Because… ABDC BACACD BDCABD AOB  DOC AO=OC DO=OB Conclusion: Now generalize the properties of a rectangle by completing the table below. A parallelogram is a rectangle if … Sides Interior angles diagonals Now that you have been writing proofs for the different properties of parallelograms, reflect on your experiences of learning to write proofs. ACTIVITY 15a Auto—math-ography What does it mean to write a mathematical proof? Write in few sentences your explanation. Your explanation here Now use the properties to answer the exercises below. Developed by the Private Education Assistance Committee 29 under the GASTPE Program of the Department of Education  ACTIVITY 16a Rectangles Challenge P R O Y A 1. Given: PA = 18, mRYA = 35. Find: RY, PR, PO, mRPA, mPOY 2. In figure below , find BD, AO,OC, DO, and OB. Given: AC = 16 A B O D C Your answers here. Developed by the Private Education Assistance Committee 30 under the GASTPE Program of the Department of Education  ACTIVITY 17a My Golden Rectangle The Golden Rectangle is proposed to be the most aesthetically pleasing of all possible rectangles. This is the reason why it has been used extensively in art and architecture. The most prominent and well known uses of the Golden Rectangle in art were created by the great Italian artist, inventor, and mathematician, Leonardo da Vinci. In da Vinci’s "Mona Lisa" Golden Rectangle frames central elements in the composition. If you draw a rectangle whose base extends from the woman's right wrist and extend the rectangle vertically until it reaches the very top of her head, you will have a Golden Rectangle. Then, if you draw squares inside this Golden Rectangle you will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth. Questions to Answer: 1. Why is it called a Golden Rectangle? 2. How can one draw a Golden Rectangle? 3. What are the other uses of Golden Rectangles in real life? 4. Make your own presentation using the golden rectangle. As soon as done share it with your teacher. Your answers here. Developed by the Private Education Assistance Committee 31 under the GASTPE Program of the Department of Education Go to the website: http://www.youtube.com/watch?v=suiDK61jAc8 (The Golden Ratio). Or go to: http://goldenratiorocks.wordpress.com/golden-ratio-real-life-examples/ You will see more examples on the uses of golden rectangles and watch a video. My Golden Rectangle When this activity is done, set an appointment with your teacher for a chat. Make sure that you did all the task and have answered questions embedded in each activity. Your teachers will ask you questions related to what you have studied. ACTIVITY 18a Let’s Chat Answer the questions of your teachers to the best you can. Don’t hesitate to ask your own questions if you have any at the end of your chat. In the next activity you will deepen your understanding on the properties of squares. ACTIVITY 19a The Power of the Square The Power of the Center explains why most painting prefers squarer shape. As we move from the center things lose importance. Also, as we'll see, a squarer format is consistent with rest, repose, dignity, and timelessness; things that artists often want their paintings to convey. But the square format has one property that the rectangular does not have; it gives a scene stillness and serenity, a calm and dignity associated with the round format. This makes it ideal for a subjects such as a Madonna. Developed by the Private Education Assistance Committee 32 under the GASTPE Program of the Department of Education Other than art the square is also used in architecture. One of the most famous structure the made use of squares is the Petronas Twin Tower of Malaysia. The design of the tower is composed of an 8-point star formed by intersecting squares. This is a common characteristic of a Muslim architecture. Questions to Answer: 1. Where else in real life do we usually find squares? 2. Cite an instance where square is a better option than the other parallelograms. 3. What are the properties of the squares that make them so useful? Your answers here. ACTIVITY 20a Closer Look at the Square Make three different sizes squares. Investigate the relationships of the sides and the angles. Developed by the Private Education Assistance Committee 33 under the GASTPE Program of the Department of Education  Questions to Answer: 1. Fold the biggest square so that the opposite sides coincide. Are the opposite sides equal? Do the same for the other squares. Is your observation the same for the smaller squares? 2. Draw the diagonals and measure. What can you say about the measures of the diagonals? 3. Fold the squares so that the four vertices coincide. Are the angles equal? What are the measures of the angles? What theorem supports your answer. 4. What kind of quadrilateral is a square? Based on this fact what are the other properties of a square? 5. Summarize your findings by completing the table below. If a parallelogram a square , then... EXAMPLE SIDES ANGLES DIAGONALS Now check your findings by going back to the interactive site: http://www.mathsisfun.com/geometry/quadrilaterals-interactive.html. Click on squares and angles. Drag the points to verify your findings. Click on diagonals. Observe. Are your findings consistent with what you have observe? Developed by the Private Education Assistance Committee 34 under the GASTPE Program of the Department of Education Now you know the special properties of the squares. These are the properties that makes them very useful. Challenge yourself to use these knowledge by doing the next activity. ACTIVITY 21a Designing with a Square Look at ads, magazines, brochures, logos, and other printed projects and try to find as many different examples of square shapes. Study the designs. Identify examples of square shapes that convey the attributes of honesty, stability, equality, comfort, or familiarity. Which designs convey rigidity or uniformity? Design your own presentation using a square. What is the most important feature of your design? Use the resources of voki.com to explain your output. When done share your design by publishing it in a social media. Tag your teacher for the evaluation of your output. Your Output here Now you know two of the very special parallelograms: rectangles and squares. The third type is the rhombus. Discover its properties by doing the next activity. Developed by the Private Education Assistance Committee 35 under the GASTPE Program of the Department of Education 1.2.3 Rhombus A rhombus is a special kind of parallelogram. Many architectural structures and art pieces make use of rhombi. Knowing about the special properties of a rhombus is important to identifying and using these special parallelograms. Access the following sites to see examples. 1. http://www.youtube.com/watch?v=i2a4B4M5L1M Watch how a rhombus can be used to make a flexible paper structure. 2. http://www.youtube.com/watch?v=S-nNib5HzUA Gives another type of flexible paper structure. 3. http://www.youtube.com/watch?v=p9xKxEV1FkY Demonstrate how to make an Origami Fireworks making use of rhombi shapes. 4. http://www.youtube.com/watch?v=knMEBSXM6WU Demonstrate how to make use of rhombi to make an origami flexiball. 5. http://rhombusspace.blogspot.com/ Gives examples of art pieces using rhombi. 6. http://www.ysjournal.com/article.asp?issn=0974- 6102;year=2009;volume=2;issue=7;spage=35;epage=46;aulast=Khair Gives examples of the uses of rhombi in architecture 7. www.photoxpress.com/photos-skyscraper-lozenge-rhombus-4723361 Gives photos depicting how rhombi are used in architecture British Museum has glass, tessellated What makes rhombus very useful as a shape? What makes it flexible? Know more about the rhombus in the next activities. Take note of these questions as you do the activities. Developed by the Private Education Assistance Committee 36 under the GASTPE Program of the Department of Education Your answers here. ACTIVITY 22a Rhombus under Scrutiny Access: http://www.mathsisfun.com/geometry/quadrilaterals- interactive.html. Click on rhombus and drag the points. Observe. Are there changes in the lengths of sides? Click on angles and drag. Observe what happens to the measures of the angles. Click on diagonals. Drag and Observe. Questions to Answer: 1. What relationships exist among the consecutive sides? The opposite sides? 2. What relationships exist between the opposite angles? Consecutive angles? 3. What relationship exist between diagonals? 4. Summarize your findings below. Developed by the Private Education Assistance Committee 37 under the GASTPE Program of the Department of Education  If a parallelogram is a rhombus, EXAMPLE then... SIDES ANGLES DIAGONALS What meaningful insight did you gain from the activity? Write this insight in your Learning Log. Have you discovered what makes rhombi flexible? The next activity will deepen your understanding of rhombi. You will investigate on the relationship between diagonals. ACTIVITY 23a Always at the Right! Given: ABCD is a rhombus. Show that its diagonals AC and BD are perpendicular. B O A C D We know that… Because… 1. AC bisect BD 2. AB = BC, AD = DC 3. ACBD You have established that diagonals of a rhombus are perpendicular to each other. You will not stop here. Investigate some more. This time on the relationship of diagonals and angles. Developed by the Private Education Assistance Committee 38 under the GASTPE Program of the Department of Education  ACTIVITY 24a Great Angle Dividers Show that Perpendiculars of a rhombus bisect the angles. B A O C D We know that… Because… 1. BCD  BAD 2. CBD ABD, CDB ADB 3. BD bisect ABC and ADC 4. ABC  ADC 5. CBD ABD, CDB ADB Thus: Now that you have investigated on the properties of the rhombi, use your knowledge by answering the next activity. Developed by the Private Education Assistance Committee 39 under the GASTPE Program of the Department of Education  ACTIVITY 25a The Rhombus Challenge Task: Given ABCD is a rhombus. Find: a. mDCE b. mBAE c. mEAD d. mECD e. mEBA f. mEBC We know that Because… 1. mBEA = 2. mDCE = 3. mDAB = mDCB Opposite angles of quadrilaterals are congruent 4. mBAE = 5. mEAD = 6. mDAB + mABC = 180 Consecutive angles of a parallelogram are supplementary 7. mABC = 8. mABE = 9. mADC = 10. mADE = Proving relationships between sides and angles is a good exercise to sharpen your reasoning skills which is a very important skill in problem solving. In the next activity you will investigate on some real life uses of a rhombus. Write down ideas as you do the activity. Developed by the Private Education Assistance Committee 40 under the GASTPE Program of the Department of Education  ACTIVITY 26a Beautiful and Flexible Rhombi The Penrose rhombuses are a pair of rhombuses with equal sides but different angles. Penrose tiling at Mitchell Institute for Fundamental Physics and Astronomy Where else in real life are rhombi used? Create a list. Now create your own design using rhombi only. What kind of rhombi did you use in your design? Share your output at the discussion forum. Make sure that you attach your output or picture of your output. Output could be an art piece , an origami or structural design. You can use tessellation tools for tessellation output. Access the following sites for practice. http://www.shodor.org/interactivate/activities/Tessellate/ You can start by selecting shape then try changing corners and edges then tessellate. Your Output here Developed by the Private Education Assistance Committee 41 under the GASTPE Program of the Department of Education Now rate your progress in understanding of the lesson based on your performance in activities 1 to 22. I need to shine Leveling up! Good job! Excellent! my star! Now that you have knowledge quadrilaterals, do the next activity to ensure that you acquire skills in the construction of the following quadrilaterals. ACTIVITY 27a Measure your Progress? Access the website below and take the quiz. How well did you perform? http://library.thinkquest.org/20991/textonly/quizzes/geo/q6/test.html Now build up your skills in construction by doing the next activity ACTIVITY 28a (Level 1 Scaffold). Let’s Construct (Your skills acquired in this activity will help you later in the making of a three dimensional model.) Use one of the web 2.0 tools to construct the following quadrilaterals. a. Geometry Skecthpad b. GeoGebra 1. General Parallelograms a. One angle measures 43˚ b. Shorter side is 2 inches and one angle measures 105˚. 2. Rectangles a. shorter side is 1 inch and longer side is 3 inches. b. longer side is 3 inches diameter is 4 inches. 3. Square Developed by the Private Education Assistance Committee 42 under the GASTPE Program of the Department of Education  a. side is 3 inches 4. Rhombus a. smallest angle is 50˚ b. one side measure 2 inches and the other measures 5 inches. How did you fare in this activity? How can you apply the skills that you have gained in construction in problem solving involving quadrilaterals? Your Answers Here It is expected that at this point you have gained deep understanding of quadrilaterals and their applications. Express this understanding by revising your original concept map to reflect your new realizations. It is expected that this time your map will be more comprehensive and will reflect the relationships between and among shapes being considered. ACTIVITY 29a Do Is See A Bigger Picture Now? Activity 29a. Do Is See A Bigger Picture Now? Go back to your concept map. Revised based on what you have learned about parallelograms and make it more comprehensive. Your Answer Here Developed by the Private Education Assistance Committee 43 under the GASTPE Program of the Department of Education  Questions to Answer: 1. What changes did you make? 2. What misconceptions were you able to correct? 3. How can your knowledge of parallelograms help in identifying the best way to solve problems involving efficiency of the use of materials and space? Your Answers Here ACTIVITY 30a Let’s Sum It Up! In the table below summarize what you have learned about parallelograms. Parallelograms Properties Where best to use it? Rectangles Squares Rhombi Questions to Answer: Given a problem involving quadrilaterals, how can your knowledge of its properties help determine the best shape to use? What other factors would you consider to identify the best solution to the problem? Your answer here. Post your answers in http://whiteboard.com Developed by the Private Education Assistance Committee 44 under the GASTPE Program of the Department of Education End of DEEPEN In this section you were able to deepen your knowledge on parallelograms by classifying them into three subsets: rectangles, squares and rhombi. You have learned to construct this figures using web 2.0 technologies. In the next section, you will use your knowledge and skills gained in many situations in life to better appreciate what you are learning. In the previous sections, you gain understanding on quadrilaterals and its sub-set parallelograms. In the next activity describe insights you have gained from the various activities. Use the questions below to guide you in your reflection. ACTIVITY 31a Journal Writing  How can you tell one quadrilateral from the other?  How are quadrilaterals used in real life?  How can models be used to show solutions to problems involving quadrilaterals? Your Answer Here Should you have questions related to the questions above, click Discussion Forum and post your question Now sum up what you have learned about quadrilaterals. Should you have questions, post these questions in the Discussion Forum. In the next activity you will challenge yourself to use what you have learn to solve the problem. Developed by the Private Education Assistance Committee 45 under the GASTPE Program of the Department of Education  ACTIVITY 32a Tiling Challenge You are task to study the number and size of tiles needed for the floor of the receiving room. The room is a square with an area of 81 square meters. The whole area must be divided into 9 congruent squares, the middle square must be divided again into 9 congruent squares and the middle square must be divided again into nine congruent squares. What is the side of the smallest middle square in the pattern? Can you use the same pattern for other number of squares? Show your solution. Your solution here. What guided you in answering this challenge? How did you identify the best solution? In the next activity use the insight that you have gained to identify the best way to solve the problem below. Developed by the Private Education Assistance Committee 46 under the GASTPE Program of the Department of Education  ACTIVITY 33a Architect’s Square Parquet Floor After making a parquet floor in an office building, the carpenters had left-over pieces of wood in the shape of right triangles with sides of 1, 2, and 5. The architect would like to use these pieces for a parquet floor in his own house. He wants to know: can he make a perfect square from 20 of these triangles? If so, what will it look like? Questions to Answer: 1. How did you solve the problem? What properties of quadrilaterals did you use? 2. What other Math concepts were useful in solving the problem? 3. Is there another way of solving the problem? Which of the process do you think is the best way to solve the problem? Justify. Your Answer Here Congratulations! You have were able to finish studying the section on parallelograms. You are now ready to study the two other subsets of quadrilaterals: trapezoids and kites. The knowledge and skills that you will gain will certainly help you acquire confidence in the use of quadrilaterals to solve problems or create designs. Developed by the Private Education Assistance Committee 47 under the GASTPE Program of the Department of Education END OF TRANSFER: In this section, your task was to make different quadrilaterals with the use of web 2.0 with different conditions. How did you find the transfer task? How did the task help you see the real world use of the topic? You have completed this lesson. But you have three more lessons before you finish this module. You need to learn more about trapezoid and kites, triangle similarity, and special right triangles to complete what you need in doing your performance task. Developed by the Private Education Assistance Committee 48 under the GASTPE Program of the Department of Education Lesson 2: Trapezoids and Kites In this lesson you will learn the following: 1. Identify trapezoid, kites and their properties 2. Explore certain websites indicated in the module that would be of great help for your better understanding of the lessons on trapezoids and kites and work on the interactive activities. 3. Take down notes of the important concepts of trapezoids and kites and follow a logical sequence of statements to come up with proofs of the different theorems about trapezoids and kites. 4. Perform the specific activities or tasks and complete the exercises and assessments provided. 5. Collaborate with the teacher and peers. You have already studied the different parallelograms, now you need to explore other kinds of quadrilaterals and how they will be of use to the real-life situations. ACTIVITY 1b What’s in me? DESCRIPTION: (Brainstorming) Take a look at the pictures (a bridge and a tiling design) and note the different polygons found in it. Share your observations with a partner. http://www.google.com/url?sa=i&rct=j&q=designs%20using%20different%20triangles%2 0and%20quadrilaterals&source=images&cd=&cad=rja&docid=ZHtUpKb7CtSd8M&tbnid =_4BboANCoJ0G_M:&ved=0CAMQjhw&url=http%3A%2F%2Fwww.mathpuzzle.com%2 FAug52001.htm&ei=bXHNUrCeBsyxrgeVk4HoBA&psig=AFQjCNHLO5aKfHFDuC4OQ- 24M5oWKkAA9Q&ust=1389277514813593 Developed by the Private Education Assistance Committee 49 under the GASTPE Program of the Department of Education http://www.google.com/url?sa=i&rct=j&q=pictures%20of%20beams%20of%20hanging% 20bridge&source=images&cd=&docid=&tbnid=&ved=0CAMQjhw&url=&ei=Hm3NUpDSL 4qJrAeW8IBQ&psig=AFQjCNHQ-BrH9gtvfkQPYCgllpXcyCtf- Q&ust=1389280206982361 After sharing your observations of the given pictures, record your ideas in the first column of the generalization table below. Write in the column your ideas about the question, what is the best way to solve problems involving quadrilaterals and similar triangles? MY INITIAL MY FINDINGS SUPPORTING QUALIFYING MY THOUGHTS AND EVIDENCE CONDITIONS GENERALIZATIONS CORRECTIONS ACTIVITY 2b I KNOW NOT Directions: Based on your observations to the pictures presented, answer each of the following process questions. Write your answer in the box below. PROCESS QUESTIONS: 1. What do you see in the pictures shown? 2. What kind/s of polygons is/are found in the picture? 3. What will happen if only one geometric figure is used in these pictures say triangles or squares? 4. What do you think are the reasons why they used those figures? 5. What makes up an architectural structure and design? 6. How do you solve problems on the efficient use of materials or spaces? Developed by the Private Education Assistance Committee 50 under the GASTPE Program of the Department of Education 7. How do you ensure the accuracy of solutions in solving problems? 8. What is the best way to solve problems involving quadrilaterals and similar triangles? End of EXPLORE: You have given your thoughts and heard the ideas of others, now you will proceed by validating or correcting your ideas by doing the next activity. The concepts that you will learn in this lesson will help you accomplish the required project/task at the end which is a miniature model of a house. In this section your goal is to explore the other types of quadrilaterals specifically trapezoids and kites and their properties and understand the proofs of their theorems. Developed by the Private Education Assistance Committee 51 under the GASTPE Program of the Department of Education  ACTIVITY 3b HANDS ON! DESCRIPTION: Investigation Activity about Trapezoids and Kites Directions: Answer each of the following questions below. Write your answer on the box provided. In a triad, make a specific parallelogram using toothpicks of the same length numbered as 1-4. Cut a portion of side 1 and record the answer of the following questions; 1. What happened to the parallelogram? 2. Can you still form a quadrilateral out of the 4 toothpicks? 3. If yes, what did you do to form the quadrilateral? 4. What kind of a quadrilateral is form? 5. What are the properties of such a quadrilateral? 6. If you lengthen one of the 3original sides using the portion being cut, what kind of a quadrilateral is formed? 7. What are the properties of the new/second quadrilateral? Cut a portion of toothpick 2 such that it should be of the same length as the first. 1. Can another quadrilateral be formed? Discuss. 2. What are its characteristics? Explain. WRITE YOUR ANSWERS HERE… Developed by the Private Education Assistance Committee 52 under the GASTPE Program of the Department of Education  ACTIVITY 4b I AM WHAT I AM Concept Attainment on trapezoids and kites You have seen three other kinds of quadrilaterals so now present your findings in tabular form; Quadrilateral Quadrilateral Quadrilateral Quadrilateral 1 2 3 4 Sketch of the figure Properties of the quadrilateral Kind of quadrilateral Parts of the quadrilateral To summarize, complete the statement by giving the definition of the given term. 1. A trapezoid is a quadrilateral with __________________________ 2. The bases of the trapezoid are _____________________________ 3. An isosceles trapezoid is a quadrilateral ______________________ 4. The legs of an isosceles trapezoid are ________________________ 5. The base angles of an isosceles trapezoid are __________________ 6. A scalene trapezoid is a quadrilateral ________________________ 7. A kite is a quadrilateral with ______________________________ Compare your answers with the concepts written inside the box below: Developed by the Private Education Assistance Committee 53 under the GASTPE Program of the Department of Education If you think that everything is clear, proceed by doing the next exercise; If you fail to get the correct answers and you need to be clarified on some things, you may refer to this site for more information. http://www.onlinemathlearning.com/properties-of-polygons.html This site contains video lessons on the properties of trapezoids and kites. Learning more on the reasons on how such properties exist and why, will easily convince you to believe and understand. To help you come up with evidence it should be back up with proofs which you will do in the next activity. ACTIVITY 5b LOOK & SEE Modelled Instruction on proving theorems about trapezoids Trapezoids have certain properties that you need to learn in order for you to have a better understanding of how they would affect their functions. You have to go through and work on the succeeding activities to come up with proofs of the different theorems. Theorem 1: Base angles of an isosceles trapezoid are congruent. Developed by the Private Education Assistance Committee 54 under the GASTPE Program of the Department of Education Take a look at the proof of theorem 1 and see how the statements flow to arrive at the final conclusion. Given: CARE is an isosceles trapezoid where AC//RE Prove: ER Statement Reason 1. CARE is an isosceles triangle Given where AC//RE 2. CE ≅ AR Definition of an isosceles trapezoid 3. Draw CS⊥ER, AD⊥ER From one point, there is only one line that can be drawn perpendicular to a given line 4. ∠CSE & ∠ADR are right Angles Definition of perpendicularity 5. ▽CSE & ▽ADR are right Definition of right triangles Triangles 6. CS≅AD Two parallel lines are everywhere equidistant 7. ▽CSE≅▽ADR H-L Congruence Theorem (If in a right triangle the hypotenuse and a leg are congruent to the corresponding hypotenuse and leg of another right triangle, then the 2 right triangles are congruent.) 8. ∠E ≅∠R Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Theorem 2: If the base angles are congruent, then the trapezoid is isosceles. Developed by the Private Education Assistance Committee 55 under the GASTPE Program of the Department of Education Fill in the missing statement or reason to complete the proof of theorem 2. Statement Reason 1. Trapezoid LIFE with LI//EF & Given ∠ELI≅∠FIL 2. Draw EA ⊥ LI and FA ⊥ LI From a point to a line there is exactly one perpendicular line that can be drawn 3. ∠EAL &∠FBI are right angles 4. Definition of a right triangle 5. EA≅FB Parallel lines are everywhere equidistant 6. ∆EAL≅∆FBI LAA Theorem (If a leg and an acute angle of one right triangle are congruent to the corresponding leg and an acute angle of another right triangle, then the 2 right triangles are congruent 7. CPCTC 8. Trapezoid LIFE is isosceles Compare your answer with the complete proof of theorem 2 and check on how far you have gone with your understanding of the concepts. Proof of Theorem 2: Statement Reason 1. Trapezoid LIFE with LI//EF & Given ∠ELI≅∠FIL 2. Draw EA ⊥ LI and FA ⊥ LI From a point to a line there is exactly one perpendicular line that can be drawn 3. ∠EAL &∠FBI are right angles Definition of perpendicularity 4. ∆EAL & ∆FBI are right triangles Definition of a right triangle 5. EA≅FB Parallel lines are everywhere equidistant 6. ∆EAL≅∆FBI LAA Theorem (If a leg and an acute angle of one right triangle are Developed by the Private Education Assistance Committee 56 under the GASTPE Program of the Department of Education  congruent to the corresponding leg and an acute angle of another right triangle, then the 2 right triangles are congruent 7. EL≅FI CPCTC 8. Trapezoid LIFE is isosceles Definition of an isosceles trapezoid Theorem 3: The diagonals of an isosceles trapezoid are congruent Another way of showing the proof is the use of flow chart. The flow of the statements is indicated by the arrows. The following is the proof of theorem 3. The flow chart is the visual and alternative way of the most common 2-column proof shown below. Developed by the Private Education Assistance Committee 57 under the GASTPE Program of the Department of Education  Statement Reason 1. LOVE is an isosceles trapezoid Given where LO//EV &OV≅LE 2. ∠OVE≅∠LEV Base angles of an isosceles trapezoid are congruent. 3. VE ≅ EV Reflexive Property 4. ▽OVE ≅▽LEV SAS Congruence Postulate 5. OE ≅ LV CPCTC To prove the next theorem you need to know the meaning of certain term/word. Read the text inside the box and proceed by doing the task that follows. Theorem 4: MIDLINE THEOREM: The median of a trapezoid is parallel to the bases and its measure is one- half the sum of the measures of the bases. This theorem can be proven to be true after you perform this modelling activity. Instructions: 1. Draw trapezoid ABCD on a piece of paper and measure the base angles. 2. Measure AD and BC in millimeter and mark the midpoint as X and Y. 3. Connect points X & Y and measure the angles with vertices X and Y 4. Measure AB, XY and DC to the nearest millimeter. 5. Make a conjecture based on your observations. 6. Verify the result using 2 other trapezoids. 7. Give your generalization. Developed by the Private Education Assistance Committee 58 under the GASTPE Program of the Department of Education Theorem 5: The diagonals of a kite are perpendicular. Developed by the Private Education Assistance Committee 59 under the GASTPE Program of the Department of Education LOOK BACK: To prove the next theorem it is important to remember the converse of the perpendicular bisector theorem which states that if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Answer the following questions to prove theorem 5. 1. What triangle is congruent to ∆EFG? Justify your answer. 2. Why is FEHDEH? 3. Why is ∆FEH∆DEH? 4. Why is FHDH? 4. Why is EHFEHD? 5. Why areEHF andEHD right angles? 6. Why is DF EG? Since you have discovered for yourself the proofs of these theorems, you are already certain that these are true statements. For ease in remembering these properties and characteristics of trapezoids and kites and be ready for their applications, it is better to outline them. ACTIVITY 6b DON’T FOOL ME! Writing Proofs Sketch the figure and write a complete proof of the following by giving the appropriate statements or reasons: (You may use a 2-column proof or flow chart.) 1. Given: FIND is an isosceles trapezoid with FD IN Prove:NFDDIN Statements Reason 1. FIND is an isosceles trapezoid with 1. FD IN Developed by the Private Education Assistance Committee 60 under the GASTPE Program of the Department of Education  Statements Reason 2. FNID 2. 3. DNND 3. 4. ∆FDN∆IND 4. 5. NFDDIN 5. 2. Given: ∆GIF ∆IGT Prove: GIFT is an isosceles trapezoid Statement Reason 1. 1. Given 2. 2. CPCTC 3. 3. Definition of an isosceles trapezoid Alternative Proof 1. 1. Given 2. 2. CPCTC 3. 3. If the base angles of a trapezoid are congruent, then it is isosceles. 3. Given: B and F are the midpoints of AC & AE of ∆ACE. AC AE Prove: BFEC is an isosceles trapezoid Statement Reason 1. 2. 3. 4. 5. Check yourself by answering the following questions. Questions to Answer: 1. How is your proving experience? Which part was clear? Which part was confusing to you? 2. Were you able to derive the correct conclusions and give the supporting reasons? Developed by the Private Education Assistance Committee 61 under the GASTPE Program of the Department of Education 3. What difficulty did you encounter? 4. What do you intend to do to cope with such difficulty? To further enhance your skills, you have to accomplish the set of practice exercises below. ACTIVITY 7b WANNA PRACTICE? Practice Exercise on drawing out conclusions and giving of reasons: To demonstrate your mastery of the concepts and skills about trapezoid, fill in the appropriate conclusion and reason. 1. Hypothesis: CORE is an isosceles trapezoid where CO//RE a. Conclusion: ___________________________________ Reason: _____________________________________ b. Conclusion: ___________________________________ Reason: _____________________________________ c. Conclusion: ___________________________________ Reason: _____________________________________ 2. Hypothesis: In trapezoid HOPE where HO//PE,H O Conclusion: ___________________________________ Reason: ____________________________________ 3. Hypothesis: DATE is a kite with DT & AE are diagonals Conclusion: ___________________________________ Reason: ____________________________________ 4. Hypothesis: In kite RULE, RU=RE Conclusion:___________________________________ Reason: _____________________________________ 5. Hypothesis: In trapezoid ACER, the diagonals AECR Conclusion: ___________________________________ Reason: ____________________________________ 6. Hypothesis: O & U are the midpoints of the legs PR & ST of trapezoid PRST a. Conclusion: ___________________________________ Reason: _____________________________________ Developed by the Private Education Assistance Committee 62 under the GASTPE Program of the Department of Education  b. Conclusion: ___________________________________ Reason: ______________________________________ ACTIVITY 8b SQUEEZE IT! Drawing out conclusions applying the different theorems on trapezoids THINK AND DISCUSS within the group of 4 members (you may sketch the figure) 1. In an isosceles trapezoid MARE where MA//ER, what is the relationship betweenM &A?E &R? Explain 2. What is the relationship between M &E? A &R? Explain. 3. WXYZ is an isosceles trapezoid, how do you compare WY & XZ? Why? 4. A & B are midpoints of the legs TQ and SR of trapezoid QRST, what is the relationship between AB & QR? AB & TS? Explain why. What do you do to determine the measure of AB? Why? 5. M & N are the midpoints of the legs BE and AS of trapezoid BASE. If MN= 25 and BA=35, what is ES? Explain. 6. In trapezoid ABCD if AC = BD, what can you say about ABCD? Why is that so? 7. If BA = DE in trapezoid BADE, what can you conclude about BADE? Why? 8. If R A in trapezoid RAIN where RA//IN, What kind of a trapezoid is RAIN? Why? 9. If HAVE is an isosceles trapezoid where HA//VE, what is the relationship between HE and VA? How did you know? What can you conclude about ∆HEV & ∆VAE? Justify your answer. Is there another way to justify your conclusion? Explain how. 10. In kite RSTV, RS=RV & TS=TV, what can you conclude about ∆RTV and ∆RTS? Justify your answer. You may now revisit your generalization table and fill up the second and third columns. Review your ideas in the explore part and compare them with your recent findings, insights and understanding. MY INITIAL MY FINDINGS SUPPORTING QUALIFYING MY THOUGHTS AND EVIDENCE CONDITIONS GENERALIZATIONS CORRECTIONS Developed by the Private Education Assistance Committee 63 under the GASTPE Program of the Department of Education END OF FIRM - UP In this section, the discussion was about trapezoids, kites, their properties and theorems. Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need revision? What new learning goal should you now try to achieve? Now that you know the important ideas about this topic, let’s go deeper by moving on to the next section. Your goal in this section is to take a closer look at some aspects of the topic and explore how these knowledge and skills be put to use. Work on the next activity and apply the skills acquired to find the measure of the specified parts. ACTIVITY 9b READY, GET SET Oral/Drill Exercise on the applications of the theorems on trapezoids (Think Aloud by Pair) Find the measure of the sides and angles of the following figures and explain why it is like that and how it should be done. A sketch of the figure will help you find the answer. 1. In an isosceles trapezoid MARE where MA//ER if the mA=740, what is the mE, mM, mR? 2. If ME= 16, what is AR? 3. If S & T are the midpoints of ME & AR respectively, what is MS? and AT? 4. If MA=20 and ER=34, what is ST? 5. If ST= 18 and MA=12, what is ER? 6. If ST= 35 and ER=45, what is MA? 7. If MR=19, what is AE? 8. In trapezoid REAP, if R&E are right angles and mA= 680, what is the mP? Developed by the Private Education Assistance Committee 64 under the GASTPE Program of the Department of Education 9. In trapezoid TUNE where TU//EN, B is the midpoint of TE and BC is a median. What is the value of x if TU=3x-8, BC=15, EN=4x + 10? 10. In kite ANTE where AN=AE & TN=TE, what angles are congruent? If the mE=1000 and the mT= 550 what is the mA? Submit your answer. Questions to Answer: 1. How did you find the activities? 2. Were you able to perform all the activities? If no, explain why. 3. What did you do to improve your performance? 4. What insights do you have about the lesson on trapezoid? 5. Do you think this would be of great help to you? Explain in what way. 6. How did you find the best solution to solve the problems? ACTIVITY 10b GOING TECHY Interactive Activity on the properties of trapezoids and kites To further improve your knowledge and skills about trapezoids and kites, visit the site given below to work on those exercises and take note of your score and do not forget to review the answers especially those items not correctly answered. Developed by the Private Education Assistance Committee 65 under the GASTPE Program of the Department of Education http://www.ixl.com/math/geometry/properties-of-trapezoids This site contains interactive exercises about trapezoids and their theorems. http://www.mathopolis.com/questions/q.php?id=621&site=1&ref=/quadrilaterals.h tml&qs=621_622_623_624_763_764_2128_2129_3230_3231 This site contains a quiz about quadrilaterals. After you have answered the quiz and found out that you have learned a lot, you now proceed to look deeper by checking on the applications of these concepts. ACTIVITY 11b CHECK ON ME! Identify objects in the surroundings or parts of a house with trapezoidal design; explain your choice of the objects. http://ph.images.search.yahoo.com/search/images;_ylt=A2oKiavkUe5SZRsAAji0Rwx.?p=real‐ life+applications+of+trapezoids+and+kites&ei=utf‐8&iscqry=&fr=sfp This site contains pictures of real‐life applications of trapezoids and kites. Also include in your sharing the answers of the follow‐up questions. Questions to Answer: 1. What do you see in these pictures? 2. Why do you think they are trapezoidal? 3. What are the advantages/disadvantages of these designs? 4. What do you think will result if different shapes are used? 5. Why is it important to use trapezoids? Developed by the Private Education Assistance Committee 66 under the GASTPE Program of the Department of Education Write the answers of the following questions in your journal using Evernote. Please refer to this site, www.Evernote.com 1. What have you realized about the lesson on trapezoids? 2. What are the benefits of learning the concepts? To prepare you with the performance task, one skill you should have is to draw figures to scale so that you will have a proportional drawing whether you reduce or enlarge a desired figure, you will need problem solving technique which is the purpose of the next activity. ACTIVITY 12b I GOT IT! Problem Solving using the concepts of trapezoids and kites To make all these concepts relevant, you have to apply these in the different real-life situations: 1. The perimeter of a kite is 64 feet. The length of one of its sides is 8 feet more than the other side. What are the lengths of each side of the kite? Let x = be the length of one side X+ 8 = the length of the other side Equation: x+x+x+8+x+8=64 4x+16=64 4x =64-16 4x =48 X = 12 Therefore, 2 sides measure 12 inches each and the other 2 sides measure 20 inches each. 2. Part of the window of the World Financial Center in New York City is made from 8 congruent isosceles trapezoids that create an illusion of a semicircle. What are the measures of the base angles? Developed by the Private Education Assistance Committee 67 under the GASTPE Program of the Department of Education Let x = be the measure of the base angles Since the measure of a semi-circle is 180 divide it by 8 = 22.5o, that is the measure of the vertex angle. Equation: x + x + 22.5 = 180 because the sum of the measure of 2x = 180-22.5 the angles of a triangle is 180 2x = 157.5 X = 78.75o Thus, the measure of each base angle is 78.75o. 3. Large sailboats have a keel to keep the boat stable in high winds. A keel is shaped like a trapezoid with its top and bottom parallel. If the root chord (the one on top) is 12.4 feet and the tip chord (the one at the bottom) is 9.6 feet, what is the length of the mid-chord? Let x = be the length of the mid-chord Equation: x = 12.4 + 9.6 2 X = 22 2 X = 11 Therefore, the length of the mid-chord of the keel is 11 feet. Before you fill-up the last 2 columns of the generalization table, take a closer look at the picture below and answer the following questions. Developed by the Private Education Assistance Committee 68 under the GASTPE Program of the Department of Education Questions to Answer: 1. Why do beams of the first bridge take the form of a trapezoid? 2. What do you see in the beams which have the shape of a parallelogram? 3. What shape is now formed with the braces? 4. What do the braces do to the structure? 5. Which shape is more flexible? 6. Which shape is more stable? 7. Which is preferred in a bridge structure, flexibility or stability? Explain. 8. What is the best solution to a problem? Fill-up the fourth column of the generalization table and submit. Developed by the Private Education Assistance Committee 69 under the GASTPE Program of the Department of Education  MY INITIAL MY FINDINGS SUPPORTING QUALIFYING MY THOUGHTS AND EVIDENCE CONDITIONS GENERALIZATIONS CORRECTIONS PAUSE AND EVALUATE YOURSELF: Draw a star below the icon that best describes your knowledge/ understanding of the lesson RATE YOURSELF I still don’t get it I acquire the basic I understand the concepts/skills of concepts of trapezoids& trapezoids & kites kites END OF DEEPEN In this section, the discussion was about the applications of the knowledge and skills pertaining to trapezoids and kites in the different real-world situations. These will help you accomplish the task in creating a miniature house model which makes use of the different quadrilaterals and similar figures. The design must be chosen in such a way to maximize spaces and efficiently use the materials. You have to answer the following questions to check on your understanding and prepare for the succeeding activities and life in general. What new realizations do you have about the topic? What new connections have you made for yourself? What helped you make these connections? Now that you have gained deeper understanding of the lesson, you are ready to use them in a particular context in the next section. Developed by the Private Education Assistance Committee 70 under the GASTPE Program of the Department of Education  Your goal in this section is apply your learning to real life situations. You will be given a practical task which will demonstrate your understanding. To test if you have already enough knowledge and skills in problem solving and posing, try to accomplish the Quiz below. ACTIVITY 13b MAKE A PROBLEM OUT OF ME Problem Posing: Follow the procedure below and answer the questions. Do your work in short type writing and then send the soft copy to your teacher through the discussion board or email. You may also submit your work face – to – face. Developed by the Private Education Assistance Committee 71 under the GASTPE Program of the Department of Education From the given situations, formulate problems, present solutions and explain. 1. Dress up an octagonal room with furniture and fixtures in such a way that the room appear to be spacious and must reflect the efficient use of materials. 2. The beams of most bridges are trapezoidal; determine the measure of the sides, braces and angles with the least number of known measures. Questions to Answer: 1. How did you go about answering the activity? 2. What is the best way to solve the problems? 2. Why is it necessary to learn about trapezoid and its properties? 3. What happens if you do not have a clear knowledge about trapezoids? To transfer your understanding you may now do the transfer task below. Developed by the Private Education Assistance Committee 72 under the GASTPE Program of the Department of Education  ACTIVITY 15b Sum it up! To reflect on the learning process, you may now complete the generalization table by writing your final answer on the last column. Fill-up the last column of the generalization table and submit. MY INITIAL MY FINDINGS SUPPORTING QUALIFYING MY THOUGHTS AND EVIDENCE CONDITIONS GENERALIZATIONS CORRECTIONS END OF TRANSFER: In this section, your task was to make trapezoid and kites with different qualifications with the use of web 2.0. How did you find the task? How did the task help you see the real world use of the topic? You have completed this lesson. But you have two more lessons before you finish this module. You need to learn more about triangles to complete what you need in doing your performance task. Developed by the Private Education Assistance Committee 73 under the GASTPE Program of the Department of Education Lesson 3: Triangle Similarity In this lesson you will learn the following: 1. Describe a proportion. 2. Illustrates similarity of figures. 3. Proves the conditions for similarity of triangles SAS Similarity Theorem SSS Similarity Theorem AA Similarity Theorem 4. Right Triangle Similarity Theorem You learned from lesson 2 the different concepts of quadrilaterals which are very essential in solving problems related to geometric figures. In this lesson, you will learn the concepts of proportion and how to use it in many situations. You will also learn the concepts of triangle similarity and the different theorems related to these lessons which are useful in solving real world problems. You will also gather ideas to answer the question “What is the best way to solve problems involving triangle similarity?” These concepts will also help you visualize situations and create solutions to the problems that you encounter. Answers to the question above will also help you do your performance task. In this section you need to analyze picture by answering different questions for you to discover important concepts. You will also do self- monitoring activity as you fill up the map of conceptual change. Let’s us start the lesson by analyzing the pictures and answering the questions that follow. Developed by the Private Education Assistance Committee 74 under the GASTPE Program of the Department of Education Triangle Similarity What is the best way to solve problems involving triangle similarity? Let’s answer these questions by doing the activities below. ACTIVITY 1c Picture Analysis (Eliciting of prior Knowledge, Motivation, Hook) Observe the pictures below and answer the questions. https://www.google.com.ph/#q=SIMILAR+PICTURES Questions to Answer: 14. Do you have a look alike? Why did you say that? 15. Can figures be that similar? In what way? Developed by the Private Education Assistance Committee 75 under the GASTPE Program of the Department of Education  16. What would you consider to determine that two figures are similar? 17. Why is it important to know two similar figures? 18. Focusing on the triangles, how would you know that two triangles are similar? 19. What is the best way to solve problems involving triangle similarity? CONCEPTUAL UNDERSTANDING CHECK In the table below, write your answers on the initial part for the question what is the best way to solve problems involving triangle similarity? INITIAL ANSWER REVISED ANSWER FINAL ANSWER Developed by the Private Education Assistance Committee 76 under the GASTPE Program of the Department of Education End of EXPLORE: You just have tried to find out how mathematics can help you determine the best way to solve problems involving triangle similarity. Let us now strengthen that insight by doing the succeeding activities. What you will be learning in this section will help you perform well in your final performance task which will challenge you to use what you know to create a model and solve problems involving structures, space and aesthetic appeal. Now move to the next activity to learn the knowledge and skills you need to be a good problem solver and respond to different situations accurately. Your goal in this section is to learn and understand key concepts of proportion which are important in solving problems involving triangle similarity. In this section there are activities which will help you discover and understand the different theorems and postulates which are useful tools in solving real life problem related to triangle similarity. ACTIVITY 2c Situational Analysis In the previous activity you are task to determine similar figures based from the given pictures. Now, let’s see if you can be able to develop the concepts that you learned to respond to the situation below. You are a newly hired employee of an organization who is working for the improvement of the environment. As your initial task, you need to estimate the number of trees of a 10 hectare forest near your place. This will be part of your company’s report to plan for improvement. When you visit the forest you observe that trees are planted consistently which is about 20 meters from each other. Developed by the Private Education Assistance Committee 77 under the GASTPE Program of the Department of Education  Questions to Answer: 1. What concept would you use to solve the problem above? How would you use it? 2. What is the estimated number of trees of the 10 hectare forest? 3. How is proportionality used in this situation? In the previous activity you learned the importance of proportion in answering problems in real life. Now, you will improve your knowledge in proportion by answering the activities below. ACTIVITY 3c Let’s Consult the Expert Directions: Click any of the videos below which explain the concepts of proportion with step by step procedure on how to solve problem related to the topic. After watching the video do the exercises below. http://www.youtube.com/watch?v=D8dA4pE5hEY http://www.youtube.com/watch?v=2d578xHNqc8 http://www.youtube.com/watch?v=G8qy4f7GKzc These sites contain videos which explain the concepts of proportion with step by step procedure on how