Mathematics 7 Unit 1: Angles and Polygons Lesson 3 PDF

Mathematics 7 Unit 1: Angles and Polygons Lesson 3 Types of Polygons Table of Contents Introduction 1 Learning Competency...

Mathematics 7 Unit 1: Angles and Polygons Lesson 3 Types of Polygons Table of Contents Introduction 1 Learning Competency 2 Learning Targets 2 Warm-Up 2 Learn about It 3 Polygons 3 Classifying Polygons 4 Classifying Polygons According to the Number of Sides 4 Classifying Triangles Based on Side Lengths 5 Classifying Polygons According to the Kinds of Angles 5 Classifying Quadrilaterals 6 Regular and Irregular Polygons 7 Concave and Convex Polygons 8 Let’s Practice 11 Check Your Understanding 17 Key Points 19 Self-Assessment 20 Reflection 21 Attributions 21 References 21 Answer Key 22 0 Mathematics 7 Unit 1: Angles and Polygons 3 Types of Polygons Figure 1. Hexagons in honeycombs Introduction Polygons have been an integral part of mathematics and its applications. They are seen in everyday life, from the designs in buildings to shapes in nature and even in computer graphics. Understanding polygons and their types can give us tools for creative expression, problem-solving, and appreciating the world around us. This lesson will guide you in identifying and classifying polygons based on their properties, from the number of sides they have to their convexity. Are you ready to explore the fascinating world of polygons? 1.3. Types of Polygons 1 Mathematics 7 Unit 1: Angles and Polygons Learning Competency At the end of this lesson, the learners should be able to classify polygons according to the number of sides, whether they are regular or irregular, and whether they are convex or non-convex. Learning Targets In this lesson, you should be able to do the following: Classify polygons based on the number of sides they have. Differentiate regular and irregular polygons. Differentiate convex and concave polygons. Warm-Up Find That Shape! 21st Century Skills Icons Legend Information, Media, and Technology Skills Communication Skills Learning and Innovation Skills Life and Career Skills Procedure 1. Draw five different polygons on a piece of paper. The polygons can have different numbers of sides. 2. Identify the number of sides for each polygon and describe them. 1.3. Types of Polygons 2 Mathematics 7 Unit 1: Angles and Polygons Guide Questions 1. How would you describe the polygons that you just drew? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ 2. Based on your descriptions, how do you think are polygons classified? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ Essential Questions What makes a polygon regular? How can you distinguish between a convex and a concave polygon? Learn about It Polygons Rocky, an art connoisseur, loves to travel across lands to witness different types of mathematical applications in real life, especially in arts and historical artifacts. On one of his travels, he went to an antique shop and noticed the souvenir section. He realized that most of the things displayed there resemble some kind of polygons. Some objects he saw are as follows. Can you name the kind of polygon in each picture? 1.3. Types of Polygons 3 Mathematics 7 Unit 1: Angles and Polygons Did You Know? The word polygon comes from the Greek words poly, meaning "many," and gonia, meaning "angles." Therefore, a polygon is essentially a shape with "many angles." Classifying Polygons Recall that polygons are closed plane figures bounded by line segments called sides. Each is classified according to the number of its sides using Latin prefixes. Classifying Polygons According to the Number of Sides The table below shows the classifications of polygons based on the number of sides. Table 1. Classification of polygons based on the number of sides Number of Sides Name of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11 undecagon 12 dodecagon 1.3. Types of Polygons 4 Mathematics 7 Unit 1: Angles and Polygons A polygon with more than 12 sides is called an 𝑛-gon, where 𝑛 is the number of sides. For instance, a polygon with 15 sides is called a 15-gon. Classifying Triangles Based on Side Lengths A triangle is a polygon with exactly three sides and three angles. There are three classifications of a triangle according to the lengths of its sides. a. A triangle that has no sides of equal length is called a scalene triangle. b. A triangle that has two sides of equal length is called an isosceles triangle. The pair of equal sides are called the legs, and the third side, which is the odd one out, is called the base. c. A triangle that has all three sides of equal length is called an equilateral triangle. Below are the visual representations of scalene, isosceles, and equilateral triangles. Classifying Polygons According to the Kinds of Angles Triangles can also be classified according to the kinds of angles they have. a. A triangle that contains a right angle is called a right triangle. b. A triangle that has an obtuse angle is called an obtuse triangle. The side that is opposite to the obtuse angle is the longest side. A triangle cannot have more than one obtuse angle. c. A triangle that has three acute angles is called an acute triangle. Notice that the visual representations that we have for the earlier categorization fall under this classification. 1.3. Types of Polygons 5 Mathematics 7 Unit 1: Angles and Polygons Below are representations of right, obtuse, and acute triangles. Classifying Quadrilaterals A polygon with exactly four sides and angles is called a quadrilateral. A special classification of a quadrilateral is a parallelogram. Parallelograms have two pairs of opposite sides parallel to each other and can be further classified according to their interior angle and length of their sides. The arrowheads indicate that the sides of the parallelogram are parallel. A parallelogram with four equal interior angles is called a rectangle. This means that each of the interior angles of the rectangle is a right angle. 1.3. Types of Polygons 6 Mathematics 7 Unit 1: Angles and Polygons A parallelogram with four equal sides is called a rhombus. Notice that the opposite angles of this figure are congruent. A parallelogram that possesses the properties of both a rectangle and a rhombus—having four equal interior angles and four equal sides—is called a square. Regular and Irregular Polygons Polygons can also be classified as regular or irregular. De nition of Terms polygons that have sides with equal regular polygons length and interior angles with equal measurement polygons that have sides with different irregular polygons length or interior angles with different measurements 1.3. Types of Polygons 7 Mathematics 7 Unit 1: Angles and Polygons The figure below is an example of a regular pentagon. Observe that the sides and angles have the same measures. The figure below is an example of an irregular pentagon. Observe that the sides and angles have different measures. Note that a regular triangle is an equilateral triangle and a regular quadrilateral is a square. Concave and Convex Polygons Polygons can also be classified as convex or concave. De nition of Terms a polygon that has all angles point convex polygon outward concave polygon a polygon that is not convex 1.3. Types of Polygons 8 Mathematics 7 Unit 1: Angles and Polygons A convex polygon should have the following properties. Properties The following are the different properties of a convex polygon: All interior angles measure less than 180°. All diagonals lie in the interior of the polygon. If the sides of the polygon are extended, the lines will not intersect on the interior of the polygon. Note that a diagonal of a polygon is a line segment connecting two nonconsecutive vertices of a polygon. The figures below are examples of a convex polygon and a concave polygon. Observe that a convex polygon follows the property of a convex polygon. Note that the measure of all interior angles of a convex polygon measure less than 180°, while a concave polygon has at least one interior angle that measures greater than 180°. Moreover, all regular polygons are convex. 1.3. Types of Polygons 9 Mathematics 7 Unit 1: Angles and Polygons Tips To quickly determine if a polygon is regular, just check if all sides and angles are equal. To find out if a polygon is convex or concave, focus on the angles. If any angle is greater than 180°, the polygon is concave. Practical Applications Understanding polygons has practical applications beyond mathematics. Science Geologists use polygons to model the shapes of crystals and natural formations. History Historians examine the polygons in ancient architecture to understand the civilizations that built them. Statistics Statisticians use polygonal shapes in graphs and models for market analysis. 1.3. Types of Polygons 10 Mathematics 7 Unit 1: Angles and Polygons Let’s Practice Example 1 Identify the polygon below. Solution Step 1: Count the number of sides. The polygon has 4 sides, so it is a quadrilateral. Step 2: Check the number of pairs of parallel sides. The arrowheads indicate the sides that are parallel. It has two pairs of parallel sides. Hence, it is a parallelogram. Step 3: Check the sides and angles. All of its angles are right angles, but its sides do not have the same length. Hence, it is a rectangle. Thus, the figure is a rectangle. 1.3. Types of Polygons 11 Mathematics 7 Unit 1: Angles and Polygons Let’s Try It Identify the polygon below. Example 2 Classify the following polygon if it is a convex or concave polygon. Solution To determine if the polygon is a convex polygon, we have to check if it satisfies the properties of a convex polygon. Otherwise, it is a concave polygon. Step 1: Check if the measures of the angles of the polygon is less than 180°. Observe that the angles of the pentagon are obtuse angles. It follows that the angles of the pentagon have measures of less than 180°. 1.3. Types of Polygons 12 Mathematics 7 Unit 1: Angles and Polygons Step 2: Check the diagonals of the polygon. Let us draw the diagonals of the polygon. Observe that in this polygon, the diagonals are in the interior of the polygon. Thus, it satisfies the properties of a convex polygon. Step 3: Extend the sides of the polygon. Observe that in this polygon, the sides, when extended, do not intersect in the interior of the polygon. Thus, it satisfies the property of a convex polygon. Thus, the polygon is a convex polygon. 1.3. Types of Polygons 13 Mathematics 7 Unit 1: Angles and Polygons Let’s Try It Determine if the following polygon is a convex or a concave polygon. Example 3 Quadrilateral 𝐴𝐵𝐶𝐷 is a rectangle. It is known that 𝐴𝐵 = 8 units and 𝐷𝐴 = 4 units. Find the sum of the lengths of 𝐵𝐶 and 𝐶𝐷. Solution Step 1: Draw a figure for the problem. Drawing a figure for the problem will help us visualize which sides are opposite which. Step 2: Determine 𝐵𝐶 and 𝐶𝐷. It is given that 𝐴𝐵𝐶𝐷 is a rectangle. Recall that a rectangle is a parallelogram; therefore, its opposite sides are congruent. 1.3. Types of Polygons 14 Mathematics 7 Unit 1: Angles and Polygons Looking at the figure, 𝐵𝐶 is opposite 𝐷𝐴, and 𝐶𝐷 is opposite 𝐴𝐵. Thus, 𝐵𝐶 = 𝐷𝐴 = 4 units, and 𝐶𝐷 = 𝐴𝐵 = 8 units. Step 3: Find the sum of lengths 𝐵𝐶 and 𝐶𝐷. Therefore, the sum of the lengths of 𝐵𝐶 and 𝐶𝐷 is 12 units. Let’s Try It ∆𝑋𝑌𝑍 is an isosceles triangle with base 𝑋𝑍 = 10 𝑢𝑛𝑖𝑡𝑠. If the sum of the lengths of the sides of ∆𝑋𝑌𝑍 is 24 units, find the length of each leg of ∆𝑋𝑌𝑍. Real-World Problems Example 4 Neil has a set of matchsticks as seen on the right. All the matchsticks have the same length. If he is to use all matchsticks, with each matchstick representing a side, what are the possible polygons that he can make? Solution There are six matchsticks in the set. Recall that a polygon with six sides is a hexagon. Thus, the first figure that Neil can form is a regular hexagon. 1.3. Types of Polygons 15 Mathematics 7 Unit 1: Angles and Polygons Recall that the least number of sides needed to form a polygon is three, to form a triangle. Thus, Neil can form two regular (or equilateral) triangles with the six matchsticks. Note that Neil cannot form a regular quadrilateral (square) with four matchsticks because the remaining two matchsticks are not enough to form a polygon. Also, he cannot form a regular pentagon with five matchsticks because the remaining one matchstick is not enough to form a polygon. Neil can form a regular hexagon and two equilateral triangles. Let’s Try It Patty has four tiles in the shape of a right triangle. All the tiles have the same measure of its sides and angles. Using all four tiles to form one polygon, what kind of polygons can she make? Give at least three examples. 1.3. Types of Polygons 16 Mathematics 7 Unit 1: Angles and Polygons Check Your Understanding A. Determine whether the given figure is a polygon. If it is, identify its name according to the number of sides. Then, check whether it is regular or irregular and whether it is convex or concave. 1. 2. 3. 4. 5. 6. 1.3. Types of Polygons 17 Mathematics 7 Unit 1: Angles and Polygons B. Identify each of the following figures. If it is a triangle, classify it according to the measure of the sides and the measure of the angles (e.g. equilateral acute triangle). If it is a quadrilateral, identify the most specific name that applies to it (i.e. parallelogram, rectangle, rhombus, or square). 1. 2. 3. 4. 5. 6. 7. 8. 9. 1.3. Types of Polygons 18 Mathematics 7 Unit 1: Angles and Polygons C. Solve the following problems. 1. It is known that triangle ∆𝐸𝐹𝐺 is an isosceles triangle. 𝐹𝐺 is the longest side and measures 10 𝑐𝑚. 𝐸𝐺 measures 8 𝑐𝑚. What is the measure of 𝐸𝐹? 2. 𝑊𝑋𝑌𝑍 is a rectangle. 𝑊𝑋 = 25 mm and 𝑊𝑍 = 14 mm. Find the lengths 𝑋𝑌 and 𝑌𝑍. 3. ∆𝐾𝐿𝑀 is an isosceles triangle with base 𝐿𝑀 = 15 units. If the sum of the lengths of the sides of ∆𝐾𝐿𝑀 is 33 units, find the length of each leg of ∆𝐾𝐿𝑀. Key Points A polygon is a closed plane figure bounded by line segments called sides.Polygons can be named based on the number of its sides. A triangle is a polygon with exactly three sides and three angles. Triangles can be classified based on the length of its sides. ○ A scalene triangle is a triangle with no sides of equal length. ○ An isosceles triangle is a triangle with two sides of equal length. The legs are a pair of equal sides, and the base is the third side. ○ An equilateral triangle is a triangle with three sides of equal length. Triangles can also be classified based on their kinds of angles. ○ A right triangle is a triangle with one right angle. ○ An acute triangle is a triangle with three acute angles. ○ An obtuse triangle is a triangle with one obtuse angle. A quadrilateral is a polygon with exactly four sides and four angles. A parallelogram is a quadrilateral with two pairs of parallel sides. It can be further classified based on its sides and angles. ○ A rectangle is a parallelogram with four right angles. ○ A rhombus is a parallelogram with four sides of equal length. ○ A square is a parallelogram with four right angles and four sides of equal length. 1.3. Types of Polygons 19 Mathematics 7 Unit 1: Angles and Polygons A regular polygon is a polygon that has all sides of equal length and all angles of equal measure. A polygon that is not a regular polygon is called an irregular polygon. A convex polygon is a polygon that has all angles pointing outward. A polygon that is not convex is called a concave polygon. The following are the properties of a convex polygon: ○ All interior angles measure less than 180°. ○ All diagonals lie in the interior of the polygon. ○ If the sides of the polygon are extended, the lines will not intersect on the interior of the polygon. Self-Assessment I think I need I have a basic I am confident Skills more time and understanding of that I can do this assistance. it. with ease. I can classify them based on the number of sides they have. I can differentiate regular and irregular polygons. I can differentiate convex and concave polygons. 1.3. Types of Polygons 20 Mathematics 7 Unit 1: Angles and Polygons Re ection I find ______________________ the most interesting because ________________________________. I need to improve on ________________________ because ___________________________________. I need to practice _________________________ because ______________________________________. I plan to ___________________________________________________________________________________. Attributions Honeycomb close up on the white by vmariia is licensed under Envato License via Envato Elements. References “Geometry Worksheets | Quadrilaterals and Polygons Worksheets.” n.d. Math-Aids.com. Accessed on September 22 2023. https://www.math-aids.com/Geometry/Polygons/. “Ideas For Teaching Your Students About Polygons.” n.d. Top Notch Teaching. Accessed on September 2023. https://topnotchteaching.com/lesson-ideas/polygons/. Jurgensen, Ray C., Richard G. Brown, and John W. Jurgensen. 2000. Geometry. 5th ed. Boston: Houghton Mifflin. 1.3. Types of Polygons 21  Math 7 Unit 1: Angles and Polygons Answer Key Warm-Up 1. How would you describe the polygons that you just drew? Possible answer: “Some polygons have many sides and some have only a few sides. Some polygons have a large interior and some do not.” 2. Based on your descriptions, how do you think are polygons classified? Possible answer: “Polygons can be classified by the number of sides that they have. They can also be classified based on the measure of their sides or angles.” Let’s Try It 1. Identify the given polygon below: Solution: Step 1: Count the number of sides. The polygon has 4 sides, so it is a quadrilateral. Step 2: Check the number of pairs of parallel sides. It has two pairs of parallel sides. Hence, it is a parallelogram. 1.3. Types of Polygons 22  Math 7 Unit 1: Angles and Polygons Step 3: Check the sides and angles. All of its sides have the same length, but its angles do not have the same measure. Hence, it is a rhombus. Answer: The figure is a rhombus. 2. Determine if the following polygon is a convex or a concave polygon. Solution: Let us extend the sides of this polygon. Observe that the sides of the polygon intersect in the interior of the polygon. This means that the polygon is concave. Answer: The polygon is a concave polygon. 1.3. Types of Polygons 23  Math 7 Unit 1: Angles and Polygons 3. ∆𝑋𝑌𝑍 is an isosceles triangle with base 𝑋𝑍 = 10 𝑢𝑛𝑖𝑡𝑠. If the sum of the lengths of the sides of ∆𝑋𝑌𝑍 is 24 units, find the length of each leg of ∆𝑋𝑌𝑍. Solution: Step 1: Sketch the image. Step 2: Get the length of the remaining sides. Since the ∆𝑋𝑌𝑍 is an isosceles triangle with base 𝑋𝑍 = 10 𝑢𝑛𝑖𝑡𝑠 and the sum of the lengths of the sides of ∆𝑋𝑌𝑍 is 24 units, subtract the 𝑋𝑍 from the sum of the length of the sides. Step 3: Find the length of the legs. Since the legs of the sides of the triangle have the same measure, we divide the remaining sides by 2. Answer: The length of each leg of ∆𝑋𝑌𝑍 is 7 units. 1.3. Types of Polygons 24  Math 7 Unit 1: Angles and Polygons 4. Patty has four tiles in the shape of a right triangle. All the tiles have the same measure of its sides and angles. Using all four tiles to form one polygon, what kind of polygons can she make? Give at least three examples. Solution: We can use two tiles to form a square, and the other two tiles to form another square. Then, we can combine the two squares to form a rectangle. Next, we can use the two tiles to form a square and place each of the remaining tiles on each side of the square. This way, we can form a parallelogram. We can also connect the tiles along their right angles to form a square. We can also form other figures such as a trapezoid and an irregular hexagon. Answer: The figures are rectangle, parallelogram, and square. There are other figures that can be formed as well. 1.3. Types of Polygons 25