Mathematics 7 Unit 1: Angles and Polygons PDF

Mathematics 7 Unit 1: Angles and Polygons Lesson 4 Interior and Exterior Angles of a Convex Polygon Table of Contents Introduction 1 Learning Competencies...

Mathematics 7 Unit 1: Angles and Polygons Lesson 4 Interior and Exterior Angles of a Convex Polygon Table of Contents Introduction 1 Learning Competencies 2 Learning Targets 2 Warm-Up 2 Learn about It 4 Interior Angles of a Polygon 4 Sum of the Measures of the Interior Angles of a Convex Polygon 5 Measure of the Interior Angles of a Regular Polygon 7 Exterior Angles of a Polygon 8 Measure of the Exterior Angles of a Regular Polygon 8 Let’s Practice 10 Check Your Understanding 13 Key Points 15 Self-Assessment 15 Reflection 16 Attribution 17 References 17 Answer Key 18 0 Mathematics 7 Unit 1: Angles and Polygons Interior and Exterior Angles of a Convex 4 Polygon A soccer ball on a field Introduction The world around us is filled with shapes, and among these shapes, polygons play a significant role. From the design of a soccer ball to the structure of houses, polygons are everywhere. In this lesson, we will take a close look at convex polygons and explore the intriguing relationship between their interior and exterior angles. By the end of this lesson, you will have a better understanding of how these angles work and why they are essential in various real-life applications. 1.4. Interior and Exterior Angles of a Convex Polygon 1 Mathematics 7 Unit 1: Angles and Polygons Learning Competencies At the end of this lesson, the learners should be able to do the following: Deduce the relationship between the exterior angle and adjacent interior angle of a polygon. Determine the measures of angles and the number of sides of polygons. Learning Targets In this lesson, you should be able to do the following: Identify the interior and exterior angles of a convex polygon. Explain the relationship between the exterior angle and the adjacent interior angle of a polygon. Determine the measures of the interior and exterior angles of a regular polygon. Warm-Up Drawing Polygons 21st Century Skills Icons Legend Information, Media, and Technology Skills Communication Skills Learning and Innovation Skills Life and Career Skills 1.4. Interior and Exterior Angles of a Convex Polygon 2 Mathematics 7 Unit 1: Angles and Polygons Procedure 1. Using a straightedge or ruler, draw the following polygons. The polygons can have any measure of its sides, but it should be convex. a. triangle b. quadrilateral c. pentagon 2. Using a protractor, measure the angles of each figure and add the measures of each figure. 3. Observe if there is a pattern between the sum of the measures of the angles and the number of its sides. Guide Questions 1. What have you observed about the sum of measures of the angles of the polygons? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ 2. What do you think is the relationship between the sum of the measures of the angles of a polygon and the number of its sides? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ Essential Questions How does the number of sides of a polygon affect the sum of the measures of its interior angles? How does the measure of the exterior angle change as the number of sides in a polygon increases? 1.4. Interior and Exterior Angles of a Convex Polygon 3 Mathematics 7 Unit 1: Angles and Polygons Learn about It In the Warm-Up activity, you were able to construct convex polygons and measure their angles. This time, we will focus more on the measure of its angles. De nition of Terms a polygon that has all angles point convex polygon outward Did You Know? All regular polygons are convex, but not all convex polygons are regular. Interior Angles of a Polygon Let us first identify what interior angles are. De nition of Terms angles that are formed by a pair of sides interior angles of a polygon and is located inside the polygon 1.4. Interior and Exterior Angles of a Convex Polygon 4 Mathematics 7 Unit 1: Angles and Polygons In the figure below, ∠𝐴𝐵𝐶 is an interior angle of ∆𝐴𝐵𝐶 because it is an angle that is formed by the sides 𝐴𝐵 and 𝐵𝐶 and it is located inside the figure. Sum of the Measures of the Interior Angles of a Convex Polygon In the Warm-Up activity, you drew a triangle and obtained the sum of the measures of its angle. What is the sum of the angles of the triangle? Property The sum of the measures of the interior angles of a triangle is 180° This property works on any kind of triangles. Using this property we can determine the sum of the measure of the interior angles of other convex polygons. Take a look at the quadrilateral below. We will draw the diagonal 𝐴𝐶 as shown below. 1.4. Interior and Exterior Angles of a Convex Polygon 5 Mathematics 7 Unit 1: Angles and Polygons Note that by drawing the diagonal, we are able to form two triangles. This means that the sum of the measures of the interior angles of a quadrilateral is 180° × 2 = 360°. Next, let us consider a pentagon. We will also draw the diagonals 𝐴𝐶 and 𝐴𝐷 from one vertex A. What do you think is the sum of the measures of the interior angles of a pentagon? With this, we can come up with a formula to determine the sum of the measures of the interior angles of a polygon given the number of its sides. Formula The sum of the measures of the interior angles of a convex polygon is where 𝑆 is the sum of the measures of the interior angles and 𝑛 is the number of sides of the polygon. 1.4. Interior and Exterior Angles of a Convex Polygon 6 Mathematics 7 Unit 1: Angles and Polygons Tips The formula is based on the number of triangles that can be formed in a triangle by drawing diagonals from one vertex of a convex polygon. The 𝑛 − 2 part of the equations means that given a convex polygon with 𝑛 sides, the number of triangles that can be formed from drawing the diagonals from a single vertex of the polygon is 𝑛 − 2. Measure of the Interior Angles of a Regular Polygon Recall that the measures of a regular polygon’s interior angles are the same. We can determine the measure of one interior angle of a regular polygon by dividing the sum of the measures of the interior angles to its number of sides. Formula The measure of one interior angle of a regular polygon is where 𝑛 is the number of sides of the polygon. 1.4. Interior and Exterior Angles of a Convex Polygon 7 Mathematics 7 Unit 1: Angles and Polygons Exterior Angles of a Polygon Consider the polygon 𝐴𝐵𝐶𝐷 below. Let us extend one side of the polygon, say 𝐴𝐵, into point 𝐸. Observe that by extending 𝐴𝐵 to 𝐸, we are able to form ∠𝐶𝐵𝐸. This angle is called the exterior angle of the polygon. De nition of Terms the adjacent angle of the interior angle exterior angles of a convex polygon formed by extending the side of the polygon Note that the interior angle and its corresponding exterior angle are supplementary, which measure that the sum of their measures is equal to 180°. For instance, if the measure of the interior angle of a regular polygon is 60°, it follows that the measure of its exterior angle is 180° − 60° = 120°. Measure of the Exterior Angles of a Regular Polygon The sum of the measures of the exterior angles of a convex polygon is 360°, regardless of the number of sides of the given polygon. 1.4. Interior and Exterior Angles of a Convex Polygon 8 Mathematics 7 Unit 1: Angles and Polygons Note that a regular polygon’s interior angles have the same measures. It follows that its exterior angles have the same measures as well. Thus, we can determine the measure of one exterior angle of a regular polygon by dividing 360° by the number of its sides. Formula The measure of one exterior angle of a regular polygon is where 𝑛 is the number of sides of the polygon. Practical Applications Interior and exterior angles of a polygon can also be applied in other fields. Science Understanding the angles of polygons can help in studying crystal structures and molecular shapes. Economics Understanding shapes and designs can aid in optimizing space for businesses, especially in packaging. 1.4. Interior and Exterior Angles of a Convex Polygon 9 Mathematics 7 Unit 1: Angles and Polygons Let’s Practice Example 1 Find the sum of the measures of the interior angles of a convex heptagon. Solution A heptagon has 7 sides. Thus, 𝑛 = 7. Let us use the formula to determine the sum of the measures of the interior angles. Thus, the sum of the measures of the interior angles of a convex heptagon is 900°. Let’s Try It Find the sum of the measures of the interior angles of a convex decagon. Example 2 Find the measure of one interior angle of a regular pentagon. Solution A pentagon has 5 sides. Thus, 𝑛 = 5. Let us use the formula to determine the measure of one interior angle of a regular pentagon. 1.4. Interior and Exterior Angles of a Convex Polygon 10 Mathematics 7 Unit 1: Angles and Polygons Thus, the measure of one interior angle of a regular pentagon is 108°. Let’s Try It Find the measure of one interior angle of a regular nonagon. Example 3 Find the measure of one exterior angle of a regular hexagon. Solution A hexagon has 6 sides. Thus, 𝑛 = 6. Let us use the formula to determine the measure of one exterior angle of a regular hexagon. Thus, the measure of one exterior angle of a regular hexagon is 60°. 1.4. Interior and Exterior Angles of a Convex Polygon 11 Mathematics 7 Unit 1: Angles and Polygons Let’s Try It Find the measure of one exterior angle of a regular dodecagon. Real-World Problems Example 4 Darla noticed that one of their plates is in the shape of a regular polygon and the measure of its interior angle is 135°. How many sides does the polygon have? Solution Step 1: Solve for the measure of the exterior angle of the polygon. Note that the interior angle of a polygon and its corresponding exterior angle are supplementary. To solve for the measure of the exterior angle, subtract the measure of the interior angle from 180°. Step 2: Solve for the number of sides of the polygon. Recall that the formula for the measure of the exterior angle of a 360° polygon is 𝑛 , where 𝑛 is the number of sides. We can use this to determine the number of sides by reversing the operation. We divide 360° by the measure of the exterior angle to determine the number of sides of the polygon. This means that the polygon has 8 sides. Thus, the plate is an octagon. 1.4. Interior and Exterior Angles of a Convex Polygon 12 Mathematics 7 Unit 1: Angles and Polygons Let’s Try It Patty owns a necklace, and its gem has the shape of a regular polygon. The interior angle of the gem is 140°. What kind of polygon is the gem on her necklace? Check Your Understanding A. Solve for the sum of the measures of the interior angles of the following convex polygons. 1. 2. 3. 4. 1.4. Interior and Exterior Angles of a Convex Polygon 13 Mathematics 7 Unit 1: Angles and Polygons 5. 6. B. Solve for the measure of one interior angle and one exterior angle of the following regular polygons. 1. square 4. regular decagon 2. regular pentagon 5. regular 15-gon 3. regular octagon 6. regular 20-gon C. Solve the following word problems. 1. The cross-section of a pencil is in the shape of a regular polygon. The interior angle of the polygon is 120°. What kind of polygon is it? 2. Twenty fences are used to assemble a playpen in the shape of a convex polygon. What is the sum of the measures of the interior angles of the playpen? 1.4. Interior and Exterior Angles of a Convex Polygon 14 Mathematics 7 Unit 1: Angles and Polygons Key Points A convex polygon is a polygon that has all angles pointing outward. The interior angles of a polygon are angles that lie between each pair of sides of a figure and are located inside the figure. ○ The sum of the measures of the interior angles of a convex polygon is with 𝑛 sides is 𝑆 = 180° × (𝑛 − 2). ○ The measure of one interior angle of a regular polygon with 𝑛 sides is 180°×(𝑛−2) 𝑛. The exterior angles of a polygon are angles adjacent to the interior angle of the convex polygon formed by extending the side of the polygon. 360° ○ The measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑛. 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 identify the interior and exterior angles of a convex polygon. I can explain the relationship between the exterior angle and the adjacent interior angle of a polygon. 1.4. Interior and Exterior Angles of a Convex Polygon 15 Mathematics 7 Unit 1: Angles and Polygons 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 determine the measures of the interior and exterior angles of a regular polygon. Re ection I find ______________________ the most interesting because ________________________________. I need to improve on ________________________ because ___________________________________. I need to practice _________________________ because ______________________________________. I plan to __________________________________________________________________________________. 1.4. Interior and Exterior Angles of a Convex Polygon 16 Mathematics 7 Unit 1: Angles and Polygons Attribution soccer ball on football field by dolgachov is licensed under Envato License via Envato Elements. References “Exterior Angles in Convex Polygons.” n.d. cK-12 Foundation. Accessed September 29, 2023. https://flexbooks.ck12.org/cbook/ck-12-basic-geometry-concepts/section/6.2/primary/lesson /exterior-angles-in-convex-polygons-bsc-geom/ “Interior Angles in Convex Polygons.” n.d. LibreTexts K12 Education. Accessed September 29, 2023. https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/05%3A_Quadrilaterals_and_P olygons/5.27%3A_Interior_Angles_in_Convex_Polygons. 1.4. Interior and Exterior Angles of a Convex Polygon 17  Math 4 Unit 1: Angles and Polygons Answer Key Warm-Up 1. What have you observed about the sum of measures of the angles of the polygons? Possible answer: “The sum of the measures of the angles of the polygons are multiples of 180°.” 2. What do you think is the relationship between the sum of the measures of the angles of a polygon and the number of its sides? Possible answer: “The sum of the measures of the angles is 180° times 2 less than the number of sides of the polygon.” Let’s Try It 1. Find the sum of the measures of the interior angles of a convex decagon. Solution: A decagon has 10 sides. Thus, 𝑛 = 10. Let us use the formula to solve for the sum of the measures of the interior angles. Answer: 1, 440° 2. Find the measure of one interior angle of a regular nonagon. Solution: A nonagon has 9 sides. Thus, 𝑛 = 9. Let us use the formula to solve for the measure of one interior angle of a regular nonagon. 1.4. Interior and Exterior Angles of a Convex Polygon 18  Math 4 Unit 1: Angles and Polygons Answer: 140° 3. Find the measure of one exterior angle of a regular dodecagon. Solution: A dodecagon has 12 sides. Thus, 𝑛 = 12. Let us use the formula to solve for the measure of one exterior angle of a regular dodecagon. Answer: 30° 4. Patty owns a necklace and its gem has the shape of a regular polygon. The interior angle of the gem is 140°. What kind of polygon is the gem on her necklace? Solution: Step 1: Solve for the measure of the exterior angle of the polygon. Note that the interior angle of a polygon and its corresponding exterior angle are supplementary. To solve for the measure of the exterior angle, subtract the measure of the interior angle from 180°. 1.4. Interior and Exterior Angles of a Convex Polygon 19  Math 4 Unit 1: Angles and Polygons Step 2: Solve for the number of sides of the polygon. Recall that the formula for the measure of the exterior angle of a 360° polygon is 𝑛 , where 𝑛 is the number of sides. We can use this to determine the number of sides by reversing the operation. We divide 360° by the measure of the exterior angle to determine the number of sides of the polygon. Answer: The gem is a nonagon. 1.4. Interior and Exterior Angles of a Convex Polygon 20