7.1 Interior and Exterior Angles

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1 COMMON CORE Locker l LESSON 7. Interior and Exterior ngles Name Class Date 7. Interior and Exterior ngles Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? Common Core Math Standards The student is expected to: COMMON CORE G-CO.C.0 Prove theorems about triangles. Mathematical Practices COMMON CORE MP.8 Patterns Language Objective Work in small groups to play angle jeopardy. ENGGE Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? The sum of the interior angle measures of a triangle is 80. You can find the sum of the interior angle measures of any n-gon, where n represents the number of sides of the polygon, by multiplying (n - ) 80. In a polygon, an exterior angle forms a linear pair with its adjacent interior angle, so the sum of their measures is 80. In a triangle, the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. Explore Exploring Interior ngles in Triangles You can find a relationship between the measures of the three angles of a triangle. n interior angle is an angle formed by two sides of a polygon with a common vertex. So, a triangle has three interior angles. Use a straightedge to draw a large triangle on a sheet of paper and cut it out. Tear off the three corners and rearrange the angles so their sides are adjacent and their vertices meet at a point. What seems to be true about placing the three interior angles of a triangle together? They form a straight angle. Make a conjecture about the sum of the measures of the interior angles of a triangle. The conjecture about the sum of the interior angles of a triangle can be proven so it can be stated as a theorem. In the proof, you will add an auxiliary line to the triangle figure. n auxiliary line is a line that is added to a figure to aid in a proof. The sum of the measures of the interior angles of a triangle is 80. The Triangle Sum Theorem The sum of the angle measures of a triangle is 80. Fill in the blanks to complete the proof of the Triangle Sum Theorem. Given: C Prove: m + m + m = 80 Statements Reasons. Draw line l through point parallel to C _.. Parallel Postulate. m = m 4 and m = m 5. lternate Interior ngles Theorem interior angles 4 5 C Resource Locker l PREVIEW: LESSON PERFORMNCE TSK View the Engage section online. Discuss the photo, asking students to recall and describe the designs of game boards of their favorite games. Then preview the Lesson Performance Task.. ngle ddition Postulate and definition of. m 4 + m + m 5 = 80 straight angle 4. m + m + m = Substitution Property of Equality Module 7 Lesson Name Class Date 7. Interior and Exterior ngles Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? Explore Exploring Interior ngles in Triangles You can find a relationship between the measures of the three angles of a triangle. n interior angle is an angle formed by two sides of a polygon with a common vertex. So, a triangle has three interior angles. G-CO.C.0 Prove theorems about triangles. Use a straightedge to draw a large triangle on a sheet of paper and cut it out. Tear off the three corners and rearrange the angles so their sides are adjacent and their vertices meet at a point. What seems to be true about placing the three interior angles of a triangle together? They form a straight angle. Make a conjecture about the sum of the measures of the interior angles of a triangle. The sum of the measures of the interior angles of a triangle is 80. The conjecture about the sum of the interior angles of a triangle can be proven so it can be stated as a theorem. In the proof, you will add an auxiliary line to the triangle figure. n auxiliary line is a line that is added to a figure to aid in a proof. The Triangle Sum Theorem The sum of the angle measures of a triangle is 80. Fill in the blanks to complete the proof of the Triangle Sum Theorem. Given: C Prove: m + m + m = 80 Statements Reasons Resource interior angles 4 5 C. Draw line l through point parallel to C _.. Parallel Postulate. m = m and m = m.. ngle ddition Postulate and definition of. m 4 + m + m 5 = 80 straight angle 4. m + m + m = lternate Interior ngles Theorem Substitution Property of Equality Module 7 Lesson HRDCOVER PGES 7 8 Turn to these pages to find this lesson in the hardcover student edition. Lesson 7.

2 Reflect. Explain how the Parallel Postulate allows you to add the auxiliary line into the triangle figure. Since there is only one line parallel to a given line that passes through a given point, I can draw that line into the triangle and know it is the only one possible.. What does the Triangle Sum Theorem indicate about the angles of a triangle that has three angles of equal measure? How do you know? _ 80 = 60, so each angle of the triangle must have a measure of 60. Explore Exploring Interior ngles in Polygons To determine the sum of the interior angles for any polygon, you can use what you know about the Triangle Sum Theorem by considering how many triangles there are in other polygons. For example, by drawing the diagonal from a vertex of a quadrilateral, you can form two triangles. Since each triangle has an angle sum of 80, the quadrilateral must have an angle sum of = 60. Draw the diagonals from any one vertex for each polygon. Then state the number of triangles that are formed. The first two have already been completed. triangle triangle quadrilateral triangles triangles quadrilateral triangles EXPLORE Exploring Interior ngles in Triangles INTEGRTE TECHNOLOGY Students have the option of completing the interior angles in triangles activity either in the book or online. QUESTIONING STRTEGIES What can you say about angles that come together to form a straight line? Why? The sum of the angle measures must be 80 by the definition of a straight angle and the ngle ddition Postulate. Is it possible for a triangle to have two obtuse angles? Why or why not? No; the sum of these angles would be greater than triangles For each polygon, identify the number of sides and triangles, and determine the angle sums. Then complete the chart. The first two have already been done for you. Polygon Number of Sides 5 triangles 6 triangles Number of Triangles Sum of Interior ngle Measures Triangle ()80 = 80 Quadrilateral 4 ()80 = 60 Pentagon 5 ( ) 80 = 540 Hexagon 6 4 ( 4 ) 80 = 70 Decagon 0 8 ( 8 ) 80 = 440 Module 7 4 Lesson PROFESSIONL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.8, which calls for students to look for and identify patterns. Throughout the lesson, students use hands-on investigations or geometry to predict patterns and relationships for the interior and exterior angles of a triangle or polygon. They prove the Triangle Sum Theorem, the Polygon ngle Sum Theorem, and the Exterior ngle Theorem. The hands-on investigations give students a chance to use inductive reasoning to make a conjecture. This is followed by a proof in which students use deductive reasoning to justify their conjectures. EXPLORE Exploring Interior ngles in Polygons VOID COMMON ERRORS When attempting to determine the sum of the interior angles of a polygon, some students may divide the figure into too many triangles. For example, a student may draw both diagonals of a quadrilateral and conclude that the sum of the interior angles of a polygon is 70. Point out that four of the angles of the triangles are not part of an interior angle of the quadrilateral, and demonstrate the correct division. Interior and Exterior ngles 4

3 QUESTIONING STRTEGIES How do you use the sum of the angles of a triangle to find the sum of the interior angle measures of a convex polygon? If a convex polygon is broken up into triangles, then the sum of the interior angles is the number of triangles times 80. EXPLIN Using Interior ngles INTEGRTE MTHEMTICL PRCTICES Focus on Math Connections MP. You may want to review with students how to evaluate algebraic expressions and how to use inverse operations to solve equations. Do you notice a pattern between the number of sides and the number of triangles? If n represents the number of sides for any polygon, how can you represent the number of triangles? n - Make a conjecture for a rule that would give the sum of the interior angles for any n-gon. Sum of interior angle measures = (n - ) 80 Reflect. In a regular hexagon, how could you use the sum of the interior angles to determine the measure of each interior angle? Since the polygon is regular, you can divide the sum by 6 to determine each interior angle measure. 4. How might you determine the number of sides for a polygon whose interior angle sum is 40? Write and solve an equation for n, where (n - ) 80 = 40. [# Explain Using Interior ngles You can use the angle sum to determine the unknown measure of an angle of a polygon when you know the measures of the other angles. Polygon ngle Sum Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n )80. Example Determine the unknown angle measures. For the nonagon shown, find the unknown angle measure x. First, use the Polygon ngle Sum Theorem to find the sum of the interior angles: n = 9 (n - )80 = (9 - )80 = (7)80 = 60 Then solve for the unknown angle measure, x : x = 60 x = 58 The unknown angle measure is x Module 7 5 Lesson COLLORTIVE LERNING Small Group ctivity Geometry software allows students to explore the theorems in this lesson. For the Triangle Sum Theorem and the Exterior ngle Theorem, students should construct a triangle, measure the three angles, and use the Calculate tool (in the Measure menu) to find the sum of the interior angle measures and also to find the sum of the exterior angles. s students drag the vertices of the triangle to change its shape, the individual angle measures will change, but the sum of the measures will remain 80 for the interior angles and 60 for the exterior angles. 5 Lesson 7.

4 Determine the unknown interior angle measure of a convex octagon in which the measures of the seven other angles have a sum of 940. n = Sum = ( - ) 80 = ( ) 80 = x = x = The unknown angle measure is QUESTIONING STRTEGIES How do you use the sum of the interior angle measures of a polygon to find the measure of an unknown interior angle? Use the Polygon Sum Theorem to find the total measure of the interior angles, then solve an algebraic equation to find the unknown angle. Reflect 5. How might you use the Polygon ngle Sum Theorem to write a rule for determining the measure of each interior angle of any regular convex polygon with n sides? (n - ) 80 You can divide the angle sum by n. n gives the measure of an interior angle for any regular polygon. Your Turn 6. Determine the unknown angle measures in this pentagon. x x n = 5 Sum = (5 - ) 80 = () 80 = x = 540 x = 70 x = 5 Each unknown angle measure is Determine the measure of the fourth interior angle of a quadrilateral if you know the other three measures are 89, 80, and 04. n = 4 Sum = (4 - ) 80 = (80 ) = x = 60 x = 87 The unknown angle measure is Determine the unknown angle measures in a hexagon whose six angles measure 69, 08, 5, 04, b, and b. n = 6 Sum = (6 - ) 80 = (4) 80 = 70 b + b = 70 b + 56 = 70 b = 04 b = 68 b = 6 The two unknown angle measures are 68 and 6. Module 7 6 Lesson DIFFERENTITE INSTRUCTION Manipulatives Have students fold and crease the four corners of a sheet of paper. Next, ask them to open the folds to reveal a creased polygon shape. Have students classify the polygon (octagon). sk them to find the sum of the interior and exterior angle measures (080 ; 60 ). Then have students measure the interior and exterior angles to verify their sums. Interior and Exterior ngles 6

5 EXPLIN Proving the Exterior ngle Theorem QUESTIONING STRTEGIES How does finding the measure of an exterior angle differ from finding the measure of an interior angle? The measure of an exterior angle is the supplement of its adjacent interior angle because the angles form linear pairs with the interior angles. The measure of an interior angle is not found by using linear pairs. Why is the Exterior ngle Theorem sometimes called a corollary of the Triangle Sum Theorem? because the Exterior ngle Theorem follows from the Triangle Sum Theorem Explain Proving the Exterior ngle Theorem n exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles. remote interior angle is an interior angle that is not adjacent to the exterior angle. Example Follow the steps to investigate the relationship between each exterior angle of a triangle and its remote interior angles. Step Use a straightedge to draw a triangle with angles,, and. Line up your straightedge along the side opposite angle. Extend the side from the vertex at angle. You have just constructed an exterior angle. The exterior angle is drawn supplementary to its adjacent interior angle. Step You know the sum of the measures of the interior angles of a triangle. m + m + m = 80 Since an exterior angle is supplementary to its adjacent interior angle, you also know: m + m 4 = 80 Make a conjecture: What can you say about the measure of the exterior angle and the measures of its remote interior angles? Conjecture: The measure of the exterior angle is the same as the sum of the measures of its two remote interior angles. The conjecture you made in Step can be formally stated as a theorem. Remote interior angles Exterior angle 4 Exterior ngle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Step Complete the proof of the Exterior ngle Theorem. 4 is an exterior angle. It forms a linear pair with interior angle. Its remote interior angles are and. 4 y the Triangle Sum Theorem, m + m + m = 80. lso, m + m 4 = 80 because they are supplementary and make a straight angle. y the Substitution Property of Equality, then, m + m + m = m + m 4. Subtracting m from each side of this equation leaves m + m = m 4. This means that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Module 7 7 Lesson LNGUGE SUPPORT Communicate Math Have students write clues about interior and exterior angles in polygons, for example: The sum of the interior angles of this three-sided polygon is 80 degrees or The sum of the exterior angles of this three-sided polygon is 60 degrees. Have each student write two clue cards about different polygons. They then read their clues to the rest of the group, and the group must decide which polygon fits the clue. 7 Lesson 7.

6 Reflect 9. Discussion Determine the measure of each exterior angle. dd them together. What can you say about their sum? Explain. The exterior angles will measure 40, 0, and 00. Their sum is 60. Each exterior angle is equal to the sum of the measures of the two remote interior angles, and the sum of all exterior angles includes each interior angle twice. 0. ccording to the definition of an exterior angle, one of the sides of the triangle must be extended in order to see it. How many ways can this be done for any vertex? How many exterior angles is it possible to draw for a triangle? for a hexagon? Two exterior angles can be drawn from any vertex by extending either side, so a triangle can have 6 exterior angles. You could draw different exterior angles for a hexagon. Explain Using Exterior ngles You can apply the Exterior ngle Theorem to solve problems with unknown angle measures by writing and solving equations. Example Find m. D Determine the measure of the specified angle. Write and solve an equation relating the exterior and remote interior angles. 45 = z + 5z - 45 = 7z - z = Now use this value for the unknown to evaluate the expression for the required angle. m = (5z - ) = (5() - ) = (05 - ) = 0 45 (5z - ) C z Find m PRS. R (x - 8) S Write an equation relating the exterior and remote interior angles. x - 8 = (x + ) + 90 Solve for the unknown (x + ) Use the value for the unknown to evaluate the expression for the required angle. Q P x - 8 = x + 9 x = 00 x = 50 m PRS = (x - 8) = ( (50) - 8) = 4 Module 7 8 Lesson VOID COMMON ERRORS Some students may confuse the theorems in this lesson and incorrectly assume that the sum of the interior angles of a polygon is 60. Remind students of the Triangle Sum Theorem. Have them draw an equilateral triangle and show that its interior angle measures add to 80 and its exterior angle measures add to 60. EXPLIN Using Exterior ngles INTEGRTE MTHEMTICL PRCTICES Focus on Patterns MP.8 Encourage students to make a table listing the sums of the exterior angles of regular triangles, quadrilaterals, pentagons, and hexagons. sk them what they notice about the sum of the exterior angles. (The sum is always 60.) sk them to find the pattern in the measure of each individual exterior angle for these regular polygons. (They each have the same measure.) CONNECT VOCULRY Help students understand the meanings of interior, exterior, and remote by writing the definitions on note cards. n interior angle is inside the figure, an exterior angle is outside the figure, and a remote interior angle is interior and away from the exterior angle. Relate the idea of a remote interior angle to a television remote control that sends a signal across the room and away from you. Interior and Exterior ngles 8

7 QUESTIONING STRTEGIES What kind of angle is formed by extending one of the sides of a triangle? What is its relationship to the adjacent interior angle? an exterior angle; the angles are supplementary. Your Turn Determine the measure of the specified angle.. Determine m N in MNP.. If the exterior angle drawn measures 50, and the measure of D is twice that of E, find the N measure of the two remote interior angles. (x + 7) D ELORTE QUESTIONING STRTEGIES How do you use the sum of the interior angle measures of a regular polygon to find the measure of each interior angle? Divide the sum of the interior angles by the number of sides. 6 (5x + 50) M P Q 5x + 50 = (x + 7) + 6 5x + 50 = x + 70 x = 0 x = 0 m N = (x + 7) = ((0) + 7) = 7 50 E F G x + x = 50 x = 50 x = 50 m E = x = 50 m D = x = 00 What happens to the measure of each exterior angle as the number of sides of a regular polygon increases? Why? The measures get smaller and smaller because the sum must remain 60. SUMMRIZE THE LESSON Have students fill out a chart to summarize the theorems in this lesson. Sample: Triangle Sum Theorem m + m + m = 80 Polygon Sum Theorem (n - ) 80 = (6 - ) 80 = 70 Elaborate. In your own words, state the Polygon ngle Sum Theorem. How does it help you find unknown angle measures in polygons? Possible answer: The sum of the measures of the interior angles of a convex polygon equals 80 (n - ). You can use it to find an unknown measure of an interior angle of a polygon when you know the measures of the other angles. 4. When will an exterior angle be acute? Can a triangle have more than one acute exterior angle? Describe the triangle that tests this. n exterior angle will be acute when paired with an obtuse adjacent interior angle; therefore, the triangle must be obtuse. Since a triangle must have two or three acute interior angles, at least two exterior angles must be obtuse. 5. Essential Question Check-In Summarize the rules you have discovered about the interior and exterior angles of triangles and polygons. The sum of the measures of the interior angles of a triangle is 80. The sum of the measures of the interior angles for any polygon can be found by the rule (n - ) 80, where n represents the number of sides of the polygon. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Module 7 9 Lesson Exterior ngle Theorem m 4 = m + m 4 9 Lesson 7.

8 Evaluate: Homework and Practice. Consider the Triangle Sum Theorem in relation to a right triangle. What conjecture can you make about the two acute angles of a right triangle? Explain your reasoning. They must be complementary. One angle of the right triangle measures 90. So the sum of the remaining two angles is = 90.. Complete a flow proof for the Triangle Sum Theorem. Given C Prove m + m + m = 80 m = m 4 lternate Interior ngles Theorem Draw l parallel to C through. Parallel Postulate m = m 5 m 4 + m + m 5 = 80 lt Int ngles Theorem Definition of straight angle m + m + m =80 Substitution Property of Equality. Given a polygon with sides, find the sum of the measures of its interior angles. (n - ) 80 = ( - ) 80 = () 80 = 980 polygon with sides has an interior angle measure sum of polygon has an interior angle sum of 060. How many sides must the polygon have? 060 = (n - ) 80 9 = n The polygon must have 9 sides. Solve for the unknown angle measures of the polygon. 6. pentagon has angle measures of 00, 05, 0 and 5. Find the fifth angle measure. (5 - ) 80 = () 80 = = x 0 = x The measure of the fifth angle is 0. Online Homework Hints and Help Extra Practice 5. Two of the angles in a triangle measure 50 and 7. Find the measure of the third angle x = 80 x = 0 7. The measures of angles of a 4-gon add up to 04. Find the fourteenth angle measure? 4 5 The measure of the third angle is 0. (4 - ) 80 = () 80 = x = 60 x = 46 The measure of the 4th angle is 46. C l EVLUTE SSIGNMENT GUIDE Concepts and Skills Explore Exploring Interior ngles in Triangles Explore Exploring Interior ngles in Polygons Example Using Interior ngles Example Proving the Exterior ngle Theorem Example Using Exterior ngles Practice Exercises Exercises 5 Exercises 6 9 Exercises 0 Exercises 4 INTEGRTE MTHEMTICL PRCTICES Focus on Technology MP.5 Students can use a spreadsheet as a reference to find the sum of the interior angles of a convex polygon with n sides for n = to 0 (or higher). They could also use the spreadsheet to give the measure of each interior and exterior angle of a regular polygon with n sides. Module 7 0 Lesson Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Skills/Concepts MP. Reasoning 5 Skills/Concepts MP.6 Precision 6 9 Skills/Concepts MP.5 Using Tools 0 6 Skills/Concepts MP.4 Modeling 7 Strategic Thinking MP. Logic 8 Strategic Thinking MP.6 Precision 9 Strategic Thinking MP. Reasoning Interior and Exterior ngles 0

9 8. Determine the unknown angle measures for the quadrilateral in the diagram. (4 - ) 80 = () 80 = 60 x = 7 x + x + x + 4x = 60 x = 08 x = 6 4x = 44 The measures of the interior angles of the quadrilateral are 6, 7, 08, and 44. 4x x x x 9. The cross-section of a beehive reveals it is made of regular hexagons. What is the measure of each angle in the regular hexagon? (n - ) 80 = (6 - ) 80 = (4) 80 = 70 6x = 70 x = 0 Each angle of a regular hexagon measures Create a flow proof for the Exterior ngle Theorem. m + m + m = 80 Triangle Sum Theorem m + m 4 = 80 Definition of supplementary 4 m + m + m = m + m 4 Substitution Property of Equality Image Credits: StudioSmart/Shutterstock m + m = m 4 Substraction Property of Equality Find the value of the variable to find the unknown angle measure(s).. Find w to find the measure of the exterior angle.. Find x to find the measure of the remote interior angle. w = x x + 46 = 4 w = 6 w x = Module 7 Lesson Lesson 7.

10 . Find m H. 4. Determine the measure of the indicated exterior angle in the diagram. H (6x - ) x 6 F G J (5x + 7) (6x - ) + (5x + 7) = 6 x = 0 m H = (6x - ) = (6 (0) - ) = Match each angle with its corresponding measure, given m = 0 and m 7 = 70. Indicate a match by writing the letter for the angle on the line in front of the corresponding angle measure.. m m 60 5 C. m 4 D 70 D. m 5 E 0 E. m 6 C 0 x 80 - (x + 4) = x + x = x 4 (x + 4)? 80 - ( () + 4) = 80 - (66 + 4) = = 0 The measure of the indicated exterior angle is 0. INTEGRTE MTHEMTICL PRCTICES Focus on Communication MP. Discuss each of the following questions about triangles as a class. Have students explain how the Triangle Sum Theorem justifies their responses.. triangle can have only one obtuse angle or only one right angle.. The acute angles of a right triangle are complementary. 6. The map of France commonly used in the 600s was significantly revised as a result of a triangulation survey. The diagram shows part of the survey map. Use the diagram to find the measure of KMJ. Note that KMJ NKM. m KMN + m MNK + m NKM = m NKM = m NKM = 80 m NKM = 44 KMJ NKM, so m KMJ = m NKM = n artistic quilt is being designed using computer software. The designer wants to use regular octagons in her design. What interior angle measures should she set in the computer software to create a regular octagon? (n - ) 80 = (8 - ) 80 = (6) 80 = 080 _ 080 = 5 8 The designer should set the interior angles of the regular octagon at Module 7 Lesson Interior and Exterior ngles

11 8. ladder propped up against a house makes a 0 angle with the wall. What would be the ladder's angle measure with the ground facing away from the house? The house is perpendicular to the ground, so the other remote interior angle is = 0, so the measure of the indicated exterior angle is 0.? ladder 0 house 9. Photography The aperture of a camera is made by overlapping blades that form a regular decagon. a. What is the sum of the measures of the interior angles of the decagon? (0 - ) 80 = (8) 80 = 440 b. What would be the measure of each interior angle? each exterior angle? = 44 ; = 6 c. Find the sum of all ten exterior angles. 6 (0) = Determine the measure of UXW in the diagram. ground Y m WUX = 90 V 80 = m UXW U W m UXW = 6 Image Credits: neyro008/istockphoto.com X Z. Determine the measures of angles x, y, and z z 55 x y 60 x = 80 - ( ) = 0 y = 80 - ( ) = 45 z = 80 - (0 + 45) = 5 Module 7 Lesson Lesson 7.

12 . Given the diagram in which D bisects C and CD bisects C, what is m DC? m C = (m DC) = (5 ) = m C + 90 _ = 80, so m C = 60. Then, m DC = (m C) _ = (60 ) = m DC + 0 = 80, so m DC = 5.. What If? Suppose you continue the congruent 4. lgebra Draw a triangle C and label the angle construction shown here. What polygon will measures of its angles a, b, and c. Draw ray D you construct? Explain. that bisects the exterior angle at vertex. Write an expression for the measure of angle CD. 5 D Possible answer: C VOID COMMON ERRORS Some students may multiply the number of sides of a polygon by 80 to find the sum of the interior angles of the polygon. Remind them that the sum is based on the number of triangles. Since the sum of the angles of a triangle ( sides) is 80, to find the sum of the interior angles, they must subtract from the number of sides before multiplying by 80. C c D 0 regular hexagon; if the construction continues and the sides are kept congruent, the polygon will include six 0 angles and six congruent sides, so it is a regular hexagon. 5. Look for a Pattern Find patterns within this table of data and extend the patterns to complete the remainder of the table. What conjecture can you make about polygon exterior angles from Column 5? Column Number of Sides Column Sum of the Measures of the Interior ngles Column verage Measure of an Interior ngle Column 4 verage Measure of an Exterior ngle Column 5 Sum of the Measures of the Exterior ngles () = (4) = a m CD = ( a + c ) 7 b 7 (5) = (6) = 60 Conjecture: It appears from the table that the sum of the measures of the exterior angles of any polygon is always 60. Module 7 4 Lesson Interior and Exterior ngles 4

13 JOURNL Have students review the Polygon ngle Sum Theorem and the Polygon Exterior ngle Theorem, and then draw a pentagon and show how to find its interior angle sum measures and its exterior angle sum measures. 540, Explain the Error Find and explain what this student did incorrectly when solving the following problem. What type of polygon would have an interior angle sum of 60? 60 = (n - ) 80 7 = n - 5 = n The polygon is a pentagon. The error is that the student subtracted from both sides instead of adding. The value of n should be 9, and the polygon is a nonagon. H.O.T. Focus on Higher Order Thinking 7. Communicate Mathematical Ideas Explain why if two angles of one triangle are congruent to two angles of another triangle, then the L R third pair of angles are also congruent. T Given: L R, M S N S M Prove: N T y the Triangle Sum Theorem, m L + m M + m N = 80 and m R + m S + m T = 80. Since each set of angle measures total 80, they are equal using the substitution property of equality. So, m L + m M + m N = m R + m S + m T. Since L R and M S, then m L = m R and m M = m S by the definition of congruence. Subtracting equals from both sides gives m N = m T. Then N T by the definition of congruence. 8. nalyze Relationships Consider a right triangle. How would you describe the measures of its exterior angles? Explain. n exterior angle will be right when paired with a right adjacent interior angle. There can be only one right angle in a triangle. Since a triangle must have two or three acute interior angles, the other two exterior angles must be obtuse. 9. Look for a Pattern In investigating different polygons, diagonals were drawn from a vertex to break the polygon into triangles. Recall that the number of triangles is always two less than the number of sides. ut diagonals can be drawn from all vertices. Make a table where you compare the number of sides of a polygon with how many diagonals can be drawn (from all the vertices). Can you find a pattern in this table? Number of Sides, n Number of Diagonals, d The number of diagonals increases by, then, 4, 5, etc. formula n (n - ) relating n and d is d = _. Module 7 5 Lesson 5 Lesson 7.

14 Lesson Performance Task You ve been asked to design the board for a new game called Pentagons. The board consists of a repeating pattern of regular pentagons, a portion of which is shown in the illustration. When you write the specifications for the company that will make the board, you include the measurements of D, C, CD and DC. Find the measures of those angles and explain how you found them. m D = m CD = 6 m C = m DC = C To find the measure of each interior angle of one of the pentagons, divide it into three triangles. This gives the sum of the measures of the five angles of the pentagon, 80 = 540. Each angle measures = 08. Draw D. m D = = 7. m C = m DC = 7 = 44 m D = m CD = 80 - (7 + 7 ) = = 6 D INTEGRTE MTHEMTICL PRCTICES Focus on Communication MP. Direct students attention to quadrilateral CD. sk: Without knowing anything about the angles of CD, how could you identify the type of quadrilateral that it is? What type is it? The sides of CD are sides of congruent regular pentagons, so they are congruent. quadrilateral with four congruent sides is a rhombus. What does the type of quadrilateral that CD is tell you about the angles of the figure? The opposite angles are congruent. The adjacent angles are supplementary. INTEGRTE MTHEMTICL PRCTICES Focus on Critical Thinking MP. The perimeter of each pentagon in the diagram is 6 cm. What is the perimeter of quadrilateral CD? Explain..8 cm; length of each side of each pentagon = 6 cm 5 =. cm; perimeter of CD = 4 x. =.8 cm Module 7 6 Lesson EXTENSION CTIVITY Have students design and draw game boards consisting of congruent quadrilaterals, congruent pentagons, and/or congruent hexagons. Each design should show at least two different classes of polygons (for example, quadrilaterals and hexagons) and a total of at least six polygons. Students should write the measure of each angle directly on the figures, and write elsewhere an explanation of how, without protractors, they found each measure. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem. Interior and Exterior ngles 6