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LESSON 7.1 Name Interior and Exterior Angles Class 7.1 Date Interior and Exterior Angles Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? Resource Locker G.6.D Verify theorems about the relationships in triangles, including…the sum of interior angles…and apply these relationships to solve problems. Also G.5.A Texas Math Standards The student is expected to: Explore 1 G.5.A Exploring Interior Angles in Triangles You can find a relationship between the measures of the three angles of a triangle. An interior angle is an angle formed by two sides of a polygon with a common vertex. So, a triangle has three interior angles. Investigate patterns to make conjectures about geometric relationships, including angles formed by . . . interior and exterior angles of polygons . . . choosing from a variety of tools. G.6.D 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? Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. They form a straight angle. Language Objective Work in small groups to play angle jeopardy. ENGAGE 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 180°. 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 - 2)180°. In a polygon, an exterior angle forms a linear pair with its adjacent interior angle, so the sum of their measures is 180°. In a triangle, the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. © Houghton Mifflin Harcourt Publishing Company 1.A, 1.F, 2.D.1, 2.D.2, 2.I.4 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 180°. 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. An 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 180°. Fill in the blanks to complete the proof of the Triangle Sum Theorem. Given: △ABC B 4 Prove: m∠1 + m∠2 + m∠3 = 180° A Module 7 ges EDIT--Chan DO NOT Key=TX-A Correction must be Name d Interior an 351 Lesson 7.1 C gles Exterior An le of a triang or angles r and exteri the interio say about can you r ion: What polygons? e sum of interio and other including…th in triangles, relationships problems. Also G.5.A about the to solve theorems ngles relationships G.6.D Verify apply these les in Tria Ang angles…and rior le. ng Inte of a triang 1 Explori three angles common vertex. res of the a Explore with measu n the polygo between sides of a a relationship formed by two You can find an angle r angle is angles. An interio has three interior le So, a triang 7.1 HARDCOVER PAGES 291304 Resource Locker Quest Essential GE_MTXESE353886_U2M07L1.indd 351 interior angles Turn to these pages to find this lesson in the hardcover student edition. le on a sheet a large triang s and tedge to draw off the three corner Use a straigh Tear nt and cut it out. are adjace of paper and so their sides the angles rearrange a point. s meet at their vertice le together? of a triang interior angles g the three about placin to be true le. angle. of a triang a straight interior angles res of the the measu le is 180°. the sum of m. In the s of a triang ture about angle as a theore aid in or conjec a interi be stated Make to so it can ures of the to a figure can be proven a line that is added of the meas a triangle The sum ry line is r angles of An auxilia the interio le figure. the sum of the triang ture about ry line to auxilia The conjec an will add proof, you a proof. Theorem gle Sum le is 180°. The Trian of a triang res measu of the angle em. The sum ℓ B le Sum Theor of the Triang 4 2 5 the proof complete blanks to Fill in the What seems They form © Houghto n Mifflin Harcour t Publishin y g Compan 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. 3 Date Class PREVIEW: LESSON PERFORMANCE TASK ℓ 5 Lesson 1 351 gh "File info" made throu 1 2 Given: △ABC m∠3 = 180° + m∠2 + Prove: m∠1 A 1 3 C Lesson 1 351 Module 7 7L1.indd 86_U2M0 ESE3538 GE_MTX 351 2/20/14 9:04 PM 2/20/14 9:04 PM Statements _ 1. Draw line ℓ through point B parallel to AC. 4 2. m∠1 = m∠ and m∠3 = m∠ 5 1 + m∠2 + m∠ EXPLORE 1 1. Parallel Postulate 2. Alternate Interior Angles Theorem Exploring Interior Angles in Triangles 3. Angle Addition Postulate and definition of straight angle 3. m∠4 + m∠2 + m∠5 = 180° 4. m∠ Reasons 3 = 180° 4. Substitution Property of Equality INTEGRATE TECHNOLOGY Students have the option of completing the interior angles in triangles activity either in the book or online. Reflect 1. 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. 2. What does the Triangle Sum Theorem indicate about the angles of a triangle that has three angles of equal measure? How do you know? 180 = 60, so each angle of the triangle must have a measure of 60°. 3 QUESTIONING STRATEGIES _ Explore 2 What can you say about angles that come together to form a straight line? Why? The sum of the angle measures must be 180° by the definition of a straight angle and the Angle Addition Postulate. Exploring Interior Angles 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 180°, the quadrilateral must have an angle sum of 180° + 180° = 360°. quadrilateral 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 180°. 2 triangles A triangle 1 triangle 4 triangles Module 7 © Houghton Mifflin Harcourt Publishing Company 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. quadrilateral 2 triangles 5 triangles 352 3 triangles EXPLORE 2 Exploring Interior Angles in Polygons AVOID COMMON ERRORS 6 triangles Lesson 1 PROFESSIONAL DEVELOPMENT GE_MTXESE353886_U2M07L1.indd 352 Integrate Mathematical Processes 2/20/14 9:04 PM 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 720°. 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. This lesson provides an opportunity to address Mathematical Process TEKS G.1.G, which calls for students to “Explain and justify arguments.” Throughout the lesson, students use hands-on investigations or geometry to predict relationships for the interior and exterior angles of a triangle or polygon. They prove the Triangle Sum Theorem, the Polygon Angle Sum Theorem, and the Exterior Angle 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 conjecture. Interior and Exterior Angles 352 B QUESTIONING STRATEGIES 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 180°. 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 EXPLAIN 1 Number of Sides Number of Triangles Triangle 3 1 (1)180° = 180° Quadrilateral 4 2 (2)180° = 360° Pentagon 5 3 ( 3 ) 180° = 540° Hexagon 6 4 ( 4 ) 180° = 720° Decagon 10 8 ( 8 ) 180° = 1440° Using Interior Angles C INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Sum of Interior Angle Measures 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-2 D 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 - 2)180° You may want to review with students how to evaluate algebraic expressions and how to use inverse operations to solve equations. Reflect 3. 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 © Houghton Mifflin Harcourt Publishing Company measure. 4. How might you determine the number of sides for a polygon whose interior angle sum is 3240°? Write and solve an equation for n, where (n - 2)180° = 3240°. [# Explain 1 Using Interior Angles 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. The Polygon Angle Sum Theorem states that the sum of the measures of the interior angles of a convex polygon can be found by multiplying 180° by 2 less than the number of sides. This can help you determine an unknown angle measurement, such as for a nonagon with eight given interior angles. Polygon Angle Sum Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2)180°. Module 7 353 Lesson 1 COLLABORATIVE LEARNING GE_MTXESE353886_U2M07L1.indd 353 Small Group Activity Geometry software allows students to explore the theorems in this lesson. For the Triangle Sum Theorem and the Exterior Angle 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. As 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 180° for the interior angles and 360° for the exterior angles. 353 Lesson 7.1 2/20/14 9:04 PM Example 1 Determine the unknown angle measures. QUESTIONING STRATEGIES For the nonagon shown, find the unknown angle measure x°. 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. First, use the Polygon Angle Sum Theorem to find the sum of the interior angles: n=9 (n - 2)180° = (9 - 2)180° = (7)180° = 1260° Then solve for the unknown angle measure, x°: 98° 125 + 130 + 172 + 98 + 200 + 102 + 140 + 135 + x = 1260 102° 125° 130° x = 158 140° The unknown angle measure is 158°. x° 135° 200° 172° 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= 8 Sum = ( ) 8 - 2 180° = 940 + x = ( 6 ) 180° = 1080° 1080 x = 140 The unknown angle measure is 140° . Reflect 5. any regular polygon. Your Turn 6. Determine the unknown angle measures in this pentagon. n=5 Sum = (5 - 2)180° = (3)180° = 540° 270 + 2x = 540 x° x° © Houghton Mifflin Harcourt Publishing Company How might you use the Polygon Angle Sum Theorem to write a rule for determining the measure of each interior angle of any regular convex polygon with n sides? (n - 2)180° gives the measure of an interior angle for You can divide the angle sum by n. __ n 2x = 270 x = 135 Each unknown angle measure is 135°. Module 7 354 Lesson 1 DIFFERENTIATE INSTRUCTION GE_MTXESE353886_U2M07L1 354 Manipulatives 2/22/14 11:17 PM 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). Ask them to find the sum of the interior and exterior angle measures (1080°; 360°). Then have students measure the interior and exterior angles to verify their sums. Interior and Exterior Angles 354 Determine the measure of the fourth interior angle of a quadrilateral if you know the other three measures are 89°, 80°, and 104°. 7. EXPLAIN 2 8. n=4 Proving the Exterior Angle Theorem Sum = (6 - 2)180° = (4)180° = 720° Sum = (4 - 2)180° = 2(180°) = 360° b + 2b + 69 + 108 + 135 + 204 = 720 89 + 80 + 104 + x = 360 QUESTIONING STRATEGIES Determine the unknown angle measures in a hexagon whose six angles measure 69°, 108°, 135°, 204°, b°, and 2b°. n=6 3b + 516 = 720 3b = 204 x = 87 b = 68 The unknown angle measure is 87°. 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. 2b = 136 The two unknown angle measures are 68° and 136°. Explain 2 Proving the Exterior Angle Theorem An 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. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Why is the Exterior Angle Theorem sometimes called a corollary of the Triangle Sum Theorem? because the Exterior Angle Theorem follows from the Triangle Sum Theorem Remote interior angles Example 2 Exterior angle Follow the steps to investigate the relationship between each exterior angle of a triangle and its remote interior angles. Step 1 Use a straightedge to draw a triangle with angles 1, 2, and 3. Line up your straightedge along the side opposite angle 2. Extend the side from the vertex at angle 3. You have just constructed an exterior angle. The exterior angle is drawn supplementary to its adjacent interior angle. ∠4 is an exterior angle. © Houghton Mifflin Harcourt Publishing Company 2 It forms a linear pair with interior angle ∠3. 1 3 4 Its remote interior angles are ∠1 and ∠2. Step 2 Use the diagram. You know the sum of the measures of the interior angles of a triangle. m∠1 + m∠2 + m∠3 = 180 ° Since an exterior angle is supplementary to its adjacent interior angle, you also know: m∠3 + m∠4 = 180 ° 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. Module 7 355 Lesson 1 LANAGUAGE SUPPORT GE_MTXESE353886_U2M07L1 355 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 180 degrees” or “The sum of the exterior angles of this three-sided polygon is 360 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. 355 Lesson 7.1 2/22/14 11:19 PM The conjecture you made in Step 2 can be formally stated as a theorem. AVOID COMMON ERRORS Exterior Angle Theorem Some students may confuse the theorems in this lesson and incorrectly assume that the sum of the interior angles of a polygon is 360°. Remind students of the Triangle Sum Theorem. Have them draw an equilateral triangle and show that its interior angle measures add to 180° and its exterior angle measures add to 360°. The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Step 3 Complete the proof of the Exterior Angle Theorem. 2 ∠4 is an exterior angle. It forms a linear pair with interior angle ∠3. 1 3 Its remote interior angles are ∠1 and ∠2. 4 By the Triangle Sum Theorem , m∠1 + m∠2 + m∠3 = 180°. Also, m∠3 + m∠4 = 180° because they are supplementary and make a straight angle. By the Substitution Property of Equality, then, m∠1 + m∠2 + m∠3 = m∠ 3 + m∠ 4 . Subtracting m∠3 from each side of this equation leaves m∠1 + m∠2 = 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. Reflect 9. Discussion Determine the measure of each exterior angle. Add them together. What can you say about their sum? Explain. 40° The exterior angles will measure 140°, 120°, and 100°. Their sum is 360°. Each exterior angle is equal to the sum of the measures of the two remote interior angles, and the sum of all 3 exterior angles includes each interior angle twice. 10. According 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 © Houghton Mifflin Harcourt Publishing Company 60° can have 6 exterior angles. You could draw 12 different exterior angles for a hexagon. Module 7 GE_MTXESE353886_U2M07L1.indd 356 356 Lesson 1 2/20/14 9:04 PM Interior and Exterior Angles 356 Explain 3 EXPLAIN 3 Using Exterior Angles You can apply the Exterior Angle Theorem to solve problems with unknown angle measures by writing and solving equations. Using Exterior Angles Example 3 Determine the measure of the specified angle. Find m∠B. INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns A 2z° Encourage students to make a table listing the sums of the exterior angles of regular triangles, quadrilaterals, pentagons, and hexagons. Ask them what they notice about the sum of the exterior angles. (The sum is always 360°.) Ask them to find the pattern in the measure of each individual exterior angle for these regular polygons. (They each have the same measure.) C B Write and solve an equation relating the exterior and remote interior angles. 145 = 2z + 5z - 2 145 = 7z - 2 z = 21 Now use this value for the unknown to evaluate the expression for the required angle. m∠B = (5z - 2)° = (5(21) - 2)° = (105 - 2)° = 103° CONNECT VOCABULARY Find m∠PRS. Q © Houghton Mifflin Harcourt Publishing Company Help students understand the meanings of interior, exterior, and remote by writing the definitions on note cards. An 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. (5z - 2)° 145° D R (3x - 8)° (x + 2)° P S Write an equation relating the exterior and remote interior angles. Solve for the unknown. 3x - 8 = (x + 2) + 90 3x - 8 = x + 92 2x = 100 x = 50 Use the value for the unknown to evaluate the expression for the required angle. m∠PRS = (3x - 8)° = (3(50) - 8)° = 142° Module 7 GE_MTXESE353886_U2M07L1.indd 357 357 Lesson 7.1 357 Lesson 1 2/20/14 9:04 PM Your Turn QUESTIONING STRATEGIES Determine the measure of the specified angle. 11. Determine m∠N in △MNP. 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. 12. If the exterior angle drawn measures 150°, and the measure of ∠D is twice that of ∠E, find the measure of the two remote interior angles. N (3x + 7)° D 150° E (5x + 50)° 63° M P F G ELABORATE x + 2x = 150 Q 3x = 150 5x + 50 = (3x + 7) + 63 QUESTIONING STRATEGIES x = 50 5x + 50 = 3x + 70 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. m∠E = x° = 50° 2x = 20 m∠D = 2x° = 100° x = 10 m∠N = (3x + 7)° = (3(10) + 7)° = 37° 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 360°. Elaborate 13. In your own words, state the Polygon Angle 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 (n - 2)180°. 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. SUMMARIZE THE LESSON Have students fill out a chart to summarize the theorems in this lesson. Sample: © Houghton Mifflin Harcourt Publishing Company 14. When will an exterior angle be acute? Can a triangle have more than one acute exterior angle? Describe the triangle that tests this. An 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. 15. 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 180°. The sum of the Triangle Sum Theorem m∠1 + m∠2 + m∠3 = 180° 2 where n represents the number of sides of the polygon. The measure of an exterior angle Polygon Sum Theorem (n - 2) 180° = (6 - 2) 180° = 720° of a triangle is equal to the sum of the measures of the two remote interior angles. Module 7 GE_MTXESE353886_U2M07L1 358 358 3 1 measures of the interior angles for any polygon can be found by the rule (n - 2)180°, Lesson 1 22/01/15 8:53 AM Exterior Angle Theorem m∠4 = m∠1 + m∠2 2 1 3 4 Interior and Exterior Angles 358