Exterior Angle Theorem

Transcription

1 7 Exterior ngle Theorem What You ll Learn You ll learn to identify exterior angles and remote interior angles of a triangle and use the Exterior ngle Theorem. Why It s Important Interior Design Designers use exterior angles to create patterns. See Exercise 8. In the figure at the right, recall that,, and are interior angles of PQR. ngle 4 is called an exterior angle of PQR. n exterior angle of a triangle is an angle that forms a linear pair with one of the angles of the triangle. Q In PQR, 4 is an exterior angle at R because it forms a linear pair with. Remote interior angles of a triangle are the two angles that do not form a linear pair with the exterior angle. In PQR, and are the remote interior angles with respect to 4. Each exterior angle has corresponding remote interior angles. How many exterior angles does YZ below have? P 4 R Vertex Exterior ngle Remote Interior ngles 4 and 9 and Y 5 and Y 6 and Z 7 and Z 8 and Y Z Notice that there are two exterior angles at each vertex and that those two exterior angles have the same remote interior angles. lso observe that an exterior angle is never a vertical angle to an angle of the triangle. Real World Example Design Link In the music stand, name the remote interior angles with respect to. ngle forms a linear pair with. Therefore, and 4 are remote interior angles with respect to. Your Turn 4 5 a. In the figure above, and are remote interior angles with respect to what angle? 8 hapter 7 Triangle Inequalities

2 You can investigate the relationships among the interior and exterior angles of a triangle. Materials: straightedge protractor Step Step Step Step 4 Use a straightedge to draw and label RPN. Extend side R N through K to form the exterior angle 4. Measure the angles of the triangle and the exterior angle. Find m m. Make a table like the one below to record the angle measures. m m m m m R P 4 N K Try These. Draw other triangles and collect the same data. Record the data in your table.. Do you see any patterns in your data? Make a conjecture that describes what you see. The relationship you investigated in the activity above suggests the following theorem. Theorem 7 Exterior ngle Theorem Words: Model: Symbols: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Y 4 Z m 4 m m Lesson 7 Exterior ngle Theorem 8

3 Examples lgebra Link If m 8 and m 4 4, what is m 5? 5 6 Examples 4 m 4 m m 5 Exterior ngle Theorem 4 8 m 5 Replace m 4 with 4 and m with m 5 8 Subtract 8 from each side. 96 m 5 Simplify. lgebra Review Solving Multi-Step Equations, p. 7 If m x 7, m x, and m 6 0, find the value of x. m 6 m m Exterior ngle Theorem 0 (x 7) x Replace m 6 with 0, m with x + 7, and 0 x 7 m with x. 0 7 x 7 7 Subtract 7 from each side. 84 x Simplify. 8 4 x Divide each side by. 8 x Simplify. Your Turn Refer to the figure above. b. What is m if m 46 and m 5 96? c. If m x, m x 4, and m 6 98, find the value of x. Then find m. There are two other theorems that relate to the Exterior ngle Theorem. In the triangle at the right, QRS is an exterior angle, and S and T are its remote interior angles. The Exterior ngle Theorem states that m QRS m S m T. Q 0 R S T 84 hapter 7 Triangle Inequalities In RST, you can see that the measure of QRS is greater than the measures of both S and T, because 0 40 and This suggests Theorem 7 4.

4 Theorem 7 4 Exterior ngle Inequality Theorem Words: Model: The measure of an exterior angle of a triangle is greater than the measure of either of its two remote interior angles. Y 4 Z Symbols: m 4 m m 4 m Example 4 Name two angles in ML that have measures less than 90. ML is a 90 exterior angle. M and are its remote interior angles. By Theorem 7 4, m ML m and m ML m. Therefore, and have measures less than 90. Your Turn d. Name two angles in VW that have measures less than 74. L 74 M V W The results of Example 4 suggest the following theorem about the angles of a right triangle. Theorem 7 5 If a triangle has one right angle, then the other two angles must be acute. heck for Understanding ommunicating Mathematics. Draw a triangle and extend all of the sides. Identify an exterior angle at each of the vertices. exterior angle remote interior angle. Trace B on a blank piece of paper and cut out the triangle. Tear off corners with and, and use the pieces to show that the Exterior ngle Theorem is true. Explain. B Lesson 7 Exterior ngle Theorem 85

5 . Maurice says that the two exterior angles at the same vertex of a triangle are always congruent. Juan says it is impossible for the angles to be congruent. Who is correct? Explain your reasoning. Guided Practice Example Example Example Example 4 Example 4 4. Name two remote interior angles with respect to KL. 5. If m 65 and m 5 4, what is m? 6. If m x 6, m x, and m 4 7, find the value of x. 7. Replace with,, or to make a true sentence. m m 8. Interior Design Refer to the floor tile at the right. a. Is an exterior angle of B? Explain. b. Which angle must have a measure greater than 5? 5 4 L K B Exercises 4 7 B Exercises Practice Homework Help For Exercises 9 7, 4 See Examples Extra Practice See page 78. Name the following angles. 9. an exterior angle of SET 0. an interior angle of ST. a remote interior angle of TE with respect to JET Find the measure of each angle.. 4. J S 4 U T J 5 86 L K T J 5 4 B E 5 S H T M 5. Find the value of x. 6. Find m. 7. Find m Y. Y x 0 T (x ) Exercises hapter 7 Triangle Inequalities

6 with,, or to make a true sentence. Replace each 9. G D 8. J H I 5 M 8 L K 7 6 E m m m G 75. Write a relationship for m B and m D using,, or.. Find the value of x. m 6 m 7 m 8 B 4 (x 8) D Exercises pplications and Problem Solving. Botany The feather-shaped leaf at the right is called a pinnatifid. In the figure, does x y? Explain. 78 y 7 x 8 Two-way mirror Movie camera x 60 Mirror Background scene projected on screen Projector 4. Entertainment For the 978 movie Superman, the flying scenes were filmed using angled mirrors as shown in the diagram at the left. Find x, the measure of the angle made by the two-way mirror and the camera projection. 5. ritical Thinking If B BD, find the measure of. D 7 4 B Mixed Review 6. Transportation orning, Red Bluff, and Redding are alifornia cities that lie on the same line, with Red Bluff in the center. Write a sentence using,, or to compare the distance from orning to Redding R and the distance from orning to Red Bluff B. (Lesson 7 ) 7. Determine whether YZ with vertices (, 6), Y(6, 4), and Z(0, ) is an isosceles triangle. Explain. (Lesson 6 7) Find the perimeter and area of each rectangle. (Lesson 6) 8. feet, w 6 feet Standardized Test Practice 9..5 meters, w. meters 0. Multiple hoice What is the solution to 60 9r 87? (lgebra Review) 9 r B 9 r 9 r D r 9 Lesson 7 Exterior ngle Theorem 87

7 hapter 7 Materials unlined paper ruler protractor uncooked linguine Investigation Measures of ngles and Sides in Triangles What happens to the length of a side of a triangle as you increase the measure of the angle opposite that side? How does this change in angle measure affect the triangle? In this investigation, you will use linguine noodles to explore this relationship. Investigate. Use uncooked linguine to investigate three different triangles. First, break a piece of linguine into two -inch lengths. a. Using a protractor as a guide, place the two -inch pieces of linguine together to form a 0 angle. Break a third piece of linguine so its length forms a triangle with the first two pieces. Trace around the triangle and label it Triangle. Measure and record the length of the third side of the triangle. Hint: Use small pieces of modeling clay or tape to hold the linguine pieces together. in. 0 in. Triangle b. Using a protractor, place the two -inch pieces of linguine together to form a 60 angle. Break another piece of linguine and use it to form a triangle with the first two pieces. Trace around the triangle and label it Triangle. Measure and record the length of the third side of your triangle. c. Using a protractor, place the two -inch pieces of linguine together to form a 90 angle. Break another piece of linguine and use it to form a triangle with the first two pieces. Trace around the triangle and label it Triangle. Measure and record the length of the third side of the triangle. d. s the angle opposite the third side of the triangle increases, what happens to the measure of the third side? 88 hapter 7 Triangle Inequalities

8 . Break four pieces of linguine so that you have the following lengths: inches, inches, 4 inches, and 5 inches. a. Use a protractor to form a 40 angle between the -inch piece and the in. -inch piece as shown at the right. Break a third piece of linguine to form 40 a triangle. Trace around the triangle in. and label it Triangle 4. Record the Triangle 4 measure of angle shown in the figure. b. In the linguine triangle from Step a, replace the -inch piece with the 4-inch piece. Keep the angle measure between the pieces 40. Break a third piece of linguine to form a triangle. Trace around the triangle and label it Triangle 5. Record the measure of angle. c. In the linguine triangle from Step b, replace the 4-inch piece with the 5-inch piece. Keep the angle measure between the pieces 40. Break a third piece of linguine to form a triangle. Trace around the triangle and label it Triangle 6. Record the measure of angle. d. In the three triangles that you formed, each contained a 40 angle. One side remained inches long, but the other side adjacent to the 40 angle increased from to 4 to 5 inches. s that side increased in length, what happened to the measure of angle? In this extension, you will further investigate the relationship between the measures of the sides and angles in triangles. Use linguine, geometry drawing software, or a graphing calculator to investigate these questions.. What happens to the length of the third side of a triangle as the angle between the other two sides ranges from 90 to 50?. What happens to the measure of an angle of a triangle as you increase the length of the side opposite that angle? Presenting Your onclusions Here are some ideas to help you present your conclusions to the class. Make a display or poster of your findings in this investigation. Write a description of the steps to follow to complete this investigation using geometry drawing software or a graphing calculator. Investigation For more information on triangle inequalities, visit: hapter 7 Investigation Linguine Triangles Hold the Sauce! 89

9 7 Inequalities Within a Triangle What You ll Learn You ll learn to identify the relationships between the sides and angles of a triangle. Why It s Important Surveying Triangle relationships are important in undersea surveying. See Example. Florists often use triangles as guides in their flower arrangements. There are special relationships between the side measures and angle measures of each triangle. You will discover these relationships in the following activity. Suppose in triangle B, the inequality > B holds true. Is there a similar relationship between the angles B and, which are across from those sides? B Graphing alculator Tutorial See pp Step Use the Triangle tool on F to draw and label B. Step Select Measure from the F5 menu. Then use the Distance & Length tool and the ngle tool on F6 to display the measures of the sides and angles of B. Try These. Refer to the triangle drawn using the steps above. a. What is the measure of the largest angle in your triangle? b. What is the measure of the side opposite the largest angle? c. What is the measure of the smallest angle in your triangle? d. What is the measure of the side opposite the smallest angle?. Drag vertex to a different location. a. What are the lengths of the longest and shortest sides of the new triangle? b. What can you conclude about the measures of the angles of a triangle and the measures of the sides opposite these angles?. Use the Perpendicular Bisector tool on F to draw the perpendicular bisector of side B. Drag vertex very close to the perpendicular bisector. What do you observe about the measures of the sides and angles? 90 hapter 7 Triangle Inequalities