Parallel Lines and Triangles. Objectives To use parallel lines to prove a theorem about triangles To find measures of angles of triangles

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1 -5 Parallel Lines and Triangles ommon ore State Standards G-O..0 Prove theorems about triangles... measures of interior angles of a triangle sum to 80. MP, MP, MP 6 Objectives To use parallel lines to prove a theorem about triangles To find measures of angles of triangles Draw and cut out a large triangle. What is the sum of the angle measures of the triangle? Explain. Do not use a protractor. (Hint: Tear off and rearrange the three corners of the triangle.) an you use your result to make a conjecture about other triangles? MTHEMTIL PRTIES In the Solve It, you may have discovered that you can rearrange the corners of the triangle to form a straight angle. You can do this for any triangle. Lesson L VocabularyV auxiliary line exterior angle of a polygon remote interior angles Essential Understanding The sum of the angle measures of a triangle is always the same. The Solve It suggests an important theorem about triangles. To prove this theorem, you will need to use parallel lines. Postulate - Parallel Postulate Through a point not on a line, there is one and only one line parallel to the given line. P There is exactly one line through P parallel to /. Lesson -5 Parallel Lines and Triangles 7

2 Theorem - Triangle ngle-sum Theorem The sum of the measures of the angles of a triangle is 80. m + m + m = 80 The proof of the Triangle ngle-sum Theorem requires an auxiliary line. n auxiliary line is a line that you add to a diagram to help explain relationships in proofs. The red line in the diagram below is an auxiliary line. Proof Proof of Theorem -: Triangle ngle-sum Theorem Given: P Prove: m + m + m = 80 R Statements Reasons ) Draw < PR > through, parallel to. ) Parallel Postulate ) P and are supplementary. ) that form a linear pair are suppl. ) m P + m = 80 ) Definition of suppl. 4) m P = m + m 4) ngle ddition Postulate 5) m + m + m = 80 5) Substitution Property 6) and 6) If lines are }, then alternate interior are. 7) m = m and m = m 7) ongruent have equal measure. 8) m + m + m = 80 8) Substitution Property When you know the measures of two angles of a triangle, you can use the Triangle ngle-sum Theorem to find the measure of the third angle. 7 hapter Parallel and Perpendicular Lines

3 Problem Using the Triangle ngle-sum Theorem 4 49 Which variable should you solve for first? From the diagram, you know two angle measures in D. The third angle is labeled x. So use what you know about the angle measures in a triangle to solve for x first. lgebra What are the values of x and y in the diagram at the right? Use the Triangle ngle-sum Theorem to write an equation involving x. Solve for x by simplifying and then subtracting 0 from each side x = x = 80 x = 78 x y 59 D z D and D form a linear pair, so they are supplementary. mjd + mjd = 80 Substitute 78 for m D and y for m D in the above equation. Solve for y by subtracting 78 from each side. x + y = y = 80 y = 0 Got It?. Use the diagram in Problem. What is the value of z? n exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side. For each exterior angle of a triangle, the two nonadjacent interior angles are its remote interior angles. In each triangle below, is an exterior angle and and are its remote interior angles. The theorem below states the relationship between an exterior angle and its two remote interior angles. Theorem - Triangle Exterior ngle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. m = m + m You will prove Theorem - in Exercise. Lesson -5 Parallel Lines and Triangles 7

4 You can use the Triangle Exterior ngle Theorem to find angle measures. Problem Using the Triangle Exterior ngle Theorem What information can you get from the diagram? The diagram shows you which angles are interior or exterior. What is the measure of j? m = Triangle Exterior ngle Theorem m = 98 Simplify What is the measure of j? 4 = 59 + m Triangle Exterior ngle Theorem = m Subtract 59 from each side. 4 Got It?. Two angles of a triangle measure 5. What is the measure of an exterior angle at each vertex of the triangle? Problem pplying the Triangle Theorems How can you apply your skills from Problem here? Look at the diagram. Notice that you have a triangle and information about interior and exterior angles. Multiple hoice When radar tracks an object, the reflection of signals off the ground can result in clutter. lutter causes the receiver to confuse the real object with its reflection, called a ghost. t the right, there is a radar receiver at, an airplane at, and the airplane s ghost at D. What is the value of x? x m + m = m D x + 0 = 80 Substitute. Triangle Exterior ngle Theorem x = 50 Subtract 0 from each side. The value of x is 50. The correct answer is. 80 Got It?. Reasoning In Problem, can you find m without using the Triangle Exterior ngle Theorem? Explain. D 74 hapter Parallel and Perpendicular Lines

5 Lesson heck Do you know HOW? Find the measure of the third angle of a triangle given the measures of two angles.. 4 and and and 0 4. x and 50 In a triangle, j is an exterior angle and j and j are its remote interior angles. Find the missing angle measure. 5. m = 4 and m = m = 70 and m = Do you UNDERSTND? 7. Explain how the Triangle Exterior ngle Theorem makes sense based on the Triangle ngle-sum Theorem. 8. Error nalysis The measures of the interior angles of a triangle are 0, x, and x. Which of the following methods for solving for x is incorrect? Explain.. x + x = 0. 4x = 0 x = 7.5 MTHEMTIL PRTIES x + x + 0 = 80 4x + 0 = 80 4x = 50 x = 7.5 Practice and Problem-Solving Exercises MTHEMTIL PRTIES Practice Find mj. See Problem lgebra Find the value of each variable x 80 x y z y 0 c Use the diagram at the right for Exercises 5 and a. Which of the numbered angles are exterior angles? b. Name the remote interior angles for each exterior angle. c. How are exterior angles 6 and 8 related? 6. a. How many exterior angles are at each vertex of the triangle? b. How many exterior angles does a triangle have in all? See Problem. 4 5 Lesson -5 Parallel Lines and Triangles 75

6 lgebra Find each missing angle measure ramp forms the angles shown at the right. What are the values of a and b? a b 7 See Problem.. lounge chair has different settings that change the angles formed by its parts. Suppose m = 7 and m = 4. Find m. pply lgebra Use the given information to find the unknown angle measures in the triangle.. The ratio of the angle measures of the acute angles in a right triangle is.. The measure of one angle of a triangle is 40. The measures of the other two angles are in a ratio of The measure of one angle of a triangle is 08. The measures of the other two angles are in a ratio of Think bout a Plan The angle measures of RST are represented by x, x + 4, and x - 8. What are the angle measures of RST? How can you use the Triangle ngle-sum Theorem to write an equation? How can you check your answer? 6. Prove the following theorem: The acute angles of Proof a right triangle are complementary. Given: with right angle Prove: and are complementary. 7. Reasoning What is the measure of each angle of an equiangular triangle? Explain. 8. Draw a Diagram Which diagram below correctly represents the following description? Explain your reasoning. Draw any triangle. Label it. Extend two sides of the triangle to form two exterior angles at vertex. I. II. III. 76 hapter Parallel and Perpendicular Lines

7 Find the values of the variables and the measures of the angles. 9. Q 0. (x 4) (8x ) P (x 9) x R (4x 7). e. E F d b c 55 H a G y 54 z D x 5. Prove the Triangle Exterior ngle Theorem (Theorem -). Proof The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. Given: is an exterior angle of the triangle. Prove: m = m + m 4 4. Reasoning Two angles of a triangle measure 64 and 48. What is the measure of the largest exterior angle of the triangle? Explain. 5. lgebra right triangle has exterior angles at each of its acute angles with measures in the ratio 4. Find the measures of the two acute angles of the right triangle. hallenge Probability In Exercises 6 40, you know only the given information about the measures of the angles of a triangle. Find the probability that the triangle is equiangular. 6. Each is a multiple of Each is a multiple of Each is a multiple of Each is a multiple of. 40. One angle is obtuse. 4. In the figure at the right, D # and D bisects. Find m DF. 4. If the remote interior angles of an exterior angle of a triangle are congruent, what can you conclude about the bisector of the exterior angle? Justify your answer. F D (x ) (5x 0) Lesson -5 Parallel Lines and Triangles 77

8 Standardized Test Prep ST/T 4. The measure of one angle of a triangle is 5. The other two angles are congruent. What is the measure of each of the congruent angles? The center of the circle at the right is at the origin. What is the approximate length of its diameter? One statement in a proof is and are supplementary angles. The next statement is m + m = 80. Which is the best justification for the second statement based on the first statement? The sum of the measures of two right angles is O 4 y (, ) x 4 ngles that form a linear pair are supplementary. Definition of supplementary angles The measure of a straight angle is 80. Extended Response 46. is an obtuse triangle with m = and is acute. a. What is m + m? Explain. b. What is the range of whole numbers for m? Explain. c. What is the range of whole numbers for m? Explain. PERFORMNE TSK pply What You ve Learned Look back at the blueprint on page 9 of the plan for a city park. a. hoose three triangles in the blueprint. Use the Triangle ngle-sum Theorem to write an equation relating the measures of the angles of each of your triangles. b. hoose one of the triangles you looked at in part (a). Find the measure of each angle of your triangle. Show all your steps and explain your reasoning. MTHEMTIL PRTIES MP 78 hapter Parallel and Perpendicular Lines