3.2. Parallel Lines and Transversals

Transcription

1 . Parallel Lines and Transversals Essential Question When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? Exploring Parallel Lines Work with a partner. Use dynamic geometry software to draw two parallel lines. Draw a third line that intersects both parallel lines. Find the measures of the eight angles that are formed. What can you conclude? E D B F ATTENDING TO PRECISION To be proficient in math, you need to communicate precisely with others. 0 0 Writing Conjectures Work with a partner. Use the results of Exploration to write conjectures about the following pairs of angles formed by two parallel lines and a transversal. A C a. corresponding angles b. alternate interior angles c. alternate exterior angles d. consecutive interior angles Communicate Your Answer. When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent?. In Exploration, m = 0. Find the other angle measures. Section. Parallel Lines and Transversals

2 . Lesson What You Will Learn Core Vocabulary Previous corresponding angles parallel lines supplementary angles vertical angles t p q Use properties of parallel lines. Prove theorems about parallel lines. Solve real-life problems. Using Properties of Parallel Lines Theorems Theorem. Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Examples In the diagram at the left, and. Proof Ex., p. 0 Theorem. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Examples In the diagram at the left, and. Proof Example, p. Theorem. Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Examples In the diagram at the left, and. Proof Ex., p. ANOTHER WAY There are many ways to solve Example. Another way is to use the Corresponding Angles Theorem to find m and then use the Vertical Angles Congruence Theorem (Theorem.) to find m and m. Theorem. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Examples In the diagram at the left, and are supplementary, and and are supplementary. Proof Ex., p. Identifying Angles The measures of three of the numbered angles are 0. Identify the angles. Explain your reasoning. 0º By the Alternate Exterior Angles Theorem, m = 0. and are vertical angles. Using the Vertical Angles Congruence Theorem (Theorem.), m = 0. and are alternate interior angles. By the Alternate Interior Angles Theorem, = 0. Chapter Parallel and Perpendicular Lines So, the three angles that each have a measure of 0 are,, and.

3 Using Properties of Parallel Lines Find the value of x. (x + ) a b Check + (x + ) = 0 + (0 + ) =? 0 0 = 0 By the Vertical Angles Congruence Theorem (Theorem.), m =. Lines a and b are parallel, so you can use the theorems about parallel lines. m + (x + ) = 0 Consecutive Interior Angles Theorem + (x + ) = 0 Substitute for m. x + 0 = 0 x = 0 So, the value of x is 0. Combine like terms. Subtract 0 from each side. Using Properties of Parallel Lines Find the value of x. c (x + 9) d Check = (x + 9) =? () + 9 = By the Linear Pair Postulate (Postulate.), m = 0 =. Lines c and d are parallel, so you can use the theorems about parallel lines. m = (x + 9) Alternate Exterior Angles Theorem = (x + 9) Substitute for m. = x Subtract 9 from each side. = x Divide each side by. So, the value of x is. Monitoring Progress Use the diagram. Help in English and Spanish at BigIdeasMath.com. Given m = 0, find m, m, and m. Tell which theorem you use in each case.. Given m = and m = (x + ), what is the value of x? Show your steps. Section. Parallel Lines and Transversals

4 Proving Theorems about Parallel Lines Proving the Alternate Interior Angles Theorem Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. STUDY TIP Before you write a proof, identify the Given and Prove statements for the situation described or for any diagram you draw. Draw a diagram. Label a pair of alternate interior angles as and. You are looking for an angle that is related to both and. Notice that one angle is a vertical angle with and a corresponding angle with. Label it. Given p q t p q Prove STATEMENTS REASONS. p q. Given.. Corresponding Angles Theorem.. Vertical Angles Congruence Theorem (Theorem.).. Transitive Property of Congruence (Theorem.) Monitoring Progress Help in English and Spanish at BigIdeasMath.com. In the proof in Example, if you use the third statement before the second statement, could you still prove the theorem? Explain. Solving Real-Life Problems Solving a Real-life Problem When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, m = 0. What is m? How do you know? Because the Sun s rays are parallel, and are alternate interior angles. By the Alternate Interior Angles Theorem,. So, by the definition of congruent angles, m = m = 0. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? In Example, yellow light leaves a drop at an angle of m =. What is m? How do you know? Chapter Parallel and Perpendicular Lines

5 . Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check. WRITING How are the Alternate Interior Angles Theorem (Theorem.) and the Alternate Exterior Angles Theorem (Theorem.) alike? How are they different?. WHICH ONE DOESN T BELONG? Which pair of angle measures does not belong with the other three? Explain. m and m m and m m and m m and m Monitoring Progress and Modeling with Mathematics In Exercises, find m and m. Tell which theorem you use in each case. (See Example.) (x + ) In Exercises and, find m, m, and m. Explain your reasoning.. 0 In Exercises 0, find the value of x. Show your steps. (See Examples and.). x. (x + ). 9.. ERROR ANALYSIS Describe and correct the error in the student s reasoning. º (x )º by the Corresponding Angles Theorem (Theorem.). Section. Parallel Lines and Transversals

6 A. HOW DO YOU SEE IT? D a. b. 9. CRITICAL THINKING Is it possible for consecutive B Use the diagram. interior angles to be congruent? Explain. C and Name two pairs of congruent angles when AD are parallel. Explain your reasoning. BC Name two pairs of supplementary angles when AB are parallel. Explain your reasoning. and DC PROVING A THEOREM In Exercises and, prove the theorem. (See Example.). Alternate Exterior Angles Theorem (Thm..) 0. THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible that a transversal intersects two parallel lines? Explain your reasoning. MATHEMATICAL CONNECTIONS In Exercises and, write and solve a system of linear equations to find the values of x and y.. (x 0).. Consecutive Interior Angles Theorem (Thm..) y. PROBLEM SOLVING A group of campers tie up their food between two parallel trees, as shown. The rope is pulled taut, forming a straight line. Find m. Explain your reasoning. (See Example.) x y (x + ) (y + ) x. MAKING AN ARGUMENT During a game of pool, your friend claims to be able to make the shot shown in the diagram by hitting the cue ball so that m =. Is your friend correct? Explain your reasoning.. DRAWING CONCLUSIONS You are designing a box like the one shown. A B C. REASONING In the diagram, and SE bisects RSF. Find m. Explain your reasoning. E a. The measure of is 0. Find m and m. F b. Explain why ABC is a straight angle. c. If m is 0, will ABC still be a straight angle? Will the opening of the box be more steep or less steep? Explain. Maintaining Mathematical Proficiency T S R Reviewing what you learned in previous grades and lessons Write the converse of the conditional statement. Decide whether it is true or false. (Section.). If two angles are vertical angles, then they are congruent.. If you go to the zoo, then you will see a tiger.. If two angles form a linear pair, then they are supplementary.. If it is warm outside, then we will go to the park. Chapter hs_geo_pe_00.indd Parallel and Perpendicular Lines /9/ 9: AM