Parallel Lines and Transversals PROPERTIES OF PARALLEL LINES
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1 . Parallel Lines and Transversals What you should learn GOAL Prove and use results about arallel lines and transversals. GOAL Use roerties of arallel lines to solve real-life roblems, such as estimating Earth s circumference in Examle 5. Why you should learn it Proerties of arallel lines hel you understand how rainbows are formed, as in Ex. 0. REAL LIFE GOAL PROPERTIES OF PARALLEL LINES In the activity on age 4, you may have discovered the following results. POSTULATE POSTULATE 5 Corresonding Angles Postulate If two arallel lines are cut by a transversal, then the airs of corresonding angles are congruent. You are ased to rove Theorems.5,.6, and.7 in Exercises 7 9. THEOREMS ABOUT ABOUT PARALLEL LINES LINES THEOREM.4 Alternate Interior Angles If two arallel lines are cut by a transversal, then the airs of alternate interior angles are congruent. 4 THEOREM.5 Consecutive Interior Angles If two arallel lines are cut by a transversal, then the airs of consecutive interior angles are sulementary. THEOREM.6 Alternate Exterior Angles If two arallel lines are cut by a transversal, then the airs of alternate exterior angles are congruent. THEOREM.7 Perendicular Transversal If a transversal is erendicular to one of two arallel lines, then it is erendicular to the other. m 5 + m 6 = fi h. Parallel Lines and Transversals 4
2 Study Ti When you rove a theorem, the hyotheses of the theorem becomes the GIVEN, and the conclusion is what you must PROVE. E X A M P L E Proving the Alternate Interior Angles Theorem Prove the Alternate Interior Angles Theorem. GIVEN PROVE Statements Reasons.. Given.. Corresonding Angles Postulate.. Vertical Angles Theorem Transitive Proerty of Congruence E X A M P L E Using Proerties of Parallel Lines Given that m 5 = 65, find each measure. Tell which ostulate or theorem you use. a. m 6 b. m 7 c. m 8 d. m a. m 6 = m 5 = 65 Vertical Angles Theorem b. m 7 = 80 º m 5 = 5 Linear Pair Postulate c. m 8 = m 5 = 65 Corresonding Angles Postulate FOCUS ON CAREERS d. m 9 = m 7 = 5 Alternate Exterior Angles Theorem E X A M P L E Classifying Leaves BOTANY Some lants are classified by the arrangement of the veins in their leaves. In the diagram of the leaf,. What is m? 0 BOTANY Botanists study lants and environmental issues such as conservation, weed control, and re-vegetation. REAL CAREER LINK INTERNET LIFE m + 0 = 80 m = 60 Consecutive Interior Angles Theorem Subtract. 44 Chater Perendicular and Parallel Lines
3 GOAL PROPERTIES OF SPECIAL PAIRS OF ANGLES xy Using Algebra E X A M P L E 4 Use roerties of arallel lines to find the value of x. Using Proerties of Parallel Lines 5 4 (x 5) m 4 = 5 m 4 + (x + 5) = 80 Corresonding Angles Postulate Linear Pair Postulate 5 + (x + 5) = 80 Substitute. x = 40 Subtract. E X A M P L E 5 Estimating Earth s Circumference APPLICATION LINK Visit our Web site for more information about Eratosthenes estimate in Examle 5. INTERNET HISTORY CONNECTION Eratosthenes was a Gree scholar. Over 000 years ago, he estimated Earth s circumference by using the fact that the Sun s rays are arallel. Eratosthenes chose a day when the Sun shone exactly down a vertical well in Syene at noon. On that day, he measured the angle the Sun s rays made with a vertical stic in Alexandria at noon. He discovered that m of a circle. 5 0 By using roerties of arallel lines, he new that m = m. So he reasoned that m of a circle. 5 0 At the time, the distance from Syene to Alexandria was believed to be 575 miles. 575 miles of a circle 5 0 Earth s circumference Earth s circumference 50(575 miles) 9,000 miles How did Eratosthenes now that m = m? center of Earth shadow Use cross roduct roerty. Because the Sun s rays are arallel, l l. Angles and are alternate interior angles, so. By the definition of congruent angles, m = m. stic well Not drawn to scale L L. Parallel Lines and Transversals 45
4 GUIDED PRACTICE Vocabulary Chec Concet Chec Sill Chec. Setch two arallel lines cut by a transversal. Label a air of consecutive interior angles.. In the figure at the right,. How many angle measures must be given in order to find the measure of every angle? Exlain your reasoning. State the ostulate or theorem that ustifies the statement m + m 5 = In the diagram of the feather below, lines and are arallel. What is the value of x? x PRACTICE AND APPLICATIONS Extra Practice to hel you master sills is on USING PARALLEL LINES Find m and m. Exlain your reasoning USING PARALLEL LINES Find the values of x and y. Exlain your reasoning HOMEWORK HELP Examle : Exs. 7 9 Examle : Exs. 8 7 Examle : Exs. 8 7 Examle 4: Exs. 8 6 Examle 5: Ex Chater Perendicular and Parallel Lines
5 7. USING PROPERTIES OF PARALLEL LINES Use the given information to find the measures of the other seven angles in the figure at the right. GIVEN, m = xy USING ALGEBRA Find the value of y y 5 5y 6y 0 xy USING ALGEBRA Find the value of x (x 4) (x 0) (x 9) 5 4. (5x 4) (x 5) 6 7(x 7) HOMEWORK HELP Visit our Web site for hel with roving theorems in Exs INTERNET 7. DEVELOPING PROOF Comlete the roof of the Consecutive Interior Angles Theorem. GIVEN PROVE and are sulementary. Statements Reasons.?. Given..?.?. Definition of congruent angles 4.? 4. Definition of linear air 5. m + m = 80 5.? 6.? 6. Substitution ro. of euality 7. and are sulementary. 7.?. Parallel Lines and Transversals 47
6 Study Ti When you rove a theorem you may use any revious theorem, but you may not use the one you re roving. PROVING THEOREMS.6 AND.7 In Exercises 8 and 9, comlete the roof. 8. To rove the Alternate Exterior Angles Theorem, first show that. Then show that. Finally, show that. GIVEN PROVE 9. To rove the Perendicular Transversal Theorem, show that is a right angle,, is a right angle, and finally that fi r. GIVEN fi, r PROVE fi r r 0. FORMING RAINBOWS When enters a dro of rain, different colors leave the dro at different angles. That s what maes a rainbow. For red light, m = 4. What is m? How do you now? rain shadow Test Prearation. MULTI-STEP PROBLEM You are designing a lunch box lie the one below. A B C a. The measure of is 70. What is the measure of? What is the measure of? b. Writing Exlain why ABC is a straight angle. Challenge EXTRA CHALLENGE. USING PROPERTIES OF PARALLEL LINES Use the given information to find the measures of the other labeled angles in the figure. For each angle, tell which ostulate or theorem you used. Æ, GIVEN PQ Æ RS LM Æ fi NK Æ, m = 48 N P R M L 7 K S 48 Chater Perendicular and Parallel Lines
7 MIXED REVIEW ANGLE MEASURES and are sulementary. Find m. (Review.6). m = m = 7 5. m = 0 6. m = m = 8. m = 8 CONVERSES Write the converse of the statement. (Review. for.4) 9. If the measure of an angle is 9, then the angle is acute. 40. I will go to the ar if you go with me. 4. I will go fishing if I do not have to wor. FINDING ANGLES Comlete the statement, given that DE Æ fi DG Æ and AB fi DC Æ. (Review.6) C G 4. If m =, then m =?. 4. If m 4 = 69, then m =?. 44. If m = 70, then m 4 =?. A E 4 D B QUIZ Self-Test for Lessons.. Comlete the statement. (Lesson.). and? are corresonding angles.. and? are consecutive interior angles.. and? are alternate interior angles. 4. and? are alternate exterior angles PROOF Write a lan for a roof. (Lesson.) GIVEN PROVE and 4 are right angles. 4 Find the value of x. (Lesson.) (x ) 8 (7x 5) 9. FLAG OF PUERTO RICO Setch the flag of Puerto Rico shown at the right. Given that m = 55, determine the measure of. Justify each ste in your argument. (Lesson.). Parallel Lines and Transversals 49
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