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hsm12gmse_0302.pdf - 3-2 Content Standard Properties of...

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148Chapter 3Parallel and Perpendicular LinesObjectivesTo prove theorems about parallel linesTo use properties of parallel lines to find angle measuresYou see thevertical angles,right? Keeplooking! There areother angle pairs.Properties ofParallel Lines3-2Look at the map of streets inClearwater, Florida. Nicholson Streetand Cedar Street are parallel.Which pairs of angles appear to becongruent?In the Solve It, you identified several pairs of angles that appear congruent. Youalready know the relationship between vertical angles. In this lesson, you will explorethe relationships between the angles you learned about in Lesson 3-1 when they areformed byparallellines and a transversal.Essential UnderstandingThe special angle pairs formed by parallel lines and atransversal are congruent, supplementary, or both.Postulate 3-1Same-Side Interior Angles PostulatePostulateIf a transversal intersectstwo parallel lines, thensame-side interior anglesare supplementary.If . . ./3mThen . . .m/41m/55180m/31m/65180<m5634Palm Bluff St.N. Garden Ave.Cedar St.Nicholson St.12345678910121119PContent StandardG.CO.9Prove theorems about lines and angles.Theorems include: . . . when a transversal crossesparallel lines, alternate interior angles arecongruent . . .MATHEMATICALPRACTICES
Problem 1Got It?Lesson 3-2Properties of Parallel Lines149Identifying Supplementary AnglesThe measure ofH3 is 55. Which angles aresupplementary toH3? How do you know?By definition, a straight angle measures 180.Ifm/a1m/b5180, then/aand/baresupplementary by definition of supplementary angles.1802555125, so any anglex, wherem/x5125, is supplementary to/3.m/45125 by the definition of a straight angle.m/85125 by the Same-Side Interior Angles Postulate.m/65m/8 by the Vertical Angles Theorem, som/65125.m/25m/4 by the Vertical Angles Theorem, som/25125.1.ReasoningIf you know the measure of one of the angles, can you alwaysfind the measures of all 8 angles when two parallel lines are cut by atransversal? Explain.56847123TheoremIf a transversal intersectstwo parallel lines, thencorresponding angles arecongruent.If . . ./3mThen . . ./1>/5/2>/6/3>/7/4>/8<m56871243You can use the Same-Side Interior Angles Postulate to prove other angle relationships.You will prove Theorem 3-2 in Exercise 25.

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Alternate Interior Angles Theorem

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