3-3 Parallel Lines and Transversals

Section 3.2

P ROPERTIES OF P ARALLEL L INES POSTULATE POSTULATE 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.4 Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.5 Consecutive Interior Angles 5 6 m 5 + m 6 = 180° If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.6 Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.7 Perpendicular Transversal j k If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Proving the Alternate Interior Angles Theorem Prove the Alternate Interior Angles Theorem. S OLUTION GIVEN p || q p || qGiven StatementsReasons PROVE  3 Corresponding Angles Postulate 3  2 Def first -Vertical Angles Theorem 1  2 Transitive property of Congruence

Using Properties of Parallel Lines S OLUTION Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use. Linear Pair Postulate m 7 = 180° – m 5 = 115° Alternate Exterior Angles Theorem m 9 = m 7 = 115° Corresponding Angles Postulate m 8 = m 5 = 65° m 6 = m 5 = 65° Vertical Angles Theorem

Using Properties of Parallel Lines Use properties of parallel lines to find the value of x. S OLUTION Corresponding Angles Postulate m 4 = 125° Linear Pair Postulate m 4 + (x + 15)° = 180° Substitute. 125° + (x + 15)° = 180° P ROPERTIES OF S PECIAL P AIRS OF A NGLES Subtract. x = 40°

Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel. When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that Estimating Earth’s Circumference: History Connection m of a circle

Estimating Earth’s Circumference: History Connection m of a circle Using properties of parallel lines, he knew that m 1 = m 2 He reasoned that m of a circle

The distance from Syene to Alexandria was believed to be 575 miles Estimating Earth’s Circumference: History Connection m of a circle Earth’s circumference 1 50 of a circle 575 miles Earth’s circumference 50(575 miles) Use cross product property 29,000 miles How did Eratosthenes know that m 1 = m 2 ?

Estimating Earth’s Circumference: History Connection How did Eratosthenes know that m 1 = m 2 ? S OLUTION Angles 1 and 2 are alternate interior angles, so 1  2 By the definition of congruent angles, m 1 = m 2 Because the Sun’s rays are parallel,