1 2 1 2 PROPERTIES OF PARALLEL LINES POSTULATE POSTULATE 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 1 2

3 4 3 4 PROPERTIES OF PARALLEL LINES 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. 3 4 3 4

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

7 8 7 8 PROPERTIES OF PARALLEL LINES 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. 7 8 7 8

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

1  3 Corresponding Angles Postulate Proving the Alternate Interior Angles Theorem Prove the Alternate Interior Angles Theorem. SOLUTION GIVEN p || q PROVE 1 2 Statements Reasons p || q Given 1 1  3 Corresponding Angles Postulate 2 3 3  2 Vertical Angles Theorem 1  2 Transitive property of Congruence 4

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

parallel lines to find the value of x. PROPERTIES OF SPECIAL PAIRS OF ANGLES Using Properties of Parallel Lines Use properties of parallel lines to find the value of x. SOLUTION Corresponding Angles Postulate m 4 = 125° Linear Pair Postulate m 4 + (x + 15)° = 180° Substitute. 125° + (x + 15)° = 180° Subtract. x = 40°

Estimating Earth’s Circumference: History Connection 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 m 2 1 50 of a circle

1 m 2 of a circle 50 m 1 = m 2 1 m 1 of a circle 50 Estimating Earth’s Circumference: History Connection m 2 1 50 of a circle Using properties of parallel lines, he knew that m 1 = m 2 He reasoned that m 1 1 50 of a circle

1 m 1 of a circle 50 1 of a circle 50 50(575 miles) 29,000 miles Estimating Earth’s Circumference: History Connection m 1 1 50 of a circle The distance from Syene to Alexandria was believed to be 575 miles 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 ?

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