Opener Alternate Interior Angles Alternate Exterior Angles Same-side Interior Angles Corresponding Angles

Sect. 3.3 Parallel Lines and Transversals. Goal 1 Properties of Parallel Lines Goal 2 Properties of Special Pairs of Angles.

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 (Same-Side) 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

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 Congruent Linear Pair Postulate m 7 = 180° – m 5 = 115° If lines || corresponding angles  m 8 = m 5 = 65° If lines || alternate exterior angles  m 9 = m 7 = 115°

parallel lines to find the value of x. Properties of Special Pairs of Angles Use properties of parallel lines to find the value of x. SOLUTION If || corr. s  m 4 = 125° If ||, Same-side ext ’s supp. m 4 + (x + 15)° = 180° Substitute. 125° + (x + 15)° = 180° Subtract. x = 40°

80° 100° 80° If EH||GI and EG||HI , find m1 = ___________ Properties of Special Pairs of Angles If EH||GI and EG||HI , find   m1 = ___________ m2 = ___________ m3 = ___________ 80° 100° 80°

Solution: x = 116 If the lines are parallel, find the value of x Properties of Special Pairs of Angles If the lines are parallel, find the value of x Solution: x = 116

x + (x - 20) = 2x – 20 because of the AAP. Properties of Special Pairs of Angles If the lines are parallel, find the value of x x + (x - 20) = 2x – 20 because of the AAP. 70° and (2x – 20)° angles are same-side int. angles. So 70 + (2x – 20) = 180, Or 2x + 50 = 180. Solution: x = 65

Given: p || q Prove: m1 + m2 = 180° Properties of Special Pairs of Angles Given: p || q Prove: m1 + m2 = 180°

1 m 2 of a circle 50 Properties of Special Pairs of Angles 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 Properties of Special Pairs of Angles 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 Properties of Special Pairs of Angles 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 ?

Homework 3.3 8-26 even, 33-44