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P ROPERTIES OF P ARALLEL L INES

P ROPERTIES OF P ARALLEL L INES. 1 2. 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. P ROPERTIES OF P ARALLEL L INES. 3 4.

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P ROPERTIES OF P ARALLEL L INES

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

  2. PROPERTIES OF PARALLEL LINES 3 4 THEOREMS ABOUT PARALLEL LINES THEOREM 3.4Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4

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

  4. PROPERTIES OF PARALLEL LINES 7 8 THEOREMS ABOUT PARALLEL LINES THEOREM 3.6Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8

  5. PROPERTIES OF PARALLEL LINES jk THEOREMS ABOUT PARALLEL LINES THEOREM 3.7Perpendicular Transversal If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.

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

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

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

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

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

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

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

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