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Geometry Angles formed by Parallel Lines and Transversals

Geometry Angles formed by Parallel Lines and Transversals. Warm Up. Give an example of each angle pair. 1) Alternate interior angles 2) Alternate exterior angles 3)Same side interior angles. 1 2. 3 4. 5 6. 7 8. Parallel, perpendicular and skew lines.

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Geometry Angles formed by Parallel Lines and Transversals

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  1. GeometryAngles formed by Parallel Lines and Transversals CONFIDENTIAL

  2. Warm Up Give an example of each angle pair. 1) Alternate interior angles 2) Alternate exterior angles 3)Same side interior angles CONFIDENTIAL

  3. 1 2 3 4 5 6 7 8 Parallel, perpendicular and skew lines When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed. There are several special pairs of angles formed from this figure. Vertical pairs: Angles1 and 4  Angles2 and 3  Angles5 and 8  Angles6 and 7 CONFIDENTIAL

  4. 1 2 3 4 5 6 7 8 Supplementary pairs: Angles1 and 2 Angles2 and 4 Angles3 and 4 Angles1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7 Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are three other special pairs of angles. These pairs are congruent pairs. CONFIDENTIAL

  5. 1 2 3 4 t 5 6 7 8 p q 1 3 2 4 5 7 6 8 Corresponding angle postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. CONFIDENTIAL

  6. A) m( ABC) C 800 x0 A B Using the Corresponding angle postulate Find each angle measure. corresponding angles x = 80 m( ABC) = 800 CONFIDENTIAL

  7. (2x-45)0 D B) m( DEF) (x+30)0 E F corresponding angles (2x-45)0 = (x+30)0 subtract x from both sides x – 45 = 30 add 45 to both sides x = 75 m( DEF) = (x+30)0 = (75+30)0 = 1050 CONFIDENTIAL

  8. Now you try! 1) m( DEF) R x0 S 1180 Q CONFIDENTIAL

  9. Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles postulate is given as a postulate, it can be used to prove the next three theorems. CONFIDENTIAL

  10. 1 2 1 3 2 4 4 3 Alternate interior angles theorem Theorem If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent. Hypothesis Conclusion CONFIDENTIAL

  11. 5 7 6 8 5 6 8 7 Alternate exterior angles theorem Theorem If two parallel lines are cut by a transversal, then the twopairs of Alternate exterior angles are congruent. Hypothesis Conclusion CONFIDENTIAL

  12. 1 2 4 3 m 1 + m 4 =1800 m 2 + m 3 =1800 Same-side interior angles theorem Theorem If two parallel lines are cut by a transversal, then the twopairs of Same-side interior angles are supplementary. Hypothesis Conclusion CONFIDENTIAL

  13. 1 2 l 3 Given: l || m Prove: 2 3 m l || m 1 3 2 3 Corresponding angles Given 2 1 Vertically opposite angles Alternate interior angles theorem Proof: CONFIDENTIAL

  14. A) m( EDF) A C 1250 B D x0 F E Finding Angle measures Find each angle measure. x = 1250 m( DEF) = 1250 Alternate exterior angles theorem CONFIDENTIAL

  15. R T 13x0 23x0 B) m( TUS) S U Same-side interior angles theorem 13x0 + 23x0 = 1800 Combine like terms 36x = 180 divide both sides by 36 x = 5 m( TUS) = 23(5)0 Substitute 5 for x = 1150 CONFIDENTIAL

  16. C B A (2x+10)0 (3x-5)0 D E Now you try! 2) Find each angle measure. CONFIDENTIAL

  17. (25x+5y)0 1250 (25x+4y)0 1200 A treble string of grand piano are parallel. Viewed from above, the bass strings form transversals to the treble string. Find x and y in the diagram. By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250 By the Corresponding Angles Postulates, (25x+4y)0 = 1200 (25x+5y)0 = 1250 - (25x+4y)0 = 1200 y = 5 25x+5(5) = 125 x = 4, y = 5 Subtract the second equation from the first equation Substitute 5 for y in 25x +5y = 125. Simplify and solve for x. CONFIDENTIAL

  18. (25x+5y)0 1250 (25x+4y)0 1200 Now you try! 3) Find the measure of the acute angles in the diagram. CONFIDENTIAL

  19. 1270 1) m( JKL) 2) m( BEF) L x0 G K J A A (7x-14)0 B C (4x+19)0 E D F H Assessment Find each angle measure: CONFIDENTIAL

  20. 3) m( 1) 4) m( CBY) 1 A X D B Y (3x+9)0 E 6x0 C Z Find each angle measure: CONFIDENTIAL

  21. M 5) m( KLM) 6) m( VYX) K Y0 L 1150 X 4a0 V Y (2a+50)0 Z W Find each angle measure: CONFIDENTIAL

  22. 4 1 3 2 5 7) m 1 = (7x+15)0 , m 2 = (10x-9)0 8) m 3 = (23x+15)0 , m 4 = (14x+21)0 State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures: CONFIDENTIAL

  23. 1 2 3 4 5 6 7 8 Let’s review Parallel, perpendicular and skew lines When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed. There are several special pairs of angles formed from this figure. Angles1 and 4  Angles2 and 3  Angles5 and 8  Angles6 and 7 Vertical pairs: CONFIDENTIAL

  24. 1 2 3 4 5 6 7 8 Supplementary pairs: Angles1 and 2 Angles2 and 4 Angles3 and 4 Angles1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7 Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are three other special pairs of angles. These pairs are congruent pairs. CONFIDENTIAL

  25. 1 2 3 4 t 5 6 7 8 p q 1 3 2 4 5 7 6 8 Corresponding angle postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. CONFIDENTIAL

  26. A) m( ABC) C 800 x0 A B Using the Corresponding angle postulate Find each angle measure. corresponding angles x = 80 m( ABC) = 800 CONFIDENTIAL

  27. (2x-45)0 D B) m( DEF) (x+30)0 E F corresponding angles (2x-45)0 = (x+30)0 subtract x from both sides x – 45 = 30 add 45 to both sides x = 75 m( DEF) = (x+30)0 = (75+30)0 = 1050 CONFIDENTIAL

  28. 1 2 1 3 2 4 4 3 Alternate interior angles theorem Theorem If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent. Hypothesis Conclusion CONFIDENTIAL

  29. 5 7 6 8 5 6 8 7 Alternate exterior angles theorem Theorem If two parallel lines are cut by a transversal, then the twopairs of Alternate exterior angles are congruent. Hypothesis Conclusion CONFIDENTIAL

  30. 1 2 4 3 m 1 + m 4 =1800 m 2 + m 3 =1800 Same-side interior angles theorem Theorem If two parallel lines are cut by a transversal, then the twopairs of Same-side interior angles are supplementary. Hypothesis Conclusion CONFIDENTIAL

  31. 1 2 l 3 Given: l || m Prove: 2 3 m l || m 1 3 2 3 Corresponding angles Given 2 1 Vertically opposite angles Alternate interior angles theorem Proof: CONFIDENTIAL

  32. You did a great job today! CONFIDENTIAL

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