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Sec 4.2 & 4.3 Parallel Lines and Transversals

Sec 4.2 & 4.3 Parallel Lines and Transversals. Guiding Questions. When two parallel lines cut by a transversal, 4 things will happen…… Corresponding angles are ______. Alternate interior angles are ____. Alternate exterior angles are _____. Consecutive interior angles are _____.

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Sec 4.2 & 4.3 Parallel Lines and Transversals

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  1. Sec 4.2 & 4.3Parallel Lines and Transversals

  2. Guiding Questions • When two parallel lines cut by a transversal, 4 things will happen…… • Corresponding angles are ______. • Alternate interior angles are ____. • Alternate exterior angles are _____. • Consecutive interior angles are _____.

  3. Parallel Lines and Transversals Transversal Line, ray, or segment that intersects two or more coplanar lines, rays, or segments, each at a different point. transversal

  4. Parallel Lines and Transversals Identify which line(s) are transversals. d c b a Line d is a transversal.

  5. Exterior Interior Exterior Special Angle Relationships

  6. Special Angle Relationships 1 2 3 4 5 6 7 8 Consecutive interior angles: 3 and 5 4 and 6

  7. Special Angle Relationships 1 2 3 4 5 6 7 8 Alternate interior angles: 3 and 6 4 and 5

  8. Special Angle Relationships 1 2 3 4 5 6 7 8 Consecutive Exterior angles: 1 and 7 2 and 8

  9. Special Angle Relationships 1 2 3 4 5 6 7 8 Alternate Exterior angles: 1 and 8 2 and 7

  10. Special Angle Relationships 1 2 3 4 5 6 7 8 1 and 5 2 and 6 Corresponding angles: 3 and 7 4 and 8

  11. b c a 2 1 Corresponding Angles Postulate • If two lines cut by a transversal are parallel, then corresponding angles are congruent. If lines a || b, then 1  2.

  12. c a 3 2 b 1 Alternate Interior Angles Theorem Given: a || b and c is the transversal. Prove: 1  2 a || b, c is transversal. Given Corresponding Angles Post. 1  3 2  3 Vertical Angles THM 1  2 Substitution Prop.

  13. Alternate Interior Angles Theorem • If two lines cut by a transversal are parallel, then alternate interior angles are congruent.

  14. c a 3 2 b 1 Alternate Exterior Angles Theorem Given: a || b and c is the transversal. Prove: 1  3 a || b, c is transversal. Given Corresponding Angles Post. 1  2 Vertical Angles Theorem 2  3 1  3 Substitution Prop.

  15. Alternate Exterior Angles Theorem • If two lines cut by a transversal are parallel, then alternate exterior angles are congruent.

  16. c a 3 2 b 1 Consecutive Interior Angles Theorem Given: a || b and c is the transversal. Prove: 1 + 2 = 180 a || b, c is transversal. Given 1  3 Corr. Angles Post. 2 + 3 = 180 Linear Pair Prop. 2 + 1 = 180 Substitution Prop. 1 + 2 = 180 Commutative Prop.

  17. Consecutive Interior Angles Theorem • If two lines cut by a transversal are parallel, then consecutive interior angles are supplementary.

  18. Special Angle Relationships If m1 = 127, find the measures of the other angles. 127 1 2 3 4 5 6 7 8 m5 = 127 m2 = 180 - 127 = 53 m6 = 53 m3 = 53 m7 = 53 m4 = 127 m8 = 127

  19. Special Angle Relationships If m1 = (3x) and m5 = (4x-24), find the measures of each angle. 1 2 3 4 5 6 3x = 4x – 24 7 8 -4x -4x 3•24 = 72 -x = -24 180 – 72 = 108 x = 24 72 108 m3 = 108 m4 = 72 m1 = m2 = m5 = 108 m7 = 108 m8 = 72 72 m6 =

  20. Special Angle Relationships m1 = (2x+7) and m2 = (5x+5), find the measures of each angle. 1 2 3 4 2x + 7 + 5x + 5 = 180 5 6 7x + 12 = 180 12 7 8 12 7x = 168 2•24+7 = 55 7 7 180 – 55 = 125 x = 24 55 125 m3 = 125 m4 = 55 m1 = m2 = m5 = 125 m7 = 125 m8 = 55 55 m6 =

  21. Same-Side Exterior Angles Theorem Given: a || b and c is the transversal. Prove: 2 + 3 = 180 c a 3 a || b, c is transversal. Given 1  3 Corr. Angles Post. b 1 2 1 + 2 = 180 Linear Pair Prop. 3 + 2 = 180 Substitution Prop. 2 + 3 = 180 Comm. Prop.

  22. Consecutive Exterior Angles Theorem • If two lines cut by a transversal are parallel, then same-side exterior angles are supplementary.

  23. 1 2 Consecutive Interior Angles 4 + 5 =180 4 3 3 + 6 = 180 5 6 8 7 Alternate Interior Angles 4  6 3  5 Corresponding Angles (Congruent) Alternate Exterior Angles 1  7 1  5 2 8 2  6 Consecutive Exterior Angles 3  7 1 + 8 =180 4  8 2 + 7 = 180

  24. Special Angle Relationships If m3 = 45, find the measures of the other angles. 45 135 1 2 45 3 4 135 5 6 135 45 7 8 45 m5 = 135 135 m1 = 180 - 45 = 135 m6 = 45 m2 = 45 m7 = 45 m4 = 135 m8 = 135

  25. Special Angle Relationships If m3 = (2x+4) and m6 = (3x-13), find the measures of each angle. 142 1 2 38 38 3 4 142 3 = 6 (Alt Int Angles Congruent) 142 5 6 38 2x + 4 = 3x – 13 7 8 38 142 -2x -2x 2•17 + 4 = 38 4 = x - 13 180 – 38 = 142 17 = x 142 38 m3 = 38 m4 = 142 m1 = m2 = m5 = 38 m7 = 38 m8 = 142 142 m6 =

  26. Special Angle Relationships m4 = (3x+17) and m6 = (2x+8), find the measures of each angle. 1 2 4 & 6 are Consecutive Int Angles (supplementary). 3 4 3x + 17 + 2x + 8 = 180 5 6 5x + 25 = 180 7 8 25 25 5x = 155 3•31+17 = 110 5 5 180 – 110 = 70 x = 31 110 70 m3 = 70 m4 = 110 m1 = m2 = m5 = 70 m7 = 70 m8 = 110 110 m6 =

  27. Angles, Parallel Lines & Transversal transversal If you can’t remember any of the angles, MEMORIZE this shortcut!!

  28. Parallel lines & Transversal

  29. Perpendicular Transversal Theorem

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