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Understanding Parallel Lines and Transversals in Geometry

Learn how corresponding angles, alternate interior/exterior angles, and same side angles are related when parallel lines are cut by a transversal in geometry. Includes proofs and theorems to deepen your understanding.

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Understanding Parallel Lines and Transversals in Geometry

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  1. Concept 18 Parallel lines and Transversals

  2. Corresponding Angles Postulate If _______ parallel lines are cut by a ________________ then each pair of ________________ angles are ___________. two transversal corresponding congruent

  3. Given: Prove: Given Cooresponding Post. Def. of Congruent Substitution Prop.

  4. Given: Prove: Given Given Cooresponding Post. Vertical Angles Thm. Transitive Prop. Def. of Cong. Segments Substitution Prop.

  5. Given: l || m Prove: ∠3 ≌ ∠7 Given Cooresponding Post. Vertical Angles Thm. Transitive Prop. Alternate Interior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ______________________ angles are _____________. two transversal alternate interior congruent

  6. Given: j || k, , Prove: x = 25 j Given Cooresponding Post. Def. of Congruent Angles Givens Substitution Prop. Distributive Prop. Addition Prop. Division Prop

  7. Given: j || k Prove: ∠1≅∠2 Given j Cooresponding Post. Vertical Angles Thm. Transitive Prop. Alternate Exterior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ___________________ angles are ____________. two transversal congruent alternate exterior

  8. Given: l || m, p || q Prove: ∠1 ≌ ∠3 l Given Alt. Ext. Angles Thm. Given Corresponding Angles Post. Transitive Prop.

  9. Given: j || k Prove: ∠1 and ∠2 are supplementary j Given Cooresponding Post. Def. of Congruent Angles Def. of Linear Pair/Given form a linear pair 3 are supplementary Linear Pair Thm Def. of Supplementary Substitution Prop. are supplementary Def of Supplementary.

  10. Same Side Interior Angle Theorem If _______ parallel lines are cut by a ______________ then each pair of ___________________________ angles are __________________. two transversal same side interior supplementary

  11. Given: p || qProve: x = 7 Given 3 are supplementary Same Side Int. Angles Thm. Def. of Supplementary Angles Vertical Angles Thm. Def. of Congruent Angles Substitution Property Givens Substitution Property 80 Simplify 80 Subtraction Prop. Division Prop

  12. Given: j || k Prove: ∠1& ∠3 are supplementary j Given Corresponding Angles Post. Def. of Congruent Angles Def of Linear Pair/Given 3 are a linear pair Linear Pair Post. 3 are supplementary Definition of Supp. Angles Substitution Prop. 3 are supplementary Def. of Supplementary

  13. Same Side Exterior Angles Theorem If _______ parallel lines are cut by a ________________ then each pair of ___________________________ angles are __________________. two transversal same side exterior supplementary

  14. Given: p || q Prove: x = 23 Given 80 Same Side Ext. Angle Thm Simplify 80 Subtraction Property Division Prop

  15. Given: p || q and p  r Prove: q  r 2 1 Given Corresponding Angles Post. Given Def of perpendicular is a right angle Def of right angle Def. of Congruent Angles Substitution Prop. Def. of Right angle is a right angle Def. of perpendicular

  16. Perpendicular Transversal Theorem If two parallel lines are cut by a transversal and one line is perpendicular to the transversal, then the other line is perpendicular to the transversal.

  17. Given: , Prove: ∠C is a right angle Given Given Perp. Transversal Thm. Def. of perpendicular C is a right angle

  18. Given: , Find the measure of each angle.

  19. Given: , , Prove: Given Alt. Int. Angles Thm. Given Alt. Int. Angles Thm. Transitive Prop. Given Alt. Int. Angles Thm. Transitive Prop.

  20. In the figure, m∠9 = 80 and m∠5 = 68. Find the measure of each angle. 1. ∠12 = 2. ∠1 = 3. ∠4 = 4. ∠3 = 5. ∠7 = 6. ∠16 = 180 – 80 80 = 100 80 80 100 68 180 – 68 68 = 112

  21. 7. In the figure, m11 = 51. Find m15. 9. If m2 = 125, find m3. • 10. Find m4. 8. Find m16.

  22. 11. If m5 = 2x – 10, and m7 = x + 15, find x. • 12. If m4 = 4(y – 25), and m8 = 4y, find y.

  23. If m1 = 9x + 6 and m2 = 2(5x – 3) find x. • 14. m3 = 5y + 14 to find y. 12

  24. Find the value of the variable(s) in each figure. Explain your reasoning. 15. 16. 106

  25. Proving Lines are parallel Concept 19

  26. If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel.

  27. Is it possible to prove that line p and q are parallel? If so explain how. Yes, because 70 =70 and the Corresponding Angles Converse Post.

  28. Given: ∠1 ≌∠2 Prove: l || m 1. 1. 2. 2. 3. 3. 4. 4. Vertical Angles Thm Given Transitive Prop. Corresponding Angles Converse Postulate

  29. If two lines are cut by a transversal and same side interior angles (angles that lie between the two lines and are on the same side of the transversal) are supplementary, then the lines are parallel If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel.

  30. Is it possible to prove that line p and q are parallel? If so explain how. NO, because using vertical angles the 75 would then make a same side interior angle pair with the 115. 115+75 = 190 and Same Side Interior Angles Converse Thmsays they should add to 180.

  31. Given: m∠1= 135, m∠4 = 45 Prove: n || o 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 7. Givens Addition Prop. Vertical Angles Thm Def. of Congruent Angles Substitution Prop. Def of supp. are supp. Same Side Interior Angles Converse Theorem

  32. If two lines are cut by a transversal and same side interior angles (angles that lie between the two lines and are on the same side of the transversal) are supplementary, then the lines are parallel If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate interior angles (angles that lie between the two lines and on opposite sides of the transversal) arecongruent, then the lines are parallel.

  33. Is it possible to prove that line p and q are parallel? If so explain how. Yes, because 115 =115 and the Alternate Interior Angles Converse Thm.

  34. Given: ∠1 ≌∠2 Prove: l || m 1. 1. 2. 2. 3. 3. 4. 4. Vertical Angles Thm Given Transitive Prop. lm Alternate Interior Angles Converse Thm.

  35. If two lines are cut by a transversal and same side interior angles (angles that lie between the two lines and are on the same side of the transversal) are supplementary, then the lines are parallel If two lines are cut by a transversal and corresponding angles (angles that have corresponding positions) are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate interior angles (angles that lie between the two lines and on opposite sides of the transversal) arecongruent, then the lines are parallel. If two lines are cut by a transversal and alternate exterior angles (angles that lie outside the two lines and on opposite sides of the transversal) arecongruent, then the lines are parallel.

  36. Is it possible to prove that line p and q are parallel? If so explain how. Yes, because 75 =75 and the Alternate Exterior Angles Converse Thm.

  37. Given: ∠3 ≌∠2 Prove: l || m 1. 1. 2. 2. 3. 3. 4. 4. Vertical Angles Thm Given Transitive Prop. Alternate Exterior Angles Converse Thm.

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