Geometry: Chapter 3 Ch. 3. 4: Prove Lines are Parallel Ch. 3.5 Using Properties of Parallel Lines

Postulate 16: Corresponding Angles Converse If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. Image taken from: Geometry. McDougal Littell: Boston, P. 161.

Ex. 1. Find the value of y that makes a || b. Image taken from: Geometry. McDougal Littell: Boston, P. 162.

Ex. 1 (cont.) Solution: Lines a and b are parallel if the marked alternate exterior angles are congruent. (5y +6) o =121 o 5y= y=115 y = 23

Theorem 3.8: Alternate Interior Angles Converse If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. Image taken from: Geometry. McDougal Littell: Boston, P. 162.

Theorem 3.9: Consecutive Interior Angles Converse If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel. Image taken from: Geometry. McDougal Littell: Boston, P. 162.

Theorem 3.10: Alternate Exterior Angles Converse If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel. Image taken from: Geometry. McDougal Littell: Boston, P. 162.

Example 2: A woman was stenciling this design on her kitchen walls. How can she tell if the top and bottom are parallel? She can measure alternate interior angles or corresponding angles and see if they are congruent. Image taken from: Geometry. McDougal Littell: Boston, P. 162.

Ex. 3: Prove that if  1 and  4 are supplementary, then a||b. Image taken from: Geometry. McDougal Littell: Boston, P. 163.

Ex. 4: In the figure, a || b and  1 is congruent to  3. Prove c || d. Use a paragraph proof.

Theorem 3.11: Transitive Property of Parallel Lines. If two lines are parallel to the same line, then they are parallel to each other.

Theorem 3.12: Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to one another. If m ┴ p and n ┴ p, then m || n. Image taken from: Geometry. McDougal Littell: Boston, P. 192.