Ch. 3-3: Prove that Lines are Parallel Mr. Schaab’s Geometry Class Our Lady of Providence Jr.-Sr. High School

Converse Theorems In section 2 of this chapter, we learned that we could prove two corresponding angles on a transversal are congruent if the two lines being cut by the transversal are parallel. In this section, we will learn the converse: how to prove that two lines cut by a transversal are parallel based on the angles that are formed.

Corresponding Angles Converse If two lines are cut by a transversal so that the pairs of corresponding angles are congruent, then the two lines are parallel. Ex.: if ∠ 3 ≅ ∠ 7, then r || s. 3 7 t r s

Alternate Interior Angles Converse If two lines are cut by a transversal so that the pairs of alternate interior angles are congruent, then the two lines are parallel. Ex.: if ∠ 3 ≅ ∠ 6, then r || s. 3 6 t r s

Alternate Exterior Angles Converse If two lines are cut by a transversal so that the pairs of alternate exterior angles are congruent, then the two lines are parallel. Ex.: if ∠ 2 ≅ ∠ 7, then r || s. 2 7 t r s

Consecutive Interior Angles Converse If two lines are cut by a transversal so that the pairs of consecutive interior angles are supplementary, then the two lines are parallel. Ex.: if ∠ 3 + ∠ 5 = 180°, then r || s. 3 5 t r s

Can you prove that lines a and b are parallel? 1.) 2.) Examples a b Yes, using Alternate Exterior Angles Converse a b 75° 95° No

Can you prove that lines a and b are parallel? 3.) 4.) Examples a b No (The congruent angles are not corresponding.) ab 42° 138° Yes; x must equal 138°, and 42°+138°=180° c d x°x°

Paragraph Proof Another way of writing a proof is called a paragraph proof, in which you simply write out your argument in sentences to explain the logical steps you took to complete the proof. The use of transitional words such as “so,” “then,” or “therefore” are used to transition from statement to statement. They help guide the reader through your paragraph proof.

Paragraph Proof Example: In the figure, r || s and ∠ 2 ≅ ∠ 15. Prove that p || q. s 6 r p q 2 15 It is given that r || s, so by the Corresponding Angles Postulate, ∠ 2 ≅ ∠ 6. It is also given that ∠ 2 ≅ ∠ 15. Then, by the Transitive Property of Angle Congruence, ∠ 6 ≅ ∠ 15. Therefore, by the Alternate Interior Angles Converse, p || q.

Examples (Def. Sheet) YES NO x = 40 x = 109 x = 115 x = 22 x = 5 x = 80

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