Question

PROOF: Complete the two-column proof to prove the perpendicular Transversal Converse.

Given: <1 and <2 are complementary; BC | CD
Prove: BA || CD

Answer 1

To prove that BA is parallel to CD, we can use the fact that angles <1 and <2 are complementary, and BC is perpendicular to CD (BC⊥CD). When two lines are perpendicular, the alternate interior angles are equal. When two lines are cut by a transversal and corresponding angles are equal, the lines are parallel. Therefore, BA is parallel to CD.

Given:

• ∠1 and ∠2 are complementary.
• BC∥CD.

To prove:

BA∥CD.

Proof:

1. Since ∠1 and ∠2 are complementary, we have ∠1+∠2=180° .
2. Consider the transversal BC intersecting parallel lines BC and CD. According to the corresponding angles postulate, ∠1 and ∠BCD are corresponding angles.
3. From step 1, ∠1+∠2=180° . Therefore, ∠BCD+∠2=180°
4. Substituting the value of ∠BCD from step 2 into step 3, we get ∠1+∠2=∠BCD+∠2.
5. Canceling ∠2 from both sides, we have, ∠1=∠BCD.
6. By the converse of the corresponding angles postulate, if two lines are cut by a transversal, and the corresponding angles are congruent, then the lines are parallel.

Therefore, BA∥CD by the converse of the corresponding angles postulate.