Question

∠1≅∠2
prove:

• l||m

complete the two - column proof of the alternate exterior angle theorem.
statements reasons

1. ∠1≅∠2 1.
2. ∠1≅∠3 2.
3. ∠2≅∠3 3.
4. l||m 4.

a vertical angles theorem
b transitive property of congruence
c given
d converse of corresponding angles postulate

Explanation:

Step1: Identify the first - given statement

The first statement $\angle1\cong\angle2$ is given in the problem, so the reason for it is "Given".

Step2: Analyze $\angle1$ and $\angle3$

$\angle1$ and $\angle3$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for $\angle1\cong\angle3$ is "Vertical Angles Theorem".

Step3: Use the transitive property

Since $\angle1\cong\angle2$ and $\angle1\cong\angle3$, by the Transitive Property of Congruence, we can conclude that $\angle2\cong\angle3$.

Step4: Prove the parallel lines

$\angle2$ and $\angle3$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. By the Converse of Corresponding Angles Postulate, since $\angle2\cong\angle3$, we have $l\parallel m$.

Answer:

1. C. Given
2. A. Vertical Angles Theorem
3. B. Transitive Property of Congruence
4. D. Converse of Corresponding Angles Postulate