Question

write an indirect proof of the converse of the corresponding angles theorem following the outline below. given: ∠1≅∠2 prove: ℓ∥n assume ℓ is not parallel to n. then it is possible to construct a line m that is parallel to ℓ and passes through the intersection of lines t and n. since line m and line n intersect and are not parallel to each other, m∠3□(do not include the degree symbol in your answer.)

Explanation:

Step1: Assume the opposite

Assume that $\ell$ is parallel to $n$. But we are given that we can construct a line $m$ parallel to $\ell$ passing through the intersection of $\ell$ and $n$. If $\ell$ were parallel to $n$, we couldn't construct such a non - parallel relationship as described in the problem setup.

Step2: Use corresponding angles property

If $\ell$ and $n$ were parallel, and we have $\angle1\cong\angle2$ (given). When we construct line $m$ parallel to $\ell$ through the intersection of $\ell$ and $n$, we get a contradiction to the properties of parallel lines and transversals. Since we can construct $m$ parallel to $\ell$ and $m$ intersects $n$, $\ell$ and $n$ cannot be parallel.

Step3: Prove the converse of corresponding angles theorem

The converse of the corresponding angles theorem states that if corresponding angles are congruent, the lines are parallel. Here, we have shown that if we assume the lines are parallel based on the congruence of $\angle1$ and $\angle2$, it leads to a contradiction with the given construction of line $m$. So, if $\angle1\cong\angle2$, then $\ell$ must be parallel to $n$.

Answer:

The proof shows that if $\angle1\cong\angle2$, then $\ell$ is parallel to $n$ by contradiction.