Question

what are the missing reasons that would complete the partial proof of the alternate exterior angle theorem (∠1≅∠8)?
statement\treason
a//b\tgiven
∠1≅∠5\t
∠5≅∠8\t
∠1≅∠8\ttransitive property of congruence

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are congruent. $\angle1$ and $\angle3$ are vertical angles, so $\angle1\cong\angle3$. Also, $\angle5$ and $\angle7$ are vertical angles, so $\angle5\cong\angle7$. And $\angle3$ and $\angle7$ are corresponding angles. If $a\parallel b$, corresponding angles are congruent.

Step2: Analyze the given statements

We know that $\angle1\cong\angle5$ is given. Since $\angle1\cong\angle3$ (vertical - angle congruence) and $\angle5\cong\angle7$ (vertical - angle congruence) and $\angle3\cong\angle7$ (corresponding - angle congruence when $a\parallel b$).
For $\angle1\cong\angle8$:

• The reason for $\angle1\cong\angle5$ being given is already stated.
• The reason for $\angle5\cong\angle8$ is vertical - angle congruence (because $\angle5$ and $\angle8$ are vertical angles).
• The reason for $\angle1\cong\angle8$ using the transitive property of congruence is valid as we have $\angle1\cong\angle5$ and $\angle5\cong\angle8$.

Answer:

The reason for $\angle5\cong\angle8$ is "Vertical - Angle Congruence". The reasoning for the overall proof is based on vertical - angle congruence and the transitive property of congruence.