Question

given: l || m, ∠1 ≅ ∠7 prove: a || b statements reasons 1. l || m 1. given 2. ∠1 ≅ ∠3 2. corresponding angle postulate 3. ∠1 ≅ ∠7 3. 4. ∠3 ≅ ∠7 4. transitive property of ≅ 5. ∠5 ≅ ∠7 5.

Explanation:

Step1: State given information

Given that $l\parallel m$ and $\angle1\cong\angle7$.

Step2: Use corresponding - angle postulate

Since $l\parallel m$, by the Corresponding Angle Postulate, $\angle1\cong\angle3$.

Step3: Recall given angle - congruence

$\angle1\cong\angle7$ is given.

Step4: Apply transitive property

By the Transitive Property of $\cong$, since $\angle1\cong\angle3$ and $\angle1\cong\angle7$, we have $\angle3\cong\angle7$.

Step5: Identify vertical - angle congruence

$\angle5$ and $\angle7$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent, so $\angle5\cong\angle7$.

Step6: Prove parallel lines

Since $\angle3\cong\angle7$ and $\angle3$ and $\angle7$ are corresponding angles for lines $a$ and $b$ with the transversal $m$, by the Converse of the Corresponding Angles Postulate, $a\parallel b$.

Answer:

1. Given; 5. Vertical Angles Theorem.