Question

which of the following statements will prove that line a is parallel to line b ?

options:

• ∠5 ≅ ∠2
• ∠7 ≅ ∠1
• ∠3 ≅ ∠6
• ∠2 ≅ ∠7

Explanation:

Step1: Recall Parallel Line Theorems

To prove two lines are parallel, we use angle relationships (corresponding, alternate interior, alternate exterior, consecutive interior, etc.). Let's analyze each option:

Step2: Analyze Option 1 ($\angle 5 \cong \angle 2$)

$\angle 5$ and $\angle 2$: $\angle 5$ is at line \( b \), $\angle 2$ at line \( a \). These are not corresponding, alternate interior, or alternate exterior angles. So this doesn't prove parallelism.

Step3: Analyze Option 2 ($\angle 7 \cong \angle 1$)

$\angle 7$ and $\angle 1$: $\angle 7$ is vertical to $\angle 5$, $\angle 1$ is vertical to $\angle 3$. Not a standard parallel - proving angle relation.

Step4: Analyze Option 3 ($\angle 3 \cong \angle 6$)

$\angle 3$ and $\angle 6$: These are alternate interior angles. When a transversal cuts two lines, if alternate interior angles are congruent, the lines are parallel. So if $\angle 3 \cong \angle 6$, by Alternate Interior Angles Theorem, \( a \parallel b \).

Step5: Analyze Option 4 ($\angle 2 \cong \angle 7$)

$\angle 2$ and $\angle 7$: These angles do not form a standard parallel - proving angle relationship (not corresponding, alternate interior, alternate exterior, or consecutive interior).

Answer:

$\boldsymbol{\angle 3 \cong \angle 6}$ (the option with $\angle 3 \cong \angle 6$)