The problem asks to identify specific angles based on their relationship to given angles in a diagram with two lines intersected by a transversal. The relationships to identify are: exterior alternate angles, interior angles on the same side of the transversal, interior alternate angles, corresponding angles, and vertically opposite angles.

1. Problem Description

The problem asks to identify specific angles based on their relationship to given angles in a diagram with two lines intersected by a transversal. The relationships to identify are: exterior alternate angles, interior angles on the same side of the transversal, interior alternate angles, corresponding angles, and vertically opposite angles.

2. Solution Steps

a) An exterior alternate angle to $∠2$ is on the exterior of the other line and on the opposite side of the transversal. Therefore, the exterior alternate angle to $∠2$ is $∠6$.

b) An interior angle on the same side of the transversal as $∠7$ is an interior angle on the same side of the transversal as $∠7$. The interior angles are $∠3, ∠4, ∠5,$ and $∠7$. Angles on the same side of the transversal are $∠3$ and $∠7$, or $∠4$ and $∠5$. Thus, the interior angle on the same side of the transversal as $∠7$ is $∠4$.

c) An interior alternate angle to $∠4$ is on the interior of the other line and on the opposite side of the transversal. Therefore, the interior alternate angle to $∠4$ is $∠3$.

d) A corresponding angle to $∠5$ is on the same side of the transversal and in the same relative position. Therefore, the corresponding angle to $∠5$ is $∠1$.

e) A vertically opposite angle to $∠3$ is the angle directly across from it at the intersection. Therefore, the vertically opposite angle to $∠3$ is $∠4$.

3. Final Answer

a) $∠6$

b) $∠4$

c) $∠3$

d) $∠1$

e) $∠1$

f) $∠4$