The image shows a problem involving angles formed by two parallel lines intersected by a transversal. The problem asks to identify which angle pairs include the named angles.

Part (a) ∠4:

To determine angle pairs for ∠4, we should identify its relationships with other angles in the diagram. Key angle pair types are:

• Corresponding angles (angles in the same relative position)
• Alternate interior angles (angles on opposite sides of the transversal and inside the parallel lines)
• Alternate exterior angles (angles on opposite sides of the transversal but outside the parallel lines)
• Same-side (consecutive) interior angles (on the same side of the transversal inside the parallel lines)
• Vertical angles (opposite each other at an intersection).

Looking at ∠4:

• Corresponding angle: ∠8 (same relative position)
• Alternate interior angle: ∠6
• Same-side interior angle: ∠5
• Vertical angle: ∠3

Thus, the angle pairs including ∠4 are:

• ∠4 and ∠8 (corresponding)
• ∠4 and ∠6 (alternate interior)
• ∠4 and ∠5 (same-side interior)
• ∠4 and ∠3 (vertical)

Part (b) ∠7:

Similarly, for ∠7:

• Corresponding angle: ∠3
• Alternate exterior angle: ∠1
• Same-side interior angle: ∠6
• Vertical angle: ∠8

The angle pairs including ∠7 are:

• ∠7 and ∠3 (corresponding)
• ∠7 and ∠1 (alternate exterior)
• ∠7 and ∠6 (same-side interior)
• ∠7 and ∠8 (vertical)

Would you like further explanation or details on any of these angle relationships?

Here are 5 related questions for practice:

1. What are the corresponding angles to ∠2?
2. Identify the alternate exterior angle of ∠5.
3. What angle is vertically opposite to ∠6?
4. Are ∠1 and ∠5 alternate interior angles? Why or why not?
5. What are the same-side interior angles with respect to ∠8?

Tip: Always look at how angles relate across the transversal line to find corresponding and alternate pairs.