The image shows a diagram of intersecting lines, forming vertical angles. We are given the measure of angle ∠6 = 83° and are asked to determine the measures of angles ∠5, ∠7, and ∠8.

Let's break this down:

1. Vertical Angles Theorem: Vertical angles are always congruent. Therefore:

   • ∠5 and ∠6 are vertical angles, so m∠5 = 83°.

2. Linear Pair: Adjacent angles on a straight line sum to 180°. This means:

   • ∠6 and ∠7 form a linear pair, so $m∠7 = 180^\circ - 83^\circ = 97^\circ$.
   • Similarly, ∠5 and ∠8 form a linear pair, so $m∠8 = 180^\circ - 83^\circ = 97^\circ$.

Correct answers:

• m∠5 = 83° (this one was correct in the image).
• m∠7 = 97° (this should replace 8°).
• m∠8 = 97° (this should replace 7°).

Do you have any questions about how I arrived at these conclusions, or would you like more details?

Here are 5 related questions to explore:

1. What is the definition of vertical angles?
2. How can you identify a linear pair of angles in an intersection?
3. Can two angles formed by intersecting lines ever add up to less than 180°?
4. What are supplementary angles, and how do they relate to a straight line?
5. How do you calculate unknown angles when given one angle in an intersecting lines problem?

Tip: Whenever you're given intersecting lines, use the Vertical Angles Theorem and the concept of supplementary angles to quickly find unknown measures.