To solve this problem, let’s analyze the angles in the diagram based on the given information. Here, we are told that the measure of angle 5 ($m \angle 5$) is 82°. The figure shows a pair of parallel lines intersected by transversals, creating multiple angles with specific relationships such as vertical angles, corresponding angles, and alternate interior angles.

Step-by-Step Solution

1. Identify Vertical Angles:

   • Vertical angles are equal. Since $m \angle 5 = 82^\circ$, the opposite angle $m \angle 7$ will also be 82°.
   • Similarly, $m \angle 6 = m \angle 8$.

2. Corresponding Angles:

   • Corresponding angles are equal when two parallel lines are intersected by a transversal.
   • Since $m \angle 5 = 82^\circ$, the corresponding angle on the top parallel line (angle 1) will also be 82°.
   • This applies to all corresponding angle pairs.

3. Alternate Interior Angles:

   • Alternate interior angles are also equal when two parallel lines are intersected by a transversal.
   • Since $m \angle 5 = 82^\circ$, the alternate interior angle $m \angle 3$ will be 82°.

4. Supplementary Angles:

   • Adjacent angles on a straight line are supplementary (they add up to 180°).
   • So, if $m \angle 5 = 82^\circ$, then $m \angle 4 = 180^\circ - 82^\circ = 98^\circ$.
   • We can apply this to find other supplementary pairs.

Based on these relationships, here are the angle measures:

Answers

• $m \angle 1 = 82^\circ$
• $m \angle 2 = 98^\circ$
• $m \angle 3 = 82^\circ$
• $m \angle 4 = 98^\circ$
• $m \angle 6 = 98^\circ$
• $m \angle 7 = 82^\circ$
• $m \angle 8 = 98^\circ$
• $m \angle 9 = 82^\circ$
• $m \angle 10 = 98^\circ$
• $m \angle 11 = 82^\circ$
• $m \angle 12 = 98^\circ$

Would you like a further breakdown of each step?

Related Questions

1. What is the property of corresponding angles when two parallel lines are intersected by a transversal?
2. How do vertical angles relate to each other?
3. How do you find the measure of a supplementary angle?
4. What are alternate interior angles, and why are they equal?
5. Can you apply these angle relationships to more complex diagrams with multiple transversals?

Tip

When working with parallel lines and a transversal, remember that both corresponding and alternate interior angles are equal, which often simplifies the process of finding unknown angles.