In problem 8, we are given that \angle 3 = \angle 6. We need to find the measures of \angle 1, \angle 3, \angle 5, and \angle 7. The lines a and b are parallel.

Problem 8

We are given that lines a and b are parallel and \angle 3 = \angle 6. We need to find the measures of \angle 1, \angle 3, \angle 5, and \angle 7.

Solution:

Since lines a and b are parallel:

• \angle 3 and \angle 5 are alternate interior angles, so \angle 3 = \angle 5.
• \angle 1 and \angle 3 are supplementary angles because they form a linear pair, so \angle 1 + \angle 3 = 180^{\circ}.
• \angle 5 and \angle 7 are supplementary angles because they form a linear pair, so \angle 5 + \angle 7 = 180^{\circ}.
• \angle 3 and \angle 6 are vertically opposite angles. The problem statement gives \angle 3 = \angle 6, which is consistent with this property.

Since we are not given any specific angle measure, we will express the angles in terms of one another.

• We are given \angle 3 = \angle 6.
• Since \angle 1 and \angle 3 form a linear pair, \angle 1 = 180^{\circ} - \angle 3.
• Since \angle 3 and \angle 5 are alternate interior angles, \angle 5 = \angle 3.
• Since \angle 5 and \angle 7 form a linear pair, \angle 7 = 180^{\circ} - \angle 5. Substituting \angle 5 = \angle 3, we get \angle 7 = 180^{\circ} - \angle 3.

Answers:

• \angle 1 = 180^{\circ} - \angle 3
• \angle 3 = \angle 3 (given)
• \angle 5 = \angle 3
• \angle 7 = 180^{\circ} - \angle 3