Two parallel lines are crossed by a transversal. What is the measure of 7?

Answer

The image shows two parallel lines intersected by a transversal. Several angles are labeled with numbers. Angle 3 and the angle labeled 90 degrees are corresponding angles. When parallel lines are intersected by a transversal, corresponding angles are congruent, meaning they have the same measure. Therefore, the measure of angle 3 is 90 degrees. Angles 3 and 7 are consecutive interior angles. Consecutive interior angles formed by a transversal intersecting parallel lines are supplementary, meaning their measures sum to 180 degrees. So, the measure of angle 3 + the measure of angle 7 = 180 degrees. Since the measure of angle 3 is 90 degrees, we have 90 degrees + the measure of angle 7 = 180 degrees. Subtracting 90 degrees from both sides gives the measure of angle 7 = 180 degrees - 90 degrees = 90 degrees. Alternatively, angles 3 and 7 are vertical angles. Vertical angles are congruent. Since angle 3 and angle 7 are vertical angles and angle 3 is corresponding to the 90 degree angle, angle 3 is 90 degrees. Therefore, angle 7 is also 90 degrees because vertical angles are congruent to each other if they are of same measure when formed by intersection of two lines. The measure of angle 2 and the angle labeled 90 degrees are consecutive interior angles. When two parallel lines are intersected by a transversal, the alternate exterior angles are equal. The alternate exterior angle to the 90 degree angle is angle 2. So measure of angle 2 is 90 degrees. Angles 2 and 7 are angles on a straight line that are adjacent to each other, so they form a linear pair. Linear pairs are supplementary, meaning their measures sum to 180 degrees. Therefore, the measure of angle 2 + the measure of angle 7 = 180 degrees. Since the measure of angle 2 is 90 degrees, we have 90 degrees + the measure of angle 7 = 180 degrees. Subtracting 90 degrees from both sides gives the measure of angle 7 = 180 degrees - 90 degrees = 90 degrees. Also, we are given that the angle labeled 90 degrees is a right angle. This angle and angle 7 are vertical angles. Vertical angles are congruent, so the measure of angle 7 is also 90 degrees.