Geometry: Understanding Corresponding Angles in Parallel Lines and Transversals

Why are the measures of corresponding angles equal when parallel lines are cut by a transversal?

Option 1: because parallel lines always have identical slopes, so a transversal will always intersect parallel lines in the same way

Option 2: because transversals always have identical slopes, so a transversal will always intersect parallel lines in the same way

Option 3: because parallel lines always have identical y-intercepts, so a transversal will always intersect parallel lines in the same way

Option 4: because transversals always have identical y-intercepts, so a transversal will always intersect parallel lines in the same way

Final answer:

The measure of corresponding angles is equal when parallel lines are cut by a transversal due to the Corresponding Angles Theorem and the congruence of alternate angles.

Explanation:

In geometry, when parallel lines are cut by a transversal, the measures of corresponding angles are equal. This is known as the Corresponding Angles Theorem. The reason behind this is that when two parallel lines are cut by a transversal, alternate angles are congruent, which creates a pattern of equal corresponding angles.

For example, if we have two parallel lines, AB and CD, and a transversal line EF, the corresponding angles would be angle AEF and angle CEF. Since AB and CD are parallel, the angle AEF and angle ADE (alternate angles) would be congruent. Similarly, angle CEF and angle CDE (alternate angles) would also be congruent. Therefore, angle AEF and angle CEF are equal.

To summarize, the measure of corresponding angles is equal when parallel lines are cut by a transversal because of the Corresponding Angles Theorem and the congruence of alternate angles.