Final answer:
Vertical angles are congruent because they are opposite angles formed when two lines intersect, and their measures are equal.
Step-by-step explanation:
The student's question addresses the concept of proving that vertical angles are congruent, which is a fundamental principle in geometry. A vertical angle is formed when two lines intersect, and the angles opposite each other at the intersection are called vertical angles. These angles are always congruent, which means they have the same measure.
To prove this, one would typically use the fact that the angles around a point sum up to 360 degrees. If two lines intersect, they form two pairs of vertical angles.
By adding the adjacent angles formed by the intersecting lines, which should equal 180 degrees for a straight line, it can be deduced that the non-adjacent, vertical angles are equal.
For example, if two lines intersect and form angles A, B, C, and D, where A and C are vertical angles and B and D are vertical angles, we can prove A equals C and B equals D.
If A plus B equals 180 degrees and C plus D equals 180 degrees (because these pairs form a straight line) and B equals D (as they are the same straight line angle), then A must equal C.