Question

Explain how to prove the Vertical Angles Theorem. Make sure you describe the steps to prove the theorem. Do not define vertical angles. Use the diagram below to explain the steps.

Answer 1

To prove the Vertical Angles Theorem, follow these steps: Start with two intersecting lines. Draw line segments to create angles. Use the properties of vertical angles and alternate interior angles to show congruence. Apply the Transitive Property of Congruence. Therefore, vertical angles are congruent.

Step-by-step explanation:

To prove the Vertical Angles Theorem, we need to show that when two lines intersect, the angles opposite each other are congruent. Let's consider the diagram below:

Here are the steps to prove the theorem:

1. Start with two lines, line AB and line CD, that intersect at point E.
2. Draw a line segment from point E to a point F on line AD, creating angle AEF and angle DEF.
3. Draw a line segment from point E to a point G on line BC, creating angle BEG and angle CEG.
4. By the definition of vertical angles and alternate interior angles, angle AEF is congruent to angle CEG and angle BEG is congruent to angle DEF.
5. Therefore, by the Transitive Property of Congruence, angle AEF is congruent to angle DEF.
6. Similarly, angle CEG is congruent to angle BEG.
7. Therefore, by the definition of vertical angles, angle AEF is congruent to angle DEF and angle CEG is congruent to angle BEG.

Thus, we have proven the Vertical Angles Theorem, which states that vertical angles are congruent.