Question

dana and joby wrote the following proofs to prove that vertical angles are congruent. which is correct?

Explanation:

To prove vertical angles are congruent, let's assume two lines \( l \) and \( m \) intersect at a point, forming vertical angles \( \angle 1 \) and \( \angle 2 \), and another pair \( \angle 3 \) and \( \angle 4 \).

Step 1: Linear Pair Postulate

A linear pair of angles is supplementary (their sum is \( 180^\circ \)). For \( \angle 1 \) and \( \angle 3 \):
\( \angle 1 + \angle 3 = 180^\circ \) (Linear Pair Postulate)

For \( \angle 2 \) and \( \angle 3 \):
\( \angle 2 + \angle 3 = 180^\circ \) (Linear Pair Postulate)

Step 2: Substitution Property

From Step 1, both \( \angle 1 + \angle 3 \) and \( \angle 2 + \angle 3 \) equal \( 180^\circ \). Thus:
\( \angle 1 + \angle 3 = \angle 2 + \angle 3 \)

Step 3: Subtract \( \angle 3 \) from Both Sides

Subtract \( \angle 3 \) from both sides of the equation:
\( \angle 1 + \angle 3 - \angle 3 = \angle 2 + \angle 3 - \angle 3 \)
Simplifying, we get:
\( \angle 1 = \angle 2 \)

This shows that vertical angles are congruent (equal in measure).

Answer:

Vertical angles are congruent (proven via linear pair postulate and substitution).