Question

given: ∠2 and ∠4 are vertical angles. prove: ∠2 ≅ ∠4
(there is a diagram with lines m and n intersecting at point p, forming angles 1,2,3,4)
statements reasons

1. ∠2 and ∠4 are vert. angles 1. given
2. ∠2 and ∠3 are a linear pair 2. def. of a linear pair
3. ∠3 and ∠4 are a linear pair 3. def. of a linear pair
4. m∠2 + m∠3 = 180 4. linear pairs are supp.
5. m∠3 + m∠4 = 180 5. linear pairs are supp.

(then there are some statement and reason boxes to fill, related to m∠2 = m∠4, ∠2 ≅ ∠4, and angle sum equalities)

Explanation:

Step1: Identify Given Information

We know that \( \angle 2 \) and \( \angle 4 \) are vertical angles (given). Also, \( \angle 2 \) and \( \angle 3 \) form a linear pair, and \( \angle 3 \) and \( \angle 4 \) form a linear pair. By the definition of a linear pair, linear pairs are supplementary, so \( m\angle 2 + m\angle 3 = 180^\circ \) and \( m\angle 3 + m\angle 4 = 180^\circ \).

Step2: Set Up Equations

From the linear pair property:
\( m\angle 2 + m\angle 3 = 180^\circ \) (Equation 1)
\( m\angle 3 + m\angle 4 = 180^\circ \) (Equation 2)

Step3: Substitute and Solve for Angles

Since both equations equal \( 180^\circ \), we can set them equal to each other:
\( m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4 \)
Subtract \( m\angle 3 \) from both sides:
\( m\angle 2 = m\angle 4 \)

Step4: Conclude Congruence

If the measures of two angles are equal, then the angles are congruent. So, \( \angle 2 \cong \angle 4 \).

Answer:

To prove \( \angle 2 \cong \angle 4 \) (vertical angles are congruent), we follow these steps:

1. Given: \( \angle 2 \) and \( \angle 4 \) are vertical angles.
2. Linear Pair Definition: \( \angle 2 \) and \( \angle 3 \) form a linear pair, so \( m\angle 2 + m\angle 3 = 180^\circ \).

\( \angle 3 \) and \( \angle 4 \) form a linear pair, so \( m\angle 3 + m\angle 4 = 180^\circ \).

1. Transitive Property of Equality: Since \( m\angle 2 + m\angle 3 = 180^\circ \) and \( m\angle 3 + m\angle 4 = 180^\circ \), we have \( m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4 \).
2. Subtract \( m\angle 3 \) from both sides: \( m\angle 2 = m\angle 4 \).
3. Definition of Congruent Angles: If \( m\angle 2 = m\angle 4 \), then \( \angle 2 \cong \angle 4 \).

Thus, \( \boldsymbol{\angle 2 \cong \angle 4} \) is proven.