Question

linear pairs and vertical angles
proving vertical angles are congruent
given: ∠2 and ∠4 are vertical angles.
prove: ∠2 ≅ ∠4
image of intersecting lines m and n at point p, forming angles 1, 2, 3, 4
select the correct reason for this statement: m∠3 + m∠4 = 180
statements

1. ∠2 and ∠4 are vert. angles
2. ∠2 and ∠3 are a linear pair
3. ∠3 and ∠4 are a linear pair
4. m∠2 + m∠3 = 180

reasons

1. given
2. def. of a linear pair
3. def. of a linear pair
4. linear pairs are supp.

reasons options: subtraction property, substitution property, linear pairs are supp., definition of ≅ angles

Explanation:

Step1: Recall Linear Pair Definition

A linear pair of angles are adjacent angles that form a straight line, so their measures sum to \(180^\circ\). For \(\angle3\) and \(\angle4\), since they form a linear pair (adjacent and on a straight line), their measures should add up to \(180^\circ\).

Step2: Match with Reason

The statement \(m\angle3 + m\angle4 = 180\) is justified by the definition of a linear pair (linear pairs are supplementary, meaning their measures sum to \(180^\circ\)). So the reason is "linear pairs are supp." (linear pairs are supplementary).

Answer:

The correct reason for the statement \(m\angle3 + m\angle4 = 180\) is "linear pairs are supp." (linear pairs are supplementary).