Question

complete the proof using the statements and reasons below. given: ∠1 and ∠2 are vertical angles m∠2 + m∠3 = 90 prove: m∠1 + m∠3 = 90 m∠1 = m∠2 m∠1 + m∠3 = 90 substitution statement reason 1. ∠1 and ∠2 are vertical angles m∠2 + m∠3 = 90 1. given 2. 2. vertical angles are congruent 3. 3.

Explanation:

Step1: Determine Statement 2

Vertical angles are congruent, so if \( \angle 1 \) and \( \angle 2 \) are vertical angles, then their measures are equal. So the statement for step 2 is \( m\angle 1 = m\angle 2 \).

Step2: Determine Statement 3 and Reason 3

We know from step 1 that \( m\angle 2 + m\angle 3 = 90 \) and from step 2 that \( m\angle 1 = m\angle 2 \). By substituting \( m\angle 2 \) with \( m\angle 1 \) in the equation \( m\angle 2 + m\angle 3 = 90 \), we get \( m\angle 1 + m\angle 3 = 90 \). The reason for this is substitution (since we substituted \( m\angle 2 \) with \( m\angle 1 \)).

Answer:

• Statement 2: \( \boldsymbol{m\angle 1 = m\angle 2} \)
• Statement 3: \( \boldsymbol{m\angle 1 + m\angle 3 = 90} \)
• Reason 3: \( \boldsymbol{\text{Substitution}} \)