We are given that $\angle 1$ and $\angle 2$ are supplementary, $\angle 3$ and $\angle 4$ are supplementary, and $\angle 1 \cong \angle 4$. We want to prove that $\angle 2 \cong \angle 3$. We are asked to fill in the reasons for each step of the proof.

1. Problem Description

We are given that $\angle 1$ and $\angle 2$ are supplementary, $\angle 3$ and $\angle 4$ are supplementary, and $\angle 1 \cong \angle 4$. We want to prove that $\angle 2 \cong \angle 3$. We are asked to fill in the reasons for each step of the proof.

2. Solution Steps

a. $\angle 1$ and $\angle 2$ are supplementary. $\angle 3$ and $\angle 4$ are supplementary. $\angle 1 \cong \angle 4$

Reason: Given

b. $m\angle 1 + m\angle 2 = 180$ and $m\angle 3 + m\angle 4 = 180$

Reason: Definition of supplementary angles. Supplementary angles are two angles whose measures add up to 180 degrees.

c. $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4$

Reason: Transitive Property of Equality. If $a = b$ and $b = c$, then $a = c$.

d. $m\angle 1 = m\angle 4$

Reason: Definition of congruent angles. Congruent angles have the same measure.

e. $m\angle 2 = m\angle 3$

Reason: Subtraction Property of Equality. If $a=b$ and $c=d$, then $a-c = b-d$. From step c, we have $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4$. From step d, we have $m\angle 1 = m\angle 4$. Therefore, $m\angle 1 + m\angle 2 - m\angle 1 = m\angle 3 + m\angle 4 - m\angle 4$ which simplifies to $m\angle 2 = m\angle 3$.

f. $\angle 2 \cong \angle 3$

Reason: Definition of congruent angles. If two angles have the same measure, then they are congruent.

3. Final Answer

a. Given

b. Definition of supplementary angles

c. Transitive Property of Equality

d. Definition of congruent angles

e. Subtraction Property of Equality

f. Definition of congruent angles