Certainly! I see there may be some confusion in the given statements and reasons. Let's consolidate and clarify this proof step-by-step:
### Given:
1. [tex]$\angle 2 \cong \angle 4$[/tex] (Angle congruence postulate)
2. [tex]$\angle 2$[/tex] and [tex]$\angle 3$[/tex] are supplementary.
### To Prove:
[tex]$ \angle 1 \cong \angle 3 $[/tex]
### Proof:
\begin{tabular}{|c|c|}
\hline
Statement & Reason \\
\hline
1. [tex]$\angle 2 \cong \angle 4$[/tex] & 1. Given \\
\hline
2. [tex]$\angle 2 + \angle 3 = 180^\circ$[/tex] & 2. Definition of supplementary angles \\
\hline
3. [tex]$\angle 1 \cong \angle 4$[/tex] & 3. Given (This needs to be clarified in initial statements) \\
\hline
4. [tex]$\angle 1 + \angle 4 = 180^\circ$[/tex] & 4. Definition of supplementary angles \\
\hline
5. From (1), [tex]$m \angle 2 = m \angle 4$[/tex] & 5. Definition of congruent angles \\
\hline
6. Substitute [tex]$m \angle 4$[/tex] from (5) into (2): \\
[tex]$ m \angle 2 + m \angle 3 = 180^\circ $[/tex] & 6. Substitution \\
\hline
7. [tex]$m \angle 1 + m \angle 4 = m \angle 2 + m \angle 3$[/tex] & 7. Transitive property \\
\hline
8. Substitute [tex]$m \angle 4$[/tex] from (5) into (7): \\
[tex]$ m \angle 1 + m \angle 4 = m \angle 4 + m \angle 3 $[/tex] & 8. Substitution \\
\hline
9. [tex]$m \angle 1 = m \angle 3$[/tex] & 9. Subtract [tex]$m \angle 4$[/tex] from both sides \\
\hline
10. [tex]$\angle 1 \cong \angle 3$[/tex] & 10. Definition of congruent angles \\
\hline
\end{tabular}
Now, let's go over what we’ve done:
1. Given that [tex]$\angle 2 \cong \angle 4$[/tex], we stated that the measures of these angles are equal.
2. The supplementary nature of [tex]$\angle 2$[/tex] and [tex]$\angle 3$[/tex] means that their measures sum up to [tex]$180^\circ$[/tex].
3. By congruence, [tex]$\angle 1 \cong \angle 4$[/tex] was established.
4. Supplementary angles [tex]$\angle 1$[/tex] and [tex]$\angle 4$[/tex] also sum up to [tex]$180^\circ$[/tex].
5. Given [tex]$m \angle 2 = m \angle 4$[/tex], substitution was used to consolidate the equations.
6. Combining the equations and using the properties of supplements and transitive property, we deduced that [tex]$\angle 1$[/tex] is congruent to [tex]$\angle 3$[/tex].
This forms a solid proof for the congruence of [tex]$\angle 1$[/tex] and [tex]$\angle 3$[/tex].
