Let's analyze the proof step-by-step to see where the justification that angles with a combined degree measure of [tex]$90^{\circ}$[/tex] are complementary is used.
\begin{tabular}{|l|l|}
\hline
Statements & Reasons \\
\hline
\hline
1. [tex]$m \angle 1 = 40^{\circ}$[/tex] & 1. Given \\
\hline
2. [tex]$m \angle 2 = 50^{\circ}$[/tex] & 2. Given \\
\hline
3. [tex]$\angle 1$[/tex] is complementary to [tex]$\angle 2$[/tex] & 3. Definition of complementary angles \\
\hline
4. [tex]$\angle 2$[/tex] is complementary to [tex]$\angle 3$[/tex] & 4. Given \\
\hline
5. [tex]$\angle 1 \cong $[/tex]\angle 3[tex]$ & 5. Congruent complements theorem \\
\hline
\end{tabular}
Now, let's break down the statements and reasons:
1. $[/tex]m \angle 1 = 40^{\circ}[tex]$:
- This is given information about the measure of angle 1.
2. $[/tex]m \angle 2 = 50^{\circ}[tex]$:
- This is given information about the measure of angle 2.
3. $[/tex]\angle 1[tex]$ is complementary to $[/tex]\angle 2[tex]$:
- Reason: This uses the definition of complementary angles. According to this definition, two angles are complementary if the sum of their measures is $[/tex]90^{\circ}[tex]$. Since $[/tex]m \angle 1 + m \angle 2 = 40^{\circ} + 50^{\circ} = 90^{\circ}[tex]$, angles $[/tex]\angle 1[tex]$ and $[/tex]\angle 2[tex]$ are complementary.
4. $[/tex]\angle 2[tex]$ is complementary to $[/tex]\angle 3[tex]$:
- This statement is given.
5. $[/tex]\angle 1 \cong \angle 3[tex]$:
- Reason: This uses the congruent complements theorem, which states that if two angles are complementary to the same angle (in this case, $[/tex]\angle 2[tex]$), then they are congruent to each other. Since $[/tex]\angle 1[tex]$ is complementary to $[/tex]\angle 2[tex]$ and $[/tex]\angle 3[tex]$ is also complementary to $[/tex]\angle 2[tex]$, it follows that $[/tex]\angle 1 \cong \angle 3[tex]$.
Hence, the justification that angles with a combined degree measure of $[/tex]90^{\circ}[tex]$ are complementary is specifically used in Statement 3: $[/tex]\angle 1[tex]$ is complementary to $[/tex]\angle 2$, and the reason provided for this is the definition of complementary angles.
