Let's begin by looking at the given information and the desired outcome of the proof. Then we can write a proof!
&Given: △ ABC with right angle C
&Prove: ∠ A and ∠ B are complementary.
Since ∠ C is a right angle, we can conclude that m ∠ C = 90.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the definition of a right angle, } \\
m \angle C=90.
\end{gathered}
Recall that the Triangle Angle-Sum Theorem tells us that the measures of the angles of a triangle is 180. Therefore, the sum of the measures of ∠ A, ∠ B, and ∠ C is 180.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Triangle Angle-Sum Theorem,} \\
m \angle A + m \angle B + m \angle C =180.
\end{gathered}
Now we have the following equations.
m ∠ C=90 and m ∠ A + m ∠ B + m ∠ C =180
By the Substitution Property of Equality, we can substitute 90 for m ∠ C in the second equation.
We got that the sum of angles A and B is 90. Therefore, by the definition of complementary angles we can conclude that ∠ A and ∠ B are complementary.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Substitution Property of Equality,} \\
m \angle A + m \angle B =90. \\ \text{ By the definition of complementary angles, } \\
\angle A \text{ and } \angle B \text{ are complementary.}
\end{gathered}
Final Proof
&Given: △ ABC with right angle C
&Prove: ∠ A and ∠ B are complementary
Prove: By the definition of a right angle, m ∠ C=90. By the Triangle Angle-Sum Theorem, m∠ A+ m∠ B+m ∠ C =190. By the Substitution Property of Equality, m ∠ A + m ∠ B =90. By the definition of complementary angles, ∠ A and ∠ B are complementary.
