Let's begin by looking at the given information and the desired outcome of the proof. Then, we can write a proof!
&Given: ∠1 is an enterior angle of a triangle.
&Prove: m ∠1 = m ∠2 + m ∠3
First, notice that ∠1 and ∠4 form a linear pair. Therefore, by the Linear Pair Postulate, they are supplementary angles. This means that the sum of their measures is 180.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Linear Pair Postulate, } \angle 1 \text{ and } \angle 4 \\
\text{are supplementary angles.} \\
\text{By the definition of supplementary angles, } \\
m \angle 1 + m \angle 4 = {\color{#FF0000}{180}}.
\end{gathered}Now, recall that the Triangle Angle-Sum Theorem tells us that the sum of the measures of the angles of a triangle is 180. Thus, the sum of the measures of ∠2, ∠3, and ∠4 is 180.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Triangle Angle-Sum Theorem,} \\
m \angle 2 + m \angle 3 + m \angle 4= {\color{#FF0000}{180}}.
\end{gathered}
Since we have 180 in both equations, we can use the Substitution Property of Equality. Let's substitute 180 with m ∠2 + m ∠3 + m ∠4 in the first equation.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Substitution Property of Equality, } \\
m \angle 1 + m \angle 4 = m \angle 2 + m \angle 3 + m \angle 4.
\end{gathered}
Finally, by the Subtraction Property of Equality we can subtract m ∠4 from both sides of our equation. This gives us m ∠1 = m ∠2+ m ∠3, and this is what we wanted to prove!
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Subtraction Property of Equality, } \\
m \angle 1 = m \angle 2+ m \angle 3.
\end{gathered}
Final Proof
&Given: ∠1 is an enterior angle of a triangle.
&Prove: m ∠1 = m ∠2 + m ∠3
Prove: By the Linear Pair Postulate , ∠1 and ∠4 are supplementary angles. By the definition of supplementary angles, m ∠1+ m ∠4 =180.
By the Triangle Angle-Sum Theorem, m ∠2 + m ∠3 + m ∠4 = 180. By the Substitution Property of Equality,
m ∠1 + m ∠4 = m ∠2 + m ∠3 + m ∠4.
By the Subtraction Property of Equality, m ∠1 = m ∠2+ m ∠3.
