Exercise 33 Page 177

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.