Question

Use the examples above to write a paragraph proof proving the Triangle Sum Theorem.

Answer 1

First, start by drawing the line EC that passes through vertex C and is parallel to AB, as showed in the question's figure.

Since AB and EC are parallel, the pairs ∠4 and ∠1 and ∠4 and ∠3 are Alternate Interior Angles, which means that the pairs are congruent, that is:

$\begin{gathered} \angle4\cong\angle1 \\ \angle5\cong\angle3 \end{gathered}$

Also, the angles ∠1, ∠2 and ∠3 are adjacent angles that make a straight line. This means that the sum of these 3 angles is 180°.

$m\angle1+m\angle2+m\angle3=180\degree$

Since congruent angles have the same measure, we can substitute:

$\begin{gathered} m\angle1=m\angle4 \\ m\angle3=m\angle5 \end{gathered}$

To obtain the Triangle Sum Theorem:

$m\angle4+m\angle5+m\angle2=180\degree$

That is, the sum of the interior angle measures of a triangle is 180°.