Question

select the correct answer from each drop - down menu.
points a, b, and c form a triangle. complete the statements to prove that the sum of the interior angles of $\triangle abc$ is $180^{circ}$.

statement	reason
let $overline{de}$ be a line passing through b and parallel to $overline{ac}$.	definition of parallel lines
$angle 3congangle 5$ and $angle 1congangle 4$
$mangle 1 = mangle 4$ and $mangle 3 = mangle 5$
$mangle 4 + mangle 2 + mangle 5 = 180^{circ}$	straight line
$mangle 1 + mangle 2 + mangle 3 = 180^{circ}$

(the options for the reasons of some statements include: alternate interior angles theorem, alternate exterior angles theorem, corresponding angles theorem, congruent angles have equal measures.)

Explanation:

1. For $\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$: When a transversal crosses parallel lines, alternate interior angles are congruent. Here, $\overline{DE} \parallel \overline{AC}$, so these pairs are alternate interior angles.
2. For $m\angle 1 = m\angle 4$ and $m\angle 3 = m\angle 5$: Congruent angles are defined as angles that have equal measures.
3. For $m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ$: The angles along a straight line form a straight angle, which measures $180^\circ$.
4. For $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$: Substitute $m\angle 4$ with $m\angle 1$ and $m\angle 5$ with $m\angle 3$ using the earlier equal angle measures.

Answer:

Statement	Reason
Let $\overline{DE}$ be a line passing through B and parallel to $\overline{AC}$.	definition of parallel lines
$\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$	Alternate Interior Angles Theorem
$m\angle 1 = m\angle 4$ and $m\angle 3 = m\angle 5$	Congruent angles have equal measures.
$m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ$	straight line (sum of angles on a straight line is $180^\circ$)
$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$	Substitution Property of Equality