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 { d e } \\) be a line passing through b and parallel to \\( \overline { a c } \\).	definition of parallel lines
\\( \angle 3 \cong \angle 5 \\) and \\( \angle 1 \cong \angle 4 \\)
\\( m \angle 1 = m \angle 4 \\) and \\( m \angle 3 = m \angle 5 \\)
\\( m \angle 4 + m \angle 2 + m \angle 5 = 180 ^ { \circ } \\)	angle addition and definition of a straight line
\\( m \angle 1 + m \angle 2 + m \angle 3 = 180 ^ { \circ } \\)	substitution

Explanation:

Step1: Alternate interior angles

Since \( \overline{DE}\parallel\overline{AC}\), by the alternate - interior angles theorem, when a transversal intersects two parallel lines, the alternate interior angles are congruent. Here, \( \overline{BC}\) is a transversal for \( \overline{DE}\) and \( \overline{AC}\), so \( \angle3\cong\angle5\), and \( \overline{AB}\) is a transversal for \( \overline{DE}\) and \( \overline{AC}\), so \( \angle1\cong\angle4\).

Step2: Congruent angles have equal measures

If two angles are congruent (\( \angle3\cong\angle5\) and \( \angle1\cong\angle4\)), then their measures are equal. That is, if \( \angle A\cong\angle B\), then \(m\angle A = m\angle B\). So, if \( \angle3\cong\angle5\), then \(m\angle3=m\angle5\), and if \( \angle1\cong\angle4\), then \(m\angle1=m\angle4\).

Answer:

For the first blank (reason for \( \angle3\cong\angle5\) and \( \angle1\cong\angle4\)): alternate - interior angles theorem.
For the second blank (reason for \(m\angle1 = m\angle4\) and \(m\angle3 = m\angle5\)): congruent angles have equal measures.