Question

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 } \\)
\\( m \angle 1 + m \angle 2 + m \angle 3 = 180 ^ { \circ } \\)

Explanation:

Step1: Alternate Interior Angles Theorem

Since \(DE\parallel AC\), by the Alternate Interior Angles Theorem, \(\angle3\cong\angle5\) (because they are alternate - interior angles formed by transversal \(BC\) with \(DE\) and \(AC\)) and \(\angle1\cong\angle4\) (because they are alternate - interior angles formed by transversal \(AB\) with \(DE\) and \(AC\)).

Step2: Congruent angles have equal measures

If \(\angle3\cong\angle5\), then \(m\angle3 = m\angle5\) (by the definition of congruent angles: congruent angles have equal measures). Similarly, if \(\angle1\cong\angle4\), then \(m\angle1=m\angle4\).

Step3: Definition of a straight line

Since \(D, B, E\) are collinear ( \(DE\) is a straight line), by the definition of a straight - line angle, \(m\angle4 + m\angle2+m\angle5=180^{\circ}\) (the sum of angles on a straight line is \(180^{\circ}\)).

Step4: Substitution

Substitute \(m\angle1\) for \(m\angle4\) and \(m\angle3\) for \(m\angle5\) in the equation \(m\angle4 + m\angle2+m\angle5=180^{\circ}\). We get \(m\angle1 + m\angle2+m\angle3=180^{\circ}\) (by substitution property of equality).

Answer:

The reason for \(\angle3\cong\angle5\) and \(\angle1\cong\angle4\) is the Alternate Interior Angles Theorem. The reason for \(m\angle1 = m\angle4\) and \(m\angle3 = m\angle5\) is that congruent angles have equal measures. The reason for \(m\angle4 + m\angle2 + m\angle5=180^{\circ}\) is the definition of a straight line. The reason for \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) is substitution.