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 } \\).

Explanation:

Step1: Determine the reason for \( \angle3\cong\angle5 \) and \( \angle1\cong\angle4 \)

Since \( \overline{DE}\parallel\overline{AC} \), by the Alternate Interior Angles Theorem, when a transversal ( \( BC \) for \( \angle3\) and \( \angle5 \), \( AB \) for \( \angle1\) and \( \angle4 \)) intersects two parallel lines, the alternate - interior angles are congruent.

Step2: Determine the reason for \( m\angle1 = m\angle4 \) and \( m\angle3 = m\angle5 \)

If two angles are congruent (\( \angle3\cong\angle5 \) and \( \angle1\cong\angle4 \)), then by the property that congruent angles have equal measures, \( m\angle1 = m\angle4 \) and \( m\angle3 = m\angle5 \).

Step3: Determine the reason for \( m\angle4 + m\angle2 + m\angle5=180^{\circ} \)

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

Step4: Determine the reason for \( m\angle1 + m\angle2 + m\angle3 = 180^{\circ} \)

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

Answer:

For \( \angle3\cong\angle5 \) and \( \angle1\cong\angle4 \): Alternate Interior Angles Theorem.
For \( m\angle1 = m\angle4 \) and \( m\angle3 = m\angle5 \): Congruent angles have equal measures.
For \( m\angle4 + m\angle2 + m\angle5 = 180^{\circ}\): Definition of a straight line.
For \( m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\): Substitution property of equality.