Question

sylvie has started a proof of the triangle angle sum theorem. which answer choice correctly completes her proof?
sylvies proof: given the diagram shown, \\( \angle 1 \cong \angle 5 ; \angle 2 \cong \angle 4 \\) because alternate interior angles are congruent when lines are parallel. then, i know that \\( m \angle 1 = m \angle 5 ; m \angle 2 = m \angle 4 \\) because congruent angles have equal measures.
(1 point)
\\( m \angle 1 + m \angle 3 + m \angle 2 = 180 ^ { \circ } \\) by the definition of a straight angle. finally, \\( m \angle 5 + m \angle 3 + m \angle 4 = 180 ^ { \circ } \\) by substitution.
\\( m \angle 5 + m \angle 3 + m \angle 4 = 180 ^ { \circ } \\) by the definition of a straight angle. finally, \\( m \angle 1 + m \angle 3 + m \angle 2 = 180 ^ { \circ } \\) by the triangle angle sum theorem.
\\( m \angle 2 + m \angle 3 + m \angle 4 = 180 ^ { \circ } \\) by the definition of a straight angle. finally, \\( m \angle 1 + m \angle 3 + m \angle 2 = 180 ^ { \circ } \\) by substitution.
\\( m \angle 5 + m \angle 3 + m \angle 4 = 180 ^ { \circ } \\) by the definition of a straight angle. finally, \\( m \angle 1 + m \angle 3 + m \angle 2 = 180 ^ { \circ } \\) by substitution

Explanation:

Step1: Analyze the first part

Since \(m\angle1 = m\angle5\) and \(m\angle2 = m\angle4\), we first consider the angles that form a straight - angle. In a straight - line, the sum of adjacent angles is \(180^{\circ}\). If we assume the line on which the angles are placed, the angles \(\angle1,\angle3,\angle2\) (or their congruent counterparts \(\angle5,\angle3,\angle4\)) form a straight - angle.
We know that by the definition of a straight - angle, the sum of angles on a straight - line is \(180^{\circ}\). If we start with the non - congruent angles (the original triangle angles), \(m\angle1 + m\angle3 + m\angle2=180^{\circ}\) (because they form a straight - angle).

Step2: Use substitution

Because \(m\angle1 = m\angle5\) and \(m\angle2 = m\angle4\) (given that \(\angle1\cong\angle5\) and \(\angle2\cong\angle4\)), we substitute \(m\angle1\) with \(m\angle5\) and \(m\angle2\) with \(m\angle4\) in the equation \(m\angle1 + m\angle3 + m\angle2 = 180^{\circ}\).
By the substitution property of equality, if \(a = b\) and \(c = d\), then \(a + e + c=b + e + d\). Here \(a=m\angle1,b = m\angle5,c=m\angle2,d = m\angle4,e=m\angle3\), so \(m\angle5 + m\angle3 + m\angle4=180^{\circ}\)

Answer:

\(m\angle1 + m\angle3 + m\angle2 = 180^{\circ}\) by the definition of a straight angle. Finally, \(m\angle5 + m\angle3 + m\angle4 = 180^{\circ}\) by substitution.