Question

use the image to answer the question
sybil has started a proof of the triangle angle sum theorem. which answer choice correctly completes her proof?
sybils proof: given the diagram shown, ∠1 ≅ ∠5, ∠2 ≅ ∠4 because alternate interior angles are congruent when lines are parallel. then, i know that m∠1 = m∠5, m∠2 = m∠4 because congruent angles have equal measures.
(1 point)
o m∠5 + m∠3 + m∠4 = 180° by the definition of a straight angle. finally, m∠1 + m∠3 + m∠2 = 180° by substitution.
o m∠1 + m∠3 + m∠2 = 180° by the definition of a straight angle. finally, m∠5 + m∠3 + m∠4 = 180° by substitution.
o m∠2 + m∠3 + m∠4 = 180° by the definition of a straight angle. finally, m∠1 + m∠3 + m∠2 = 180° by substitution.
o m∠5 + m∠3 + m∠4 = 180° by the definition of a straight angle. finally, m∠1 + m∠3 + m∠2 = 180° by the triangle angle sum theorem.

Explanation:

Step1: Analyze the straight - angle property

Since \(\angle5\), \(\angle3\), and \(\angle4\) form a straight line, by the definition of a straight angle, \(m\angle5 + m\angle3 + m\angle4=180^{\circ}\).

Step2: Use the angle - congruence substitution

Given that \(\angle1\cong\angle5\) (so \(m\angle1 = m\angle5\)) and \(\angle2\cong\angle4\) (so \(m\angle2 = m\angle4\)), substitute \(m\angle5\) with \(m\angle1\) and \(m\angle4\) with \(m\angle2\) in the equation \(m\angle5 + m\angle3 + m\angle4 = 180^{\circ}\). We get \(m\angle1 + m\angle3 + m\angle2 = 180^{\circ}\).

Answer:

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