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, ∠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) m∠1 + m∠3 + m∠2 = 180° by the definition of a straight angle. finally, m∠5 + m∠3 + m∠4 = 180° by substitution. 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. m∠5 + m∠3 + m∠4 = 180° by the definition of a straight angle. finally, m∠1 + m∠3 + m∠2 = 180° by substitution. m∠2 + m∠3 + m∠4 = 180° by the definition of a straight angle. finally, m∠1 + m∠3 + m∠2 = 180° by substitution.

Explanation:

Step1: Recall straight - angle property

The sum of angles on a straight line is $180^{\circ}$. In the context of the proof, $\angle1$, $\angle3$, and $\angle2$ (or $\angle5$, $\angle3$, and $\angle4$) form a straight - line.

Step2: Use substitution

Since $\angle1\cong\angle5$ and $\angle2\cong\angle4$ (alternate interior angles of parallel lines) and congruent angles have equal measures ($m\angle1 = m\angle5$, $m\angle2 = m\angle4$), we can substitute the angle measures. First, we know that $m\angle1 + m\angle3 + m\angle2=180^{\circ}$ by the definition of a straight angle. Then, substituting $\angle1$ with $\angle5$ and $\angle2$ with $\angle4$, we get $m\angle5 + m\angle3 + m\angle4 = 180^{\circ}$.

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.