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) 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∠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∠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: Recall straight - angle property

$\angle5$, $\angle3$, and $\angle4$ form a straight - angle. By the definition of a straight angle, the sum of the measures of angles that form a straight line is $180^{\circ}$. So, $m\angle5 + m\angle3 + m\angle4=180^{\circ}$.

Step2: Use angle - congruence substitution

Since $m\angle1 = m\angle5$ and $m\angle2 = m\angle4$ (because $\angle1\cong\angle5$ and $\angle2\cong\angle4$ as alternate interior angles of parallel lines), we can substitute $\angle5$ with $\angle1$ and $\angle4$ with $\angle2$. Then we get $m\angle1 + m\angle3 + m\angle2 = 180^{\circ}$.

Answer:

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