Question

1. complete the following plan to prove that ∠4 ≅ ∠8.

use the same-side interior angles postulate to show that ∠4 is supplementary to.
show ∠8 and are because they form a.
since ∠4 and ∠8 are to the same angle, they are congruent to each other.

Explanation:

Step1: Apply Same - Side Interior Angles Postulate

The Same - Side Interior Angles Postulate states that if two parallel lines are cut by a transversal, then the same - side interior angles are supplementary. When considering the angles formed by a transversal with two parallel lines, \(\angle4\) and \(\angle5\) are same - side interior angles. So, \(\angle4\) is supplementary to \(\angle5\) (i.e., \(m\angle4 + m\angle5=180^{\circ}\)).

Step2: Analyze linear pair

\(\angle8\) and \(\angle5\) form a linear pair. By the definition of a linear pair, if two adjacent angles form a straight line (a linear pair), then they are supplementary. So, \(m\angle8 + m\angle5 = 180^{\circ}\), which means \(\angle8\) and \(\angle5\) are supplementary.

Step3: Use Congruent Supplements Theorem

The Congruent Supplements Theorem states that if two angles are supplementary to the same angle (or to congruent angles), then the two angles are congruent. Since \(\angle4\) and \(\angle8\) are both supplementary to \(\angle5\), we can conclude that \(\angle4\cong\angle8\).

Answer:

\(\angle4\) is supplementary to \(\angle5\). \(\angle8\) and \(\angle5\) are supplementary because they form a linear pair. Since \(\angle4\) and \(\angle8\) are supplementary to the same angle, they are congruent to each other.