Question

1. complete the following plan to prove that $\angle4\cong\angle8$. use the same - side interior angles postulate to show that $\angle4$ is supplementary to __. show $\angle8$ and are because they form a . since $\angle4$ and $\angle8$ are __ to the same angle, they are congruent to each other.

Explanation:

Step1: Use the 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 relationship for \(\angle4\), we know that \(\angle4\) and \(\angle5\) are same - side interior angles. So, \(\angle4\) is supplementary to \(\angle5\) (i.e., \(\angle4+\angle5 = 180^{\circ}\)).

Step2: Analyze the linear pair relationship

A linear pair of angles is formed when two adjacent angles form a straight line. \(\angle8\) and \(\angle5\) form a linear pair. By the definition of a linear pair, \(\angle8\) and \(\angle5\) are supplementary (i.e., \(\angle8+\angle5=180^{\circ}\)).

Step3: Use the congruent - supplements theorem

The congruent - supplements theorem states that if two angles are supplementary to the same angle, then the two angles are congruent. Since \(\angle4\) and \(\angle8\) are both supplementary to \(\angle5\) (\(\angle4+\angle5 = 180^{\circ}\) and \(\angle8+\angle5 = 180^{\circ}\)), we can conclude that \(\angle4\cong\angle8\).

Answer:

\(\angle5\); \(\angle5\); supplementary; linear pair; supplementary