Question

complete the following plan to prove that ∠3≅∠6. use the same - side interior angles postulate to show that ∠3 is supplementary to ∠5. since ∠3 and ∠6 are to the same angle, they are congruent to each other. show ∠6 and ∠5 are

Explanation:

Step1: Recall angle - pair relationships

By the Same - Side Interior Angles Postulate, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. In the given figure, if we assume the two horizontal lines \(l\) and \(m\) are parallel and \(t\) is the transversal, \(\angle3\) and \(\angle6\) are same - side interior angles. So \(\angle3\) is supplementary to \(\angle6\).

Step2: Use the property of supplementary angles

We know that if \(\angle3+\angle6 = 180^{\circ}\) and also, if we consider another pair of angles such that \(\angle3\) and some other angle \(\angle x\) and \(\angle6\) and \(\angle x\) satisfy the supplementary condition. Since \(\angle3\) and \(\angle6\) are supplementary to the same angle (we can think of the linear - pair relationships around the intersection points), we can use the fact that angles supplementary to the same angle are congruent.
We know that \(\angle3\) and \(\angle6\) are supplementary because they are same - side interior angles. And we show that \(\angle6\) and \(\angle3\) are supplementary to the same angle (for example, if we consider the linear pairs formed at the intersection of the transversal and the parallel lines). Since \(\angle3\) and \(\angle6\) are supplementary to the same angle, they are congruent to each other.

Answer:

1. supplementary
2. the same angle
3. congruent