Question

what is the name of the pair of angles ∠dbe and ∠ebc whose noncommon sides are opposite rays that form a straight line?

Explanation:

To determine the name of the angle pair \(\angle DBE\) and \(\angle EBC\) with non - common sides as opposite rays forming a straight line, we recall angle pair definitions:

1. Adjacent Angles: These are angles that share a common side and a common vertex, and their non - common sides are adjacent (not opposite rays in the way we need here for this specific type of pair).
2. Vertical Angles: Vertical angles are formed by two intersecting lines and are opposite each other. They are not adjacent and do not have non - common sides forming a straight line in the way described.
3. Complementary Angles: Complementary angles are two angles whose sum is \(90^{\circ}\). There is no indication here that the sum of \(\angle DBE\) and \(\angle EBC\) is \(90^{\circ}\) (in fact, since their non - common sides form a straight line, their sum should be \(180^{\circ}\)).
4. Supplementary Angles: Supplementary angles are defined as two angles whose sum is \(180^{\circ}\). When two angles have a common side and their non - common sides are opposite rays (forming a straight line), the sum of the measures of the two angles is \(180^{\circ}\) (because a straight line is a \(180^{\circ}\) angle). Also, since \(\angle DBE\) and \(\angle EBC\) share the common side \(BE\) and their non - common sides \(BD\) and \(BC\) are opposite rays (forming a straight line \(DC\) passing through \(B\)), they are adjacent supplementary angles. But the key property here is that their sum is \(180^{\circ}\) because their non - common sides form a straight line, so they are supplementary angles. Also, since they share a common side and vertex, they are adjacent supplementary angles, but the main category for angles with non - common sides as opposite rays forming a straight line (and thus summing to \(180^{\circ}\)) is supplementary angles (and specifically adjacent supplementary angles in this case as they are adjacent).

Answer:

The pair of angles \(\angle DBE\) and \(\angle EBC\) is called a linear pair (or adjacent supplementary angles). The key characteristic is that they are supplementary (sum to \(180^{\circ}\)) because their non - common sides form a straight line, and they are adjacent (share a common side and vertex). So the name of this pair of angles is a linear pair (or adjacent supplementary angles). If we consider the general term for angles that sum to \(180^{\circ}\) and are adjacent with non - common sides as opposite rays, it's a linear pair.