Question

identify the pair(s) of congruent angles in the figure. explain how you know they are congruent.
∠abc is supplementary to ∠cbd.
∠cbd is supplementary to ∠def.
∠cbd≅∠fea by the congruent supplements theorem
∠def≅∠fea by the linear pair postulate
∠def≅∠cbd by the vertical angles congruence theorem
∠abc≅∠def by the congruent supplements theorem

Explanation:

Step1: Recall the Congruent Supplements Theorem

If two angles are supplementary to the same angle, then they are congruent.

Step2: Analyze the given information

We know that \(\angle ABC\) is supplementary to \(\angle CBD\) (so \(\angle ABC+\angle CBD = 180^{\circ}\)), and \(\angle CBD\) is supplementary to \(\angle DEF\) (so \(\angle CBD+\angle DEF=180^{\circ}\)).
Since \(\angle ABC\) and \(\angle DEF\) are both supplementary to \(\angle CBD\), by the Congruent Supplements Theorem, \(\angle ABC\cong\angle DEF\).

Answer:

\(\angle ABC\cong\angle DEF\) by the Congruent Supplements Theorem.