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 congruent - supplements theorem

The congruent - supplements theorem states that if two angles are supplementary to the same angle, then they are congruent.
We know that \(\angle ABC\) is supplementary to \(\angle CBD\) and \(\angle CBD\) is supplementary to \(\angle DEF\). So, \(\angle ABC\cong\angle DEF\) by the congruent - supplements theorem.

Step2: Analyze other options

• \(\angle CBD\) and \(\angle FEA\): There is no information given to suggest they are related by the congruent - supplements theorem or any other valid geometric relationship.
• \(\angle DEF\) and \(\angle FEA\): They are a linear pair. By the linear - pair postulate, they are supplementary (\(\angle DEF+\angle FEA = 180^{\circ}\)), not congruent.
• \(\angle DEF\) and \(\angle CBD\): They are supplementary, not congruent by the vertical - angles congruence theorem (they are not vertical angles).

Answer:

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