Question

which statements regarding the diagram of δebc are true? select three options. ∠bec is an exterior angle. ∠dec is an exterior angle. ∠abe and ∠ebc are supplementary angles. ∠bcf and ∠dec are supplementary angles. ∠bec is a remote interior angle to exterior ∠bcf.

Explanation:

1. For ∠BEC being an exterior angle: An exterior angle of a triangle is formed by one side and the extension of another side. ∠BEC is an interior angle of ΔEBC, not exterior. So this is false.
2. For ∠DEC being an exterior angle: ∠DEC is formed by extending side EC of ΔEBC (from E to D), so it is an exterior angle. This is true.
3. For ∠ABE and ∠EBC being supplementary: ∠ABE and ∠EBC form a linear pair (they are adjacent and their non - common sides form a straight line), so they are supplementary (sum to 180°). This is true.
4. For ∠BCF and ∠DEC being supplementary: ∠BCF is an exterior angle at C, and ∠DEC is an exterior angle at E. There is no reason for them to be supplementary. In fact, in triangle EBC, ∠BCF is supplementary to ∠ECB, and ∠DEC is supplementary to ∠BEC. So this is false.
5. For ∠BEC being a remote interior angle to exterior ∠BCF: A remote interior angle to an exterior angle is an interior angle that is not adjacent to the exterior angle. For exterior angle ∠BCF (at vertex C), the remote interior angles are ∠BEC and ∠EBC. So ∠BEC is a remote interior angle to ∠BCF. This is true.

Answer:

• ∠DEC is an exterior angle.
• ∠ABE and ∠EBC are supplementary angles.
• ∠BEC is a remote interior angle to exterior ∠BCF.