Question

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

Explanation:

Step1: Analyze exterior angles

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. $\angle DEC$ is formed by side $EC$ of $\triangle EBC$ and the extension of side $EB$ (since $DE$ is a straight - line extension). So, $\angle DEC$ is an exterior angle. $\angle BEC$ is an interior angle of $\triangle EBC$, not an exterior angle.

Step2: Analyze supplementary angles

Supplementary angles add up to $180^{\circ}$. $\angle ABE$ and $\angle EBC$ form a linear pair (they are adjacent angles on a straight line $AD$). By the definition of a linear pair, $\angle ABE+\angle EBC = 180^{\circ}$, so they are supplementary.

Step3: Analyze remote interior angles

A remote interior angle of an exterior angle of a triangle is an interior angle that is not adjacent to the exterior angle. For exterior angle $\angle BCF$ of $\triangle EBC$, $\angle BEC$ is an interior angle of $\triangle EBC$ and is not adjacent to $\angle BCF$. So, $\angle BEC$ is a remote interior angle to exterior $\angle BCF$.

Answer:

B. $\angle DEC$ is an exterior angle, C. $\angle ABE$ and $\angle EBC$ are supplementary angles, E. $\angle BEC$ is a remote interior angle to exterior $\angle BCF$