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: Recall the definition of an exterior angle

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. In \(\triangle EBC\), \(\angle DEC\) is formed by side \(EC\) and the extension of side \(BE\). So \(\angle DEC\) is an exterior angle. \(\angle BEC\) is an interior angle of \(\triangle EBC\), so the statement “\(\angle BEC\) is an exterior angle” is false.

Step2: Recall the definition of supplementary angles

Two angles are supplementary if their sum is \(180^{\circ}\). \(\angle ABE+\angle EBC = 180^{\circ}\) since they form a linear - pair. So \(\angle ABE\) and \(\angle EBC\) are supplementary angles.

Step3: Recall the relationship between angles and the concept of supplementary angles

For \(\angle BCF\) and \(\angle DEC\), we know that \(\angle BCF=\angle BEC + \angle EBC\) (by the exterior - angle theorem of a triangle: an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles). Also, \(\angle DEC+\angle BEC = 180^{\circ}\) (linear - pair). But \(\angle BCF
eq180^{\circ}-\angle DEC\).
For \(\angle BEC\) and \(\angle BCF\), by the exterior - angle theorem of a triangle (\(\angle BCF=\angle BEC+\angle EBC\)), \(\angle BEC\) 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 \(\angle BCF\).