Question

which statements regarding the diagram of △ebc 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:

Step1: Recall definitions

• Remote interior angle: For an exterior angle of a triangle, the two non - adjacent interior angles are remote interior angles.
• Exterior angle: An angle formed by one side of a triangle and the extension of an adjacent side.
• Supplementary angles: Two angles whose sum is \(180^{\circ}\).

Step2: Analyze each option

• For \(\angle BEC\) and \(\angle BCF\):
• In \(\triangle EBC\), \(\angle BEC\) is a non - adjacent interior angle with respect to the exterior angle \(\angle BCF\). So, \(\angle BEC\) is a remote interior angle to exterior \(\angle BCF\).
• For \(\angle ABE\) and \(\angle EBC\):
• \(\angle ABE+\angle EBC = 180^{\circ}\) (linear pair). So, \(\angle ABE\) and \(\angle EBC\) are supplementary angles.
• For \(\angle BEC\):
• \(\angle BEC\) is formed by one side \(EC\) of \(\triangle EBC\) and the extension of \(EB\) (not a side of the triangle). \(\angle DEC\) is an exterior angle of \(\triangle EBC\) (formed by side \(EC\) and extension of \(ED\) which is not relevant here). \(\angle BEC\) is an interior angle of \(\triangle EBC\).
• For \(\angle BCF\) and \(\angle DEC\):
• There is no direct relationship (like supplementary) based on the triangle \(\triangle EBC\) properties. \(\angle BCF\) is an exterior angle of \(\triangle EBC\) and \(\angle DEC\) is not related to \(\angle BCF\) in a supplementary way for \(\triangle EBC\).
• For \(\angle DEC\):
• \(\angle DEC\) is not an exterior angle of \(\triangle EBC\). The exterior angles of \(\triangle EBC\) are angles like \(\angle BCF\) (formed by side \(BC\) and its extension \(CF\))

Answer:

\(\angle BEC\) is a remote interior angle to exterior \(\angle BCF\), \(\angle ABE\) and \(\angle EBC\) are supplementary angles.