Triangle E B C is shown with its exterior angles. Line E C extends through point F. Line C B extends through point A. Line B E extends through point D.

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.

Answer 1

To evaluate which statements regarding triangle \( \Delta EBC \) are true, we need to consider the properties of angles in a triangle and their exterior angles.

1. ∠BEC is an exterior angle.
   True. The angle \(\angle BEC\) is formed by extending line \(EC\) at point \(C\), making it an exterior angle of triangle \(EBC\).

2. ∠DEC is an exterior angle.
   True. The angle \(\angle DEC\) is formed by extending line \(BE\) at point \(E\), making it another exterior angle of triangle \(EBC\).

3. ∠ABE and ∠EBC are supplementary angles.
   True. Since \(\angle ABE\) is the angle formed between line segment \(AB\) and line segment \(BE\) (which is a side of triangle \(EBC\)), and these two angles combine to form a straight line along \(BE\), they are supplementary.

4. ∠BCF and ∠DEC are supplementary angles.
   This is also true because \(\angle BCF\) is an angle formed by the extension of line \(CB\), which is adjacent to \(\angle DEC\), thereby making these two angles supplementary.

5. ∠BEC is a remote interior angle to exterior ∠BCF.
   False. The remote interior angles of a triangle are the ones that are not adjacent to a given exterior angle. \(\angle BEC\) is actually an exterior angle itself; hence it cannot be a remote interior angle to \(\angle BCF\).

Based on this analysis, the three correct statements regarding triangle \( \Delta EBC \) are:

• ∠BEC is an exterior angle.
• ∠DEC is an exterior angle.
• ∠ABE and ∠EBC are supplementary angles.

So the true options are the first three statements.