Question
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.
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.
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.
Answers
GPT-4o mini
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.
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.