Which statements regarding the diagram Triangle EBC are true? Select three options: • Angle BEC is an exterior angle • Angle DEC is an exterior angle • Angle ABE and EBC are supplementary angles • Angle BCF and DEC are supplementary angles • Angle BEC is a remote interior angle to exterior angle BCF
The Correct Answer and Explanation is:
Based on the diagram of Triangle EBC, the three correct statements are:
- Angle DEC is an exterior angle
- Angle ABE and EBC are supplementary angles
- Angle BEC is a remote interior angle to exterior angle BCF
Here is a detailed explanation of why these three are correct:
1. Angle DEC is an exterior angle: An exterior angle is formed when a side of a triangle is extended and the angle lies outside the triangle. In this case, segment EC is part of triangle EBC, and it has been extended to point D. Angle DEC lies outside triangle EBC and is adjacent to angle BEC, which makes angle DEC an exterior angle to triangle EBC.
2. Angle ABE and EBC are supplementary angles: Two angles are supplementary if the sum of their measures is 180 degrees. In the diagram, angle ABE and angle EBC share a common ray BE and lie on a straight line from point A through B to C. Since they lie along a straight angle, their sum is 180 degrees. This makes them supplementary.
3. Angle BEC is a remote interior angle to exterior angle BCF: A remote interior angle is one that is inside the triangle but not adjacent to the given exterior angle. In triangle EBC, angle BCF is an exterior angle formed by extending side BC. The interior angles of triangle EBC are angles EBC, BEC, and CEB. Of these, angle BEC is not adjacent to angle BCF, and it lies inside the triangle. Thus, BEC is a remote interior angle relative to exterior angle BCF.
The remaining two statements are false because angle BEC is inside the triangle (not exterior), and angle BCF and DEC lie on separate parts of the diagram and do not form a supplementary pair.
