Question

Consider triangle WXY.

Triangle W X Y is shown. The length of side W X is 5% 2 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.
A. ∠BEC is an exterior angle.
B. ∠DEC is an exterior angle.
C. ∠ABE and ∠EBC are supplementary angles.
D. ∠BCF and ∠DEC are supplementary angles.
E. ∠BEC is a remote interior angle to exterior ∠BCF.

Answer 1

The true statements regarding the diagram of triangle EBC are that ∆DEC is an exterior angle, ∆ABE and ∆EBC are supplementary angles, and ∆BCF and ∆DEC are supplementary angles.

Step-by-step explanation:

Based on the information provided, we can infer that the question is asking about the properties of exterior and interior angles of a triangle and their relationships. In triangle geometry, an exterior angle is equal to the sum of the two remote interior angles. Meanwhile, a linear pair of angles, such as one formed by extending one side of a triangle, consist of a pair of supplementary angles which add up to 180 degrees.

Given this understanding, let's evaluate the options:

A. ∆BEC is an exterior angle. (False, ∆BEC is actually an interior angle of the triangle.)

B. ∆DEC is an exterior angle. (True, this is formed by extending side BE through D.)

C. ∆ABE and ∆EBC are supplementary angles. (True, because they form a linear pair.)

D. ∆BCF and ∆DEC are supplementary angles. (True, since ∆BCF is an exterior angle and ∆DEC is a remote interior angle, and together with angle EBC, they sum to 180 degrees.)

E. ∆BEC is a remote interior angle to exterior ∆BCF. (False, ∆BEC is adjacent to ∆BCF making it not remote.)

Therefore, the true statements regarding the diagram of ∆EBC are B, C, and D.

Answer 2

The correct statements regarding the diagram of ΔEBC are A. ∠BEC is an exterior angle, C. ∠ABE and ∠EBC are supplementary angles, and E. ∠BEC is a remote interior angle to exterior ∠BCF.

Step-by-step explanation:

In triangle EBC, an exterior angle is formed by extending one side of the triangle. Statement A (∠BEC is an exterior angle) is true because ∠BEC is formed by extending side BE beyond the vertex E.

Statements C (∠ABE and ∠EBC are supplementary angles) and E (∠BEC is a remote interior angle to exterior ∠BCF) are also correct. Supplementary angles add up to 180 degrees, and in this case, ∠ABE and ∠EBC together form a straight line, making them supplementary. Additionally, ∠BEC is a remote interior angle to exterior ∠BCF, as it is an interior angle opposite the exterior ∠BCF formed by extending side BC beyond the vertex B.

The other options, B and D, are not true based on the given information. ∠DEC is not an exterior angle, and ∠BCF and ∠DEC are not supplementary angles. Therefore, the correct options are A, C, and E, as they accurately describe the relationships between angles in triangle EBC.