which statements regarding the diagram of $ \triangle ebc $ are true? select three options. $ \angle bec $ is an exterior angle. $ \angle dec $ is an exterior angle. $ \angle abe $ and $ \angle ebc $ are supplementary angles. $ \angle bcf $ and $ \angle dec $ are supplementary angles. $ \angle bec $ is a remote interior angle to exterior $ \angle bcf $.

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# Brief Explanations:
1. **Analyze ∠BEC as an exterior angle**: An exterior angle of a triangle is formed by one side and the extension of another side. ∠BEC is an interior angle of ΔEBC, not exterior. So this statement is false.
2. **Analyze ∠DEC as an exterior angle**: For ΔEBC, side EB is extended to A, and side EC is extended to D? Wait, no. Wait, ∠DEC: Let's see, in ΔEBC, the sides are EB, BC, EC. The angle ∠DEC is formed by extending side DE (wait, D-E-C is a straight line? Wait, D, E, C: D---E---C? No, the diagram shows D---E---C? Wait, no, the diagram has D, E, then B, C, F. Wait, ∠DEC: the angle at E between D and C? Wait, no, ΔEBC has vertices E, B, C. So the exterior angle at E would be formed by extending EB or EC. Wait, ∠DEC: if we consider side EC, and extend it to D, then ∠DEC is adjacent to ∠BEC. Wait, maybe I got it wrong. Wait, actually, for triangle EBC, the exterior angle at E: the two sides at E are EB and EC. So an exterior angle at E would be formed by extending EB or EC. If we extend EC to D, then ∠DEB would be an exterior angle? Wait, no, the options are ∠DEC. Wait, maybe I misread. Wait, the diagram: points D, E, then B, C, F. So D---E---(B? No, E is connected to B and C. So E is a vertex, B and C are the other two vertices. So the angles at E: ∠BEC (between EB and EC) and if we extend EB to A, then ∠ABE is adjacent. Wait, maybe I need to recall the definition: an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Let's check each option again.

3. **Analyze ∠ABE and ∠EBC**: ∠ABE and ∠EBC form a linear pair (they are adjacent and their non - common sides form a straight line, since A---B---E? Wait, A, B, and then E? Wait, no, A---B--- (E? No, the diagram has A on a line going up from B, and D on a line going left from E. Wait, ∠ABE and ∠EBC: if A, B, and C are such that B is a point, A is on one line, E is on another, and C is on the third. Wait, actually, ∠ABE and ∠EBC: since A, B, and E? No, A---B--- (E? Wait, the two angles ∠ABE and ∠EBC: their sum is 180° because they form a linear pair (they are adjacent angles on a straight line). So they are supplementary. So this statement is true.

4. **Analyze ∠BCF and ∠DEC as supplementary**: ∠BCF is an exterior angle at C for ΔEBC (formed by extending BC to F). ∠DEC: let's see, ∠DEC and ∠BEC are supplementary (since D, E, C are colinear? Wait, D---E---C? No, D, E, and then C? Wait, maybe D, E, C are colinear? If D, E, C are colinear, then ∠DEC and ∠BEC are supplementary. But ∠BCF: ∠BCF is an exterior angle at C, and ∠ECB is its adjacent interior angle, so ∠BCF + ∠ECB = 180°. ∠DEC and ∠BEC = 180° (if D, E, C are colinear). But ∠ECB and ∠BEC are interior angles of ΔEBC, along with ∠EBC. So ∠BCF and ∠DEC: are they supplementary? Let's see, ∠BCF + ∠ECB = 180°, ∠DEC + ∠BEC = 180°. There's no direct relation to make them supplementary unless ∠ECB = ∠DEC, which is not necessarily true. So this statement is false.

5. **Analyze ∠BEC as a remote interior angle to exterior ∠BCF**: The exterior angle ∠BCF is at vertex C of ΔEBC. The remote interior angles to ∠BCF are the interior angles that are not adjacent to ∠BCF. The adjacent interior angle to ∠BCF is ∠ECB. So the remote interior angles are ∠BEC and ∠EBC. So ∠BEC is a remote interior angle to exterior ∠BCF. This statement is true.

6. **Re - analyze ∠DEC as an exterior angle**: Wait, maybe I made a mistake earlier. For ΔEBC, the exterior angle at E: if we consider side EB, and extend it to A, then ∠ABE is adjacent to ∠EBC. Wait, no, the exterior angle at E: the two sides at E are EB and EC. So when we extend EC to D, the angle ∠DEC is formed. Wait, ∠DEC and ∠BEC: if D, E, C are colinear, then ∠DEC is supplementary to ∠BEC. But is ∠DEC an exterior angle? The exterior angle of a triangle is formed by one side and the extension of another side. So for ΔEBC, side EC is extended to D, so the angle between ED (extension of EC) and EB is an exterior angle? Wait, no, the angle between ED and EB would be ∠DEB. But the option is ∠DEC. Wait, maybe the diagram shows that D, E, C are colinear, and E is connected to B. So ∠DEC: the angle at E between D and C, and E is connected to B. So ∠DEC is adjacent to ∠BEC. Wait, maybe the correct way: in triangle EBC, the exterior angle at E is ∠DEC? Wait, no, the interior angle at E is ∠BEC, and if we extend EC to D, then ∠DEC is supplementary to ∠BEC, but is it an exterior angle? An exterior angle should be equal to the sum of the two non - adjacent interior angles. Let's calculate: if ∠DEC is an exterior angle, then ∠DEC = ∠EBC+∠ECB. But ∠BEC + ∠EBC+∠ECB = 180°, and ∠DEC + ∠BEC = 180° (if D, E, C are colinear). So ∠DEC = 180° - ∠BEC, and ∠EBC + ∠ECB = 180° - ∠BEC. So ∠DEC = ∠EBC + ∠ECB. So yes, ∠DEC is an exterior angle of ΔEBC. So my earlier analysis was wrong. So ∠DEC is an exterior angle (formed by extending side EC to D, and the side EB). So this statement is true.

So the true statements are:
- ∠DEC is an exterior angle.
- ∠ABE and ∠EBC are supplementary angles.
- ∠BEC is a remote interior angle to exterior ∠BCF.

# Answer:
- ∠DEC is an exterior angle.
- ∠ABE and ∠EBC are supplementary angles.
- ∠BEC is a remote interior angle to exterior ∠BCF.

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