here is a diagram.
overline{ab}congoverline{dc}
overline{ac}congoverline{db}
angle acongangle d
select all statements that are true about the triangles.
a triangles abc and dcb are congruent by the angle - angle triangle congruence theorem.
b triangles abc and bcd are congruent by the angle - side - angle triangle congruence theorem.
c triangles abc and bcd are congruent by the side - side - side triangle congruence theorem.
d triangles abc and dcb are congruent by the side - angle - side triangle congruence theorem.
e triangles abc and dcb are congruent by the side - side - side triangle congruence theorem.
f there is not enough information to determine if the triangles are congruent.

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Accepted Answer

# Brief Explanations:
- For option A: The Angle - Angle (AA) congruence theorem requires two angles, but here we have side - angle - side or side - side - side conditions, so A is wrong.
- For option B: The triangles in question are $ \triangle ABC $ and $ \triangle DCB $ (not $ \triangle BCD $ in the way the option describes), and the ASA theorem is not applicable here as the sides and angles don't match the ASA configuration, so B is wrong.
- For option C: Similar to B, the triangles are $ \triangle ABC $ and $ \triangle DCB $, and SSS is not the correct theorem here (though we do have three sides, but the correct correspondence is with SAS or SSS? Wait, let's re - evaluate. Wait, we know $ \overline{AB}\cong\overline{DC} $, $ \overline{AC}\cong\overline{DB} $, and $ \overline{BC} $ is common? Wait no, in $ \triangle ABC $ and $ \triangle DCB $, $ \overline{AB}\cong\overline{DC} $, $ \angle A\cong\angle D $, $ \overline{AC}\cong\overline{DB} $, so it's SAS. Also, if we consider $ \overline{AB}\cong\overline{DC} $, $ \overline{BC}\cong\overline{CB} $ (common side), $ \overline{AC}\cong\overline{DB} $, then it's SSS. Wait, let's check the given information again. The problem states $ \overline{AB}\cong\overline{DC} $, $ \overline{AC}\cong\overline{DB} $, and $ \angle A\cong\angle D $. Also, $ \overline{BC} $ is a common side for $ \triangle ABC $ and $ \triangle DCB $. So for $ \triangle ABC $ and $ \triangle DCB $:
    - SSS: $ \overline{AB}\cong\overline{DC} $, $ \overline{AC}\cong\overline{DB} $, $ \overline{BC}\cong\overline{CB} $ (so E is correct as it says triangles $ ABC $ and $ DCB $ are congruent by SSS).
    - SAS: $ \overline{AB}\cong\overline{DC} $, $ \angle A\cong\angle D $, $ \overline{AC}\cong\overline{DB} $ (so D is correct as it says triangles $ ABC $ and $ DCB $ are congruent by SAS). Wait, maybe I made a mistake earlier. Let's re - check the options:
        - Option D: Triangles $ ABC $ and $ DCB $ are congruent by SAS. We have $ \overline{AB}\cong\overline{DC} $, $ \angle A\cong\angle D $, $ \overline{AC}\cong\overline{DB} $, which is SAS (side - angle - side: two sides and the included angle).
        - Option E: Triangles $ ABC $ and $ DCB $ are congruent by SSS. We have $ \overline{AB}\cong\overline{DC} $, $ \overline{AC}\cong\overline{DB} $, $ \overline{BC}\cong\overline{CB} $ (three sides), so SSS holds.
    - Option C is wrong because it refers to $ \triangle ABC $ and $ \triangle BCD $ (incorrect triangle names, should be $ DCB $).
    - Option F is wrong because we have enough information.
- So, option D (SAS) and option E (SSS) are correct. Wait, let's re - examine the options:
    - Option D: "Triangles $ ABC $ and $ DCB $ are congruent by the Side - Angle - Side Triangle Congruence Theorem." Given $ \overline{AB}\cong\overline{DC} $, $ \angle A\cong\angle D $, $ \overline{AC}\cong\overline{DB} $, this is SAS (the angle is included between the two sides).
    - Option E: "Triangles $ ABC $ and $ DCB $ are congruent by the Side - Side - Side Triangle Congruence Theorem." Since $ \overline{AB}\cong\overline{DC} $, $ \overline{AC}\cong\overline{DB} $, and $ \overline{BC}\cong\overline{CB} $ (common side), this is SSS.

# Answer:
D. Triangles $ ABC $ and $ DCB $ are congruent by the Side - Angle - Side Triangle Congruence Theorem.
E. Triangles $ ABC $ and $ DCB $ are congruent by the Side - Side - Side Triangle Congruence Theorem.

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