Question

Line segments AD and BE intersect at C, and triangles ABC and DEC are formed. They have the following characteristics:



∠ACB and ∠DCE are vertical angles

∠B ≅ ∠E

BC ≅ EC

Which congruence theorem can be used to prove △ABC ≅ △DEC?

HL
ASA
SSS
SAS

Answer 1

To determine which congruence theorem can be used to prove triangles \( \triangle ABC \) and \( \triangle DEC \) congruent based on the given information, let's analyze the characteristics provided:

1. **Vertical angles**: \( \angle ACB \) and \( \angle DCE \) are vertical angles, so they are congruent:
\[
\angle ACB \cong \angle DCE
\]

2. **Angle congruence**: \( \angle B \cong \angle E \) is given.

3. **Side congruence**: \( BC \cong EC \) is also given.

Now, let's summarize what we have:

- Two angles in triangle \( ABC \) are \( \angle ACB \) and \( \angle B \).
- Two angles in triangle \( DEC \) are \( \angle DCE \) and \( \angle E \).
- One side in each triangle is \( BC \) and \( EC \).

Using this information:
- We have \( \angle ACB \cong \angle DCE \) (vertical angles),
- We also know \( \angle B \cong \angle E \) (given),
- And we know \( BC \cong EC \) (given).

With two angles and the included side congruent in both triangles, we can apply the **Angle-Side-Angle (ASA)** congruence theorem.

Thus, the answer is:
\[
\text{ASA}
\]