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

Explanation:

To determine the congruence theorem for \(\triangle ABC \cong \triangle DEC\), we analyze the given information:

• \(\angle ACB\) and \(\angle DCE\) are vertical angles, so \(\angle ACB \cong \angle DCE\) (vertical angles are congruent).
• \(\angle B \cong \angle E\) (given).
• \(\overline{BC} \cong \overline{EC}\) (given).

The ASA (Angle - Side - Angle) congruence theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. Here, we have two angles (\(\angle B \cong \angle E\), \(\angle ACB \cong \angle DCE\)) and the included side (\(\overline{BC} \cong \overline{EC}\)) congruent, so ASA applies.

HL (Hypotenuse - Leg) is for right triangles, SSS (Side - Side - Side) requires three sides, and SAS (Side - Angle - Side) requires two sides and the included angle (here, the angle is not between the two sides in the given info).

Answer:

ASA (the option with "ASA" text)