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
\overline{bc} ≅ \overline{ec}
which congruence theorem can be used to prove △abc ≅ △dec?
○ hl
○ asa
○ sss
○ sas

Explanation:

Step1: Recall the ASA congruence theorem

ASA (Angle - Side - Angle) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Step2: Identify the elements in \(\triangle ABC\) and \(\triangle DEC\)

• We know that \(\angle ACB\cong\angle DCE\) (vertical angles are congruent).
• \(\angle B\cong\angle E\) (given).
• \(\overline{BC}\cong\overline{EC}\) (given).

The side \(\overline{BC}\) is between \(\angle B\) and \(\angle ACB\) in \(\triangle ABC\), and the side \(\overline{EC}\) is between \(\angle E\) and \(\angle DCE\) in \(\triangle DEC\). So, we have two angles and the included side congruent.

HL (Hypotenuse - Leg) is for right - angled triangles. SSS (Side - Side - Side) requires three pairs of congruent sides. SAS (Side - Angle - Side) requires two sides and the included angle. Here, ASA is the correct congruence theorem.

Answer:

ASA