Question

triangle congruence
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?

Explanation:

Step1: Recall congruence theorems

• \(HL\) (Hypotenuse - Leg) is for right - angled triangles. We have no information about right - angles here.
• \(ASA\) (Angle - Side - Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
• \(SAS\) (Side - Angle - Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
• \(SSS\) (Side - Side - Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Step2: Analyze given information

We know that \(\angle ACB\) and \(\angle DCE\) are vertical angles (\(\angle ACB=\angle DCE\)), \(\angle B=\angle E\), and \(BC = EC\). The side \(BC\) is between \(\angle B\) and \(\angle ACB\) in \(\triangle ABC\), and the side \(EC\) is between \(\angle E\) and \(\angle DCE\) in \(\triangle DEC\).

Answer:

ASA