Question

line segments ad and be intersect at c, and triangles abc and dec are formed. they have the following characteristics: \\( \angle acb \\) and \\( \angle dce \\) are vertical angles \\( \angle b \cong \angle e \\) \\( \overline{bc} \cong \overline{ec} \\) which congruence theorem can be used to prove \\( \triangle abc \cong \triangle dec \\)? hl asa sss sas

Explanation:

Step1: Recall congruence theorems

• HL (Hypotenuse - Leg): For right - angled triangles. There is no information about right - angles in \(\triangle ABC\) and \(\triangle DEC\), so \(HL\) is not applicable.
• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
• We know \(\angle ACB=\angle DCE\) (vertical angles are equal), \(\angle B = \angle E\), 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 \(\triangle DEC\).
• SSS (Side - Side - Side): Three sides of one triangle are equal to three sides of another triangle. We only know one pair of sides (\(BC = EC\)) are equal, so \(SSS\) is not applicable.
• SAS (Side - Angle - Side): Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle. We don't have information about two pairs of sides being equal, so \(SAS\) is not applicable.

Answer:

ASA