Question

use the given information to prove that δabc ≅ δdec. given: bc ≅ ec, ac ≅ dc prove: δabc ≅ δdec

Explanation:

Step1: Analyze ∠ACB and ∠DCE

∠ACB and ∠DCE are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for ∠ACB ≅ ∠DCE is "Vertical Angles are Congruent".

Step2: Prove Triangle Congruence

We have \( BC \cong EC \) (given), \( AC \cong DC \) (given), and \( \angle ACB \cong \angle DCE \) (vertical angles). By the Side - Angle - Side (SAS) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. So for \( \triangle ABC \cong \triangle DEC \), the reason is "SAS Congruence Postulate".

Answer:

• For step 3 (∠ACB ≅ ∠DCE), Reason: Vertical Angles are Congruent.
• For step 4 (△ABC ≅ △DEC), Reason: SAS Congruence Postulate.