Question

complete the proof.
given: ∠b ≅ ∠d
\\(overline{ac} \cong overline{ec}\\)
prove: \\(overline{ab} \cong overline{ed}\\)
image of two triangles intersecting at point c, with ∠b and ∠d marked congruent, and \\(overline{ac}\\) and \\(overline{ec}\\) marked congruent
statements | reasons
∠b ≅ ∠d |
\\(overline{ac} \cong overline{ec}\\) | given
∠acb ≅ ∠ecd | dropdown
△abc ≅ △edc | dropdown
\\(overline{ab} \cong overline{ed}\\) | dropdown

Explanation:

Step1: Identify Vertical Angles

$\angle ACB$ and $\angle ECD$ are vertical angles, so they are congruent by the Vertical Angles Theorem.

Step2: Prove Triangle Congruence

We have $\angle B \cong \angle D$ (given), $\overline{AC} \cong \overline{EC}$ (given), and $\angle ACB \cong \angle ECD$ (vertical angles). By the AAS (Angle - Angle - Side) Congruence Theorem, $\triangle ABC \cong \triangle EDC$.

Step3: Corresponding Parts of Congruent Triangles

Since $\triangle ABC \cong \triangle EDC$, their corresponding sides $\overline{AB}$ and $\overline{ED}$ are congruent by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

Answer:

• For $\angle ACB \cong \angle ECD$: Vertical Angles Theorem
• For $\triangle ABC \cong \triangle EDC$: AAS (Angle - Angle - Side) Congruence Theorem
• For $\overline{AB} \cong \overline{ED}$: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)