Question

complete the proofs using the most appropriate method. some may require cpctc.

1. given: $\angle bac \cong \angle edc$, $\overline{bc} \cong \overline{ec}$

prove: $\delta abc \cong \delta dec$
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$$\begin{tabular}{|l|l|} \\hline statements & reasons \\\\ \\hline 1. $\\angle bac \\cong \\angle edc$ & 1. given \\\\ \\hline 2. $\\overline{bc} \\cong \\overline{ec}$ & 2. given \\\\ \\hline 3. & 3. \\\\ \\hline 4. & 4. \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Identify vertical angles

$\angle ACB \cong \angle DCE$

Step2: State vertical angles theorem

Vertical angles are congruent.

Step3: State triangle congruence

$\triangle ABC \cong \triangle DEC$

Step4: Apply AAS congruence rule

Two angles and a non-included side of one triangle are congruent to the corresponding parts of the other triangle.

Answer:

Statements	Reasons
2. $\overline{BC} \cong \overline{EC}$	2. Given
3. $\angle ACB \cong \angle DCE$	3. Vertical Angles Theorem
4. $\triangle ABC \cong \triangle DEC$	4. AAS Congruence Rule