Question

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

1. given: \\( \angle b a c \cong \angle e d c, \overline{b c} \cong \overline{e c} \\)

prove: \\( \triangle a b c \cong \triangle d e c \\)

1. given: \\( \overline{j k} \cong \overline{l m}, \angle j k m \cong \angle l m k \\)

prove: \\( \triangle j m k \cong \triangle l k m \\)

Explanation:

Step1: Identify vertical angles

In problem 7, $\angle ACB$ and $\angle DCE$ are vertical angles. By the Vertical Angles Theorem, $\angle ACB\cong\angle DCE$.

Step2: Apply the AAS (Angle - Angle - Side) congruence criterion

For $\triangle ABC$ and $\triangle DEC$, we have $\angle BAC\cong\angle EDC$ (given), $\angle ACB\cong\angle DCE$ (vertical angles), and $\overline{BC}\cong\overline{EC}$ (given). So, by AAS, $\triangle ABC\cong\triangle DEC$.

Step3: Identify vertical angles for problem 8

In problem 8, $\angle JMK$ and $\angle LKM$ are vertical angles. By the Vertical Angles Theorem, $\angle JMK\cong\angle LKM$.

Step4: Apply the AAS congruence criterion

For $\triangle JMK$ and $\triangle LKM$, we have $\overline{JK}\cong\overline{LM}$ (given), $\angle JKM\cong\angle LMK$ (given), and $\angle JMK\cong\angle LKM$ (vertical angles). So, by AAS, $\triangle JMK\cong\triangle LKM$.

Answer:

For problem 7:

1. $\angle ACB\cong\angle DCE$ (Vertical Angles Theorem)

For problem 8:

1. $\angle JMK\cong\angle LKM$ (Vertical Angles Theorem)