Question

complete the proof by dragging the statemts and reasons in the correct order.

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 \\)

Explanation:

Step1: Fill in the first two statements

Since the first two reasons are "Given", the first two statements should be the given information. So statement 1 is $\angle BAC\cong\angle EDC$ and statement 2 is $\overline{BC}\cong\overline{EC}$.

Step2: Determine the reason for $\angle ACB\cong\angle DCE$

$\angle ACB$ and $\angle DCE$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for $\angle ACB\cong\angle DCE$ is "Vertical Angles".

Step3: Prove $\triangle ABC\cong\triangle DEC$

We have two angles ($\angle BAC\cong\angle EDC$ and $\angle ACB\cong\angle DCE$) and a non - included side ($\overline{BC}\cong\overline{EC}$) congruent. By the Angle - Angle - Side (AAS) congruence criterion, $\triangle ABC\cong\triangle DEC$. So statement 4 is $\triangle ABC\cong\triangle DEC$ and the reason 4 is "Angle Angle Side (AAS)".

Answer:

1. $\angle BAC\cong\angle EDC$; 2. $\overline{BC}\cong\overline{EC}$; 3. Vertical Angles; 4. $\triangle ABC\cong\triangle DEC$, Angle Angle Side (AAS)