Chapter 3.2, Problem 17PSB

To determine

To prove : The congruency of $\Delta ABC\cong \Delta DEC$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\angle 1\cong \angle 6$

  $\overrightarrow{BC}\cong \overrightarrow{EC}$

Formula used : If for any two triangles, the two angles and a side between them are congruent, then the triangles are said to be congruent by ASA property

Proof : We know that $\angle 3$ and $\angle 4$ are vertically opposite angles. Hence, we can say that

  $\angle 3\cong \angle 4$

Now, we also know that supplementary of congruent angles are congruent angles. Thus, we know that

  $\angle 1$ is supplementary to $\angle 2$ ,

  $\angle 6$ is supplementary to $\angle 5$ ,

And

  $\angle 1\cong \angle 6$

Hence, we can say that

  $\angle 2\cong \angle 5$

Thus, using the information that

  $\angle 2\cong \angle 5$

  $\angle 3\cong \angle 4$

  $\overrightarrow{BC}\cong \overrightarrow{EC}$

We can say that $\Delta ABC\cong \Delta DEC$ by ASA property

Hence, given congruency is proved