Chapter 3.2, Problem 12PSA

To determine

To prove : The congruency of $\Delta ZOA\cong \Delta XOY$ is to be proved

Explanation of Solution

Given information : The following information has been given

O is the mid-point of AY

O is the mid-point of ZX

Formula used : If two sides and the angle between them are congruent for any two triangles, then the triangles are said to be congruent by SAS property

Proof : we know that O is the mid-point of AY. Hence, we can say that

  $\overrightarrow{AO}\cong \overrightarrow{YO}$

And, we also know that O is the mid-point of ZX. Hence, we can also say that

  $\overrightarrow{ZO}\cong \overrightarrow{XO}$

Also, we can say that

  $\angle AOZ\cong \angle XOY$

Because they are vertically opposite angles

Now, as we know that

  $\overrightarrow{AO}\cong \overrightarrow{YO}$

  $\overrightarrow{ZO}\cong \overrightarrow{XO}$

  $\angle AOZ\cong \angle XOY$

Hence, we can say that the two triangles are congruent by SAS property mentioned above