Chapter 7.2, Problem 10PSA

To determine

To prove : The incongruency of $\angle H\ne \angle M$ .

Explanation of Solution

Given information : The following information has been given

  $\angle J=\angle O$

  $\overrightarrow{JK}\cong \overrightarrow{OP}$

  $\overrightarrow{HK}\ne \overrightarrow{MP}$

Formula used : Two triangles are congruent by SAS property iff they have two congruent sides, and the angle containing them is also congruent

Proof : We know that in triangles HJK and MOP,

  $\angle J=\angle O$

  $\overrightarrow{JK}\cong \overrightarrow{OP}$

But, $\overrightarrow{HK}\ne \overrightarrow{MP}$

Hence, we know that the two triangles are not congruent

Now, if the two triangles are not congruent, then by SAS property, if they have

  $\angle J=\angle O$

  $\overrightarrow{JK}\cong \overrightarrow{OP}$

  $\angle K\ne \angle P$

Now, we know that sum of the angles both triangles is equal to ${180}^0$ . Hence, we can say that

If $\angle K\ne \angle P$ , then $\angle H\ne \angle M$ (because $\angle J=\angle O$ is already given)

Thus, proved that $\angle H\ne \angle M$