Chapter 3.2, Problem 18PSB

To determine

To prove : The congruency of $\Delta GHJ\cong \Delta MKJ$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{FH}\cong \overrightarrow{FK}$

  $\angle H\cong \angle K$

G is midpoint of FH

M is midpoint of FK

J is midpoint of HK

Formula used : If two triangles have two congruent sides and the angle containing them is also congruent, then we can say that the triangles are congruent by SAS (side-angle-side) property

Proof : We know that

J is midpoint of HK, thus, we can say that

  $\overrightarrow{HJ}\cong \overrightarrow{JK}$

Now, we know that $\overrightarrow{FH}\cong \overrightarrow{FK}$ , and

G is midpoint of FH

M is midpoint of FK

Thus, from this set of information, we can say that

  $\overrightarrow{FG}\cong \overrightarrow{GH}\cong \overrightarrow{MK}\cong \overrightarrow{FM}$

So, we can also say that

  $\overrightarrow{GH}\cong \overrightarrow{MK}$

Now, using the information

  $\overrightarrow{GH}\cong \overrightarrow{MK}$

  $\overrightarrow{HJ}\cong \overrightarrow{JK}$

  $\angle H\cong \angle K$

We can say that $\Delta GHJ\cong \Delta MKJ$ by SAS property

Hence, the given congruency is proved