Chapter 3.2, Problem 27PSC

To determine

To show that two triangles are congruent.

Answer to Problem 27PSC

  $\Delta FJKand\Delta HGMarecongruent.$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{JH} \cong \overline{FG};\\ KandMaremidpoints.\end{array}$

  $\begin{array}{l}\angle HKF \cong \angle FMH,\hfill \\ \angle KJG \cong \angle MGJ,\hfill \\ \angle JGH \cong \angle FJG.\hfill \end{array}$

Formula used:

The below properties are used:

Halves of congruent segments are congruent.

If two angles form a straight angle, they are supplementary.

Supplementary of congruent angles are congruent.

Two triangles are congruent by ASA congruence rule.

Proof:

The below two statements are given,

  $\begin{array}{l}\overline{JH} \cong \overline{FG};\\ KandMaremidpoints.\end{array}$

  

Halves of congruent segments are congruent.

  $\overline{JK} \cong \overline{MG}$

It is given that,

  $\angle HKF \cong \angle FMH$

If two angles form a straight angle, they are supplementary.

  $\begin{array}{l}\angle 1issupplementaryto\angle HKF.\hfill \\ \angle 2issupplementaryto\angle FMH.\hfill \end{array}$

Supplementary of congruent angles are congruent.

  $\angle 1\cong \angle 2$

The below two statements are given:

  $\begin{array}{l}\angle KJG\cong \angle MGJ\\ \angle FJG\cong \angle JGH\end{array}$

By addition property,

  $\angle KJF\cong \angle HGM$

By ASA congruence rule,

  $\Delta FJK \cong \Delta HGM$