Chapter 12.5, Problem 31E

To determine

To show: Whether the $△JKL\text{ is congruent to }△\text{LMJ }$ by SSS Congruence Theorem or $△JKL\text{ is congruent to }△\text{LMJ }$ by the HL Congruence Theorem statement is true.

Answer to Problem 31E

Both the ways are correct.

Explanation of Solution

Given information:

  

Formula used:

The Hypotenuse-Leg Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle , then the two triangles are congruent.

Side-Side-Side Congruence Theorem

If three sides of one triangle are congruent to three sides of a second triangle, then two triangles are congruent.

Calculation:

According to the question,

There are two triangles $△JKL\text{ and }△LMJ\text{.}$

Also, $△JKL\cong △LJM\text{.}$

Now,

Statements	Reasons
S 1. $\overline{JK}=\overline{LM}$	Given
S 2. $\overline{KL}=\overline{MJ}$	Given
S 3. $\overline{JL}=\overline{LJ}$	Reflexive Property of Congruence
4. $△JKL\cong △LMJ$	SSS Congruence Theorem

Therefore, the triangles are congruent by SSS Congruence Theorem .

Now, $△JKL\cong △LJM\text{.}$

Statements	Reasons
H  1. $\overline{JL}=\overline{LJ}$   2. $\overline{JM}\perp \overline{ML}$ , $\overline{LK}\perp \overline{KJ}$   3. $\angle M\text{ and }\angle K\text{ are right angles}\text{.}$   4. $△JKL\text{ and }△LMJ\text{ are right triangles}\text{.}$	1.Given 2.Given 3. Definition of $\perp$ lines. 4. Definition of a right triangle.
L  5. $\overline{JM}=\overline{KL}$   6. $△JKL\cong △LMJ$	5. Reflexive Property of Congruence 6. HL Congruence Theorem

Therefore, the triangles are congruent by HL Congruence Theorem .

Hence, both the ways are correct.