Chapter 12.5, Problem 23E

a.

To determine

To show: The additional information to prove $△JKL\cong △MKL$ by HL Congruence Theorem.

Answer to Problem 23E

The additional information is $JL\cong ML$ .

Explanation of Solution

Given information:

  

To support a tree, wires have been attached from the trunk of the tree to stakes in the ground.

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.

Calculation:

According to the question,

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

Now,

Statements	Reasons
H 1. $\overline{LK}\perp \overline{JK}$ , $\overline{LK}\perp \overline{KM}$ 2. $\angle JKL\text{ and }\angle MKL\text{ are right angles}\text{.}$ 3. $△ABC\text{ and }△FED\text{ are right triangles}\text{.}$	1. Given 2. Definition of $\perp$ lines. 3. Definition of a right triangle.
L 4. $\overline{KL}=\overline{KL}$	4.Reflexive Property of Congruence

The additional information to prove $△JKL\cong △MKL$ is :-

The hypotenuse should be equal.

  $JL\cong ML$

Hence, the additional information is $JL\cong ML$ .

b.

To determine

To name: The theorem used to prove $△JKL\cong △MKL$ .

Answer to Problem 23E

The SAS Congruence Theorem is used .

Explanation of Solution

Given information:

  

To support a tree, wires have been attached from the trunk of the tree to stakes in the ground.

K is the midpoint of JM.

Formula used:

Side-Angle-Side Congruence Theorem

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then two triangles are congruent.

Calculation:

According to the question,

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

Now,

Statements	Reasons
S 1. $\overline{JK}=\overline{MK}$	Given, K is the midpoint of JM.
A 2. $\angle JKL=\angle MKL$	Given, $\overline{LK}\perp \overline{JK}$ , $\overline{LK}\perp \overline{KM}$
S 3. $\overline{KL}=\overline{KL}$	4.Reflexive Property of Congruence
4. $△JKL\cong △MKL$	SAS Congruence Theorem

The SAS Congruence theorem is used to prove $△JKL\cong △MKL$ .