Question

For the following figure, is the statement “ $\overline{MK}\cong \overline{NK}$if and only if $\angle 3\cong \angle 4$” is true or false.

Short answer

The statement is a true statement.

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Apply the concept of isosceles triangle.

Property of isosceles triangle states that in an isosceles triangle, sides opposite to equal opposite angles are also congruent.

Step 3. Step description.

From the figure, consider the triangle $\mathrm{\Delta }KMN$and $\overline{MK}\cong \overline{NK}$.

Since from isosceles triangle theorem if two sides of a triangle are congruent, then the angles opposite those sides are congruent thus $\angle 3\cong \angle 4$.

Therefore, $\angle 3\cong \angle 4$.

Prove the converse as follows:

In $\mathrm{\Delta }KMN$, $\angle 3\cong \angle 4$that is the base angles are congruent.

Thus$\mathrm{\Delta }KMN$ is an isosceles triangle.

Since in an isosceles triangle, sides opposite to these congruent angles will also be congruent thus$\overline{MK}\cong \overline{NK}$

Therefore, the congruent sides are $\overline{MK}\cong \overline{NK}$.

Thus, the statement is true statement.