Use the Pythagorean Theorem to find the base of the right triangle.

Now, because two of the angles in this triangle are the same, this is an isosceles triangle. In an isosceles triangle, the sides that are directly across from the congruent angles are also congruent.

In addition, the height in an isosceles triangle will always cut the 3rd side in half. With this information, fill out the triangle as shown below:

To find the perimeter, add up all the sides.

A triangle with three sides of different length is, by definition, scalene.

Below is regular Pentagon  with center , a segment drawn from  to each vertex - that is, each of its radii drawn.

By symmetry, all of the radii of a regular pentagon are congruent - specifically, . This triangle has at least two congruent sides, so it is isosceles.

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of  is the only choice that fits this criteria:

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of  is the only choice that fits this criteria:

The sum of the measures of the interior angles of a triangle is , so 

Substitute the given two angle measures and solve for :

Subtract  from both sides:

, so, by the Converse of the Isosceles Triangle Theorem, their opposite sides are also congruent - that is,

We are given that, in , two angles are congruent; specifically, . It is a consequence of the Converse of the Isosceles Triangle Theorem that the sides opposite the angles are also congruent - that is, ..

Having three sides of different lengths, this triangle is scalene. In any scalene triangle, the angle with greatest measure is opposite the longest side, and the angle with least measure is opposite the shortest side. Therefore, since , their opposite angles would be in order from greatest to least measure - that is, .

The dashes on the two sides of the triangle indicate that those two sides are congruent. Thus, the length of the missing side is  also.

Now, use the information given about the perimeter to set up an equation to solve for . The sum of all the sides will give the perimeter.

The dashes on the two sides of the triangle indicate that those two sides are congruent. Thus, the length of the missing side is  also.

Now, use the information given about the perimeter to set up an equation to solve for . The sum of all the sides will give the perimeter.