Exercise 20 Page 290

We are given a triangle and a pair of its interior angle measures. We also know that two of the side lengths are congruent.

We want to find the value of the variable $x.$ To do so, we will start by identifying the type of triangle. Let's recall the classification of triangles.

Classification of Triangles
Scalene Triangle	A scalene triangle is a triangle in which all three sides have different lengths.
Isosceles Triangle	An isosceles triangle is a triangle that has two congruent sides and two base angles with the same measure.
Equilateral Triangle	An equilateral triangle is a triangle in which all the sides are congruent.
Acute Triangle	An acute triangle is a triangle where all angles are less than $90^{\circ }$ or $\frac{\pi }{2}.$
Obtuse Triangle	An obtuse triangle is a triangle with exactly one an angle whose measure is greater than $90^{\circ }$ or $\frac{\pi }{2}.$
Right Triangle	A right triangle is a specific type of triangle that contains one angle of $90^{\circ }.$

Since the given triangle has two congruent sides, triangle $EHG$ is an isosceles triangle. Then, to find $x,$ we will first consider Isosceles Triangle Theorem.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Using the theorem, let's label the congruent angles.

Now, we can find $x$ by using the Triangle Angle-Sum Theorem. Recall that this theorem states that the measures of the interior angles of a triangle add up to $180^{\circ }.$