Exercise 39 Page 291

Consider the given triangle.

We want to find the value of $y.$ 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, the given triangle is an isosceles triangle. Then, we can consider the Isosceles Triangle Theorem to find $y.$

Isosceles Triangle Theorem

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

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

Now we can find $y$ by using the Triangle Sum Theorem. As a result, the value of $y$ will be either $\text{-}18$ or $14.$ However, $\text{-}18$ may not satisfy $2y-5$ because it is negative. Let's check! Since an angle measure cannot be negative, the only possible value of $y$ is $14.$