Exercise 3 Page 289

Consider the given triangle.

Looking at the labels on the given figure, we can see that $m\angle GFH=m\angle HGF.$ We can write this by using a congruence statement. Now, 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 have two congruent angles, triangle $FGH$ is an isosceles triangle. We want to find $\overline{FH}.$ To do so, we will use the Converse of the Isosceles Triangle Theorem.

Converse of the Isosceles Triangle Theorem

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

Using this theorem, let's show the congruent sides.

As a result, $\overline{GH}\cong \overline{FH}.$ Therefore, $FH=12.$