We are given that triangle DEF has two congruent sides adjacent to the vertex angle of 90.° Because of Thai fact, we can apply the Isosceles Triangle Theorem and the Triangle Sum Theorem.
Here are these two theorems:
Isosceles Triangle Theorem: if two sides of a triangle are congruent, then the base angles opposite those sides are congruent.
Triangle Sum Theorem: all three interior angles of a triangle sum up to 180.°
Because we have two congruent sides in the triangle, the base angles opposite these sides are congruent.
This means that m∠DEF ≅ m∠FDE
We also know that the vertex angle is 90° and that the two base angles are congruent, so we can create an equation to solve for x:
2x+90=180
Since the two base angles are congruent, they are denoted with the same variable, “x,” because they are the same measure. Since there are two congruent angles, 2x can represent x+x.
Now, solve for x:
Subtract 90 from both sides:
2x=180-90
2x=90
Divide by 2 on both sides:
x=90/2
x=45
Therefore, x=45°