The first statement, XZ ⊥ XY and XY = XZ = 5 proves that the triangle XYZ is isosceles.
One isosceles triangle has two congruent sides and two congruent angles.
And you should also know the theorem that states that if two sides of a triangle are congruent then the two angles opposite that sides are also congruent.
So, given that the two sides XZ and XY are congruent, their opposite sides are also congruent, and that proves that the the triangle is isosceles.