Let's begin by drawing △ XYZ and labeling the lengths of the sides. We will also color code the opposite angles and sides. It will help us use the Law of Sines and Law of Cosines later.
Let's find the measures of XY, ∠ X, and ∠ Y one at a time.
Finding XY
The measures of two sides and one angle of the triangle are given. Therefore, we can use the Law of Cosines to find XY.
X Y^2=X Z^2+Y Z^2 -2 * X Z * Y Z * cos Z
Let's substitute X Z= 8.9, Y Z= 9.9, and Z= 63^(∘) to isolate X Y.
Now that we know the triangle is isosceles, we know that measure of ∠ X is the same as measure of ∠ Z, so m∠ X=63^(∘).
Finding m ∠ Y
Finally, to find m ∠ Y we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180.
m ∠ Y+ 63+ 63 = 180 ⇔ m ∠ Y ≈ 54
Completing the Triangle
With all of the angle measures, we can complete our diagram.
