Questions: Use the diagram to answer the question. Find each of the following. xz ≈ m angle X ≈ m angle Z ≈

Solution Steps

The sum of angles in a triangle is 180°. We are given angle Y = 99° and side XY=10.5, side YZ=8. We are looking for XZ, angle X, and angle Z. Since we know one angle, we can find the measure of angle Z using the Law of Sines:

sin(Z)/XY = sin(Y)/XZ However, we don't yet know XZ.

Let's use the law of cosines to find XZ first.

XZ² = XY² + YZ² - 2(XY)(YZ)cos(Y)

XZ² = 10.5² + 8² - 2(10.5)(8)cos(99)

XZ² = 110.25 + 64 - 168(-0.156)

XZ² ≈ 174.25 + 26.208

XZ² ≈ 200.458

XZ ≈ √200.458

XZ ≈ 14.16

Use the Law of Sines to find m∠Z:

sin(Z)/10.5 = sin(99)/14.16

sin(Z) ≈ (10.5 * 0.988)/14.16

sin(Z) ≈ 0.73

Z = arcsin(0.73)

Z ≈ 46.9°

We know that the sum of angles in a triangle is 180°. Therefore:

m∠X + m∠Y + m∠Z = 180°

m∠X + 99° + 46.9° = 180°

m∠X = 180° - 99° - 46.9°

m∠X ≈ 34.1°

Final Answer: