Question: 16. $\overrightarrow{PD}$ is an angle bisector of $\angle BPE$,
m$\angle 4 = x + 30$ and m$\angle BPD = 3x - 10$. Find $x$.
17. $\overrightarrow{PY}$ bisects $\angle XYZ$, m$\angle XYP = 4x - 58$, and
m$\angle PYZ = 2x - 14$. Solve for $x$ and find
m$\angle XYZ$
1. **Problem 16:** Given that $\overrightarrow{PD}$ bisects $\angle BPE$, the two angles formed by the bisector are equal.
2. This means:
$$m\angle BPD = m\angle 4$$
3. Substitute the given expressions:
$$3x - 10 = x + 30$$
4. Solve for $x$:
$$3x - 10 = x + 30$$
$$3x - x = 30 + 10$$
$$2x = 40$$
$$x = \frac{40}{2}$$
$$x = 20$$
5. **Answer for problem 16:** $x = 20$
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6. **Problem 17:** $\overrightarrow{PY}$ bisects $\angle XYZ$, so:
$$m\angle XYP = m\angle PYZ$$
7. Substitute the given expressions:
$$4x - 58 = 2x - 14$$
8. Solve for $x$:
$$4x - 58 = 2x - 14$$
$$4x - 2x = -14 + 58$$
$$2x = 44$$
$$x = \frac{44}{2}$$
$$x = 22$$
9. Find $m\angle XYZ$ by adding the two equal angles:
$$m\angle XYZ = m\angle XYP + m\angle PYZ = (4x - 58) + (2x - 14)$$
$$= 6x - 72$$
10. Substitute $x=22$:
$$m\angle XYZ = 6(22) - 72 = 132 - 72 = 60$$
11. **Answer for problem 17:** $x = 22$, $m\angle XYZ = 60$ degrees