Question: If ray $p$ bisects angle $a p c$ and if the measure of angle $a p b$ equals $2x + 9$ and the measure of angle $a p c$ equals $3x + 35$, then finds the measure of angle $a p c$
1. **State the problem:**
We are given that ray $p$ bisects angle $a p c$. The measure of angle $a p b$ is $2x + 9$ and the measure of angle $a p c$ is $3x + 35$. We need to find the measure of angle $a p c$.
2. **Understand the bisector property:**
If ray $p$ bisects angle $a p c$, then it divides angle $a p c$ into two equal angles. This means:
$$\text{measure of } \angle a p b = \text{measure of } \angle b p c$$
3. **Express the relationship:**
Since $p$ bisects $\angle a p c$, the whole angle $a p c$ is twice the measure of $a p b$:
$$\angle a p c = 2 \times \angle a p b$$
4. **Set up the equation:**
Given:
$$\angle a p b = 2x + 9$$
$$\angle a p c = 3x + 35$$
Using the bisector property:
$$3x + 35 = 2(2x + 9)$$
5. **Solve for $x$:**
$$3x + 35 = 4x + 18$$
Subtract $3x$ from both sides:
$$\cancel{3x} + 35 = \cancel{3x} + 4x - 3x + 18 \Rightarrow 35 = x + 18$$
Subtract $18$ from both sides:
$$35 - 18 = x + \cancel{18} - 18 \Rightarrow 17 = x$$
6. **Find the measure of angle $a p c$:**
Substitute $x = 17$ into $3x + 35$:
$$3(17) + 35 = 51 + 35 = 86$$
**Final answer:**
$$\boxed{86^\circ}$$