Question: Prove this property of angle bisectors: If you know $\overrightarrow{MP}$ bisects $\angle LMN$, prove that two times $m\angle LMP$ is $m\angle LMN$.
1. **State the problem:**
We are given that $\overrightarrow{MP}$ bisects $\angle LMN$. We need to prove that $$2 \times m\angle LMP = m\angle LMN$$.
2. **Recall the definition of angle bisector:**
By definition, if $\overrightarrow{MP}$ bisects $\angle LMN$, then it divides the angle into two equal angles:
$$m\angle LMP = m\angle PMN$$
3. **Use the Angle Addition Postulate:**
The measure of the whole angle $\angle LMN$ is the sum of the two adjacent angles formed by the bisector:
$$m\angle LMN = m\angle LMP + m\angle PMN$$
4. **Substitute the equality from step 2:**
Since $m\angle LMP = m\angle PMN$, substitute to get:
$$m\angle LMN = m\angle LMP + m\angle LMP$$
5. **Simplify the right side:**
$$m\angle LMN = 2 \times m\angle LMP$$
6. **Conclusion:**
We have shown that twice the measure of $\angle LMP$ equals the measure of $\angle LMN$, which completes the proof.
**Final answer:** $$2 \times m\angle LMP = m\angle LMN$$