given: $overline{su}$ bisects $angle tsw$ prove: $mangle usv + mangle vsw = mangle tsu$

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# Explanation:
## Step1: Define angle - bisector property
Since $\overline{SU}$ bisects $\angle TSW$, by the definition of an angle - bisector, $m\angle TSU=m\angle USW$.
## Step2: Express $\angle USW$ as a sum of angles
We know that $\angle USW=\angle USV+\angle VSW$ (by the Angle - Addition Postulate, which states that if point $V$ lies in the interior of $\angle USW$, then $m\angle USW = m\angle USV+m\angle VSW$).
## Step3: Substitute $\angle USW$
Substitute $\angle USW$ in the equation $m\angle TSU=m\angle USW$ with $\angle USV+\angle VSW$. So, $m\angle TSU=m\angle USV + m\angle VSW$.

# Answer:
| Statement | Justification |
|--|--|
| 1. $\overline{SU}$ bisects $\angle TSW$ | Given |
| 2. $m\angle TSU=m\angle USW$ | Definition of angle - bisector |
| 3. $\angle USW=\angle USV+\angle VSW$ | Angle - Addition Postulate |
| 4. $m\angle USW=m\angle USV + m\angle VSW$ | Definition of angle measure (if $\angle A=\angle B+\angle C$, then $m\angle A=m\angle B + m\angle C$) |
| 5. $m\angle TSU=m\angle USV + m\angle VSW$ | Substitution Property of Equality (substitute $m\angle USW$ in statement 2 with $m\angle USV + m\angle VSW$ from statement 4) |

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Question

Explanation:

Step1: Define angle - bisector property

Step2: Express $\angle USW$ as a sum of angles

Step3: Substitute $\angle USW$

Answer:

Statement	Justification
2. $m\angle TSU=m\angle USW$	Definition of angle - bisector
3. $\angle USW=\angle USV+\angle VSW$	Angle - Addition Postulate
4. $m\angle USW=m\angle USV + m\angle VSW$	Definition of angle measure (if $\angle A=\angle B+\angle C$, then $m\angle A=m\angle B + m\angle C$)
5. $m\angle TSU=m\angle USV + m\angle VSW$	Substitution Property of Equality (substitute $m\angle USW$ in statement 2 with $m\angle USV + m\angle VSW$ from statement 4)