Question

1. given: ∠1 is supplementary to ∠2

∠3 is supplementary to ∠2
prove: m∠1 = m∠3

statements reasons

1. ∠1 is supplementary to ∠2 1. given
2. 2. definition of supplementary
3. 3. given
4. 4. definition of supplementary
5. 5. transitive property of equality
6. 6. subtraction property of equality

1: $overrightarrow{qs}$ bisects ∠pqt
$overrightarrow{qt}$ bisects ∠sqr
∠pqs ≅ ∠tqr

statements reasons

1. given

2.

Explanation:

Step1: Recall supplementary - angle definition

If $\angle1$ is supplementary to $\angle2$, then $m\angle1 + m\angle2=180^{\circ}$.

Step2: Apply to $\angle3$ and $\angle2$

Since $\angle3$ is supplementary to $\angle2$, we have $m\angle3 + m\angle2 = 180^{\circ}$.

Step3: Set up equations

Let $a=m\angle1$, $b = m\angle2$, $c=m\angle3$. So $a + b=180$ and $c + b=180$.

Step4: Use subtraction property

From $a + b=180$ and $c + b=180$, we can subtract the second - equation from the first: $(a + b)-(c + b)=180 - 180$.
Simplifying gives $a - c+(b - b)=0$, which simplifies to $a=c$. So $m\angle1=m\angle3$.

Step1: Use angle - bisector definition

Since $\overrightarrow{QS}$ bisects $\angle PQT$, then $\angle PQS=\angle SQT$. Since $\overrightarrow{QT}$ bisects $\angle SQR$, then $\angle SQT=\angle TQR$.

Step2: Use transitive property

Given $\angle PQS\cong\angle TQR$ (already given), and from the angle - bisector properties $\angle PQS=\angle SQT$ and $\angle SQT=\angle TQR$, we can use the transitive property of equality to make further conclusions about the relationships of the angles.

Answer:

The proof is completed as shown above.

For the second part about angle - bisectors: