Question

t is in the interior of ∠rsu. find each of the following. 15. m∠rsu if m∠rst = 38° and m∠tsu = 28.6°. 16. m∠rst if m∠tsu = 46.7° and m∠rsu = 83.5°. multi - step $overrightarrow{sp}$ bisects ∠rst. find each of the following. 17. m∠rst if m∠rsp=(3x - 2)° and m∠pst=(9x - 26)°. 18. m∠rsp if m∠rst = $\frac{5}{2}y$° and m∠pst=(y + 5)°

Explanation:

Step1: Recall angle - addition postulate

If a point $T$ is in the interior of $\angle RSU$, then $m\angle RSU=m\angle RST + m\angle TSU$.

Step2: Solve for 15

Given $m\angle RST = 38^{\circ}$ and $m\angle TSU=28.6^{\circ}$, then $m\angle RSU=38^{\circ}+ 28.6^{\circ}=66.6^{\circ}$.

Step3: Solve for 16

Since $m\angle RSU=m\angle RST + m\angle TSU$, then $m\angle RST=m\angle RSU - m\angle TSU$. Substituting $m\angle TSU = 46.7^{\circ}$ and $m\angle RSU = 83.5^{\circ}$, we get $m\angle RST=83.5^{\circ}-46.7^{\circ}=36.8^{\circ}$.

Step4: Recall angle - bisector property

If $\overrightarrow{SP}$ bisects $\angle RST$, then $m\angle RSP=m\angle PST$.

Step5: Solve for 17

Set $m\angle RSP=m\angle PST$. So, $3x - 2=9x - 26$.
Subtract $3x$ from both sides: $- 2=6x - 26$.
Add 26 to both sides: $24 = 6x$.
Divide both sides by 6: $x = 4$.
Then $m\angle RSP=3x - 2=3\times4 - 2=10^{\circ}$ and $m\angle RST=m\angle RSP + m\angle PST=2m\angle RSP=2\times10^{\circ}=20^{\circ}$.

Step6: Solve for 18

Since $m\angle RSP=m\angle PST$ and $m\angle RST=m\angle RSP + m\angle PST$, and $m\angle RST=\frac{5}{2}y^{\circ}$ and $m\angle PST=(y + 5)^{\circ}$, then $\frac{5}{2}y=2(y + 5)$.
Multiply both sides by 2 to get $5y = 4(y + 5)$.
Expand: $5y=4y + 20$.
Subtract $4y$ from both sides: $y = 20$.
So $m\angle RSP=y + 5=20+5 = 25^{\circ}$.

Answer:

1. $66.6^{\circ}$
2. $36.8^{\circ}$
3. $20^{\circ}$
4. $25^{\circ}$