Question

2 find m∠rqt. (4x + 15)° (10x - 3)° a) 59° b) 61° c) 63° d) 117° e) 119°

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. $\angle PQS$ and $\angle RQT$ are vertical angles. So, $4x + 15=10x - 3$.

Step2: Solve for $x$

Subtract $4x$ from both sides: $15 = 10x-4x - 3$, which simplifies to $15 = 6x - 3$. Then add 3 to both sides: $15 + 3=6x$, so $18 = 6x$. Divide both sides by 6: $x=\frac{18}{6}=3$.

Step3: Find the measure of $\angle RQT$

Substitute $x = 3$ into the expression for $\angle RQT$ (which is $10x - 3$). So, $m\angle RQT=10\times3 - 3=30 - 3 = 27$. But this is wrong. Let's use the fact that $\angle PQR$ and $\angle SQT$ are also vertical - angles and $\angle PQS+\angle SQT = 180^{\circ}$ (linear pair). Also, $\angle PQS = 4x + 15$ and $\angle SQT=10x - 3$. So, $(4x + 15)+(10x - 3)=180$.

Step4: Simplify the equation

Combine like - terms: $4x+10x+15 - 3 = 180$, which gives $14x+12 = 180$.

Step5: Solve for $x$ again

Subtract 12 from both sides: $14x=180 - 12=168$. Divide both sides by 14: $x = 12$.

Step6: Find $m\angle RQT$

Substitute $x = 12$ into the expression for $\angle RQT$ (which is $10x - 3$). So, $m\angle RQT=10\times12-3=120 - 3=117^{\circ}$.

Answer:

D. $117^{\circ}$