We are given that $\overrightarrow{PQ} \perp \overrightarrow{QR}$, which means that $\angle PQR$ is a right angle and $m\angle PQR = 90^\circ$. We are also given that point S lies in the interior of $\angle PQR$, $m\angle PQS = 4 + 7a$ and $m\angle SQR = 9 + 4a$. We want to find $m\angle PQS$ and $m\angle SQR$.

1. Problem Description

We are given that $\overrightarrow{PQ} \perp \overrightarrow{QR}$, which means that $\angle PQR$ is a right angle and $m\angle PQR = 90^\circ$. We are also given that point S lies in the interior of $\angle PQR$, $m\angle PQS = 4 + 7a$ and $m\angle SQR = 9 + 4a$. We want to find $m\angle PQS$ and $m\angle SQR$.

2. Solution Steps

Since point S lies in the interior of $\angle PQR$, we have:

$m\angle PQS + m\angle SQR = m\angle PQR$

We are given $m\angle PQS = 4 + 7a$, $m\angle SQR = 9 + 4a$, and we know $m\angle PQR = 90^\circ$.

Substituting the given values into the equation, we get:

$(4 + 7a) + (9 + 4a) = 90$

Combining like terms, we get:

$13 + 11a = 90$

Subtracting 13 from both sides, we get:

$11a = 90 - 13$

$11a = 77$

Dividing both sides by 11, we get:

$a = \frac{77}{11}$

$a = 7$

Now, we can find $m\angle PQS$ and $m\angle SQR$ by substituting $a = 7$ into the given expressions:

$m\angle PQS = 4 + 7a = 4 + 7(7) = 4 + 49 = 53$

$m\angle SQR = 9 + 4a = 9 + 4(7) = 9 + 28 = 37$

3. Final Answer

$m\angle PQS = 53^\circ$ and $m\angle SQR = 37^\circ$