Rays PQ and QR are perpendicular. p;oint S lies in the interior of angle PQR. If angle PQS equals 4+7a and angle SQR equals 9+4a, find the angle PQS and angle SQR.

Can you please help me, I don't understand it.

the two angles must add up to 90º

so solve 4+7a + 9+4a = 90

(I got the two angles to be 53 and 37 degrees)

how do you solve the equation tho. I don't understand how to do that part. I got the answer i just don't know how to solve the equation.

4+7a + 9+4a = 90

11a = 90 - 4 - 9
11a = 77
a = 7

if a = 7, then 4+7a = 4+7(7) = 53
and 9+4a = 9 + 4(7) = 37

Thank you SOOOOOOOOOOOOOOOO much i really appreciate it. I really understand it much more now.

Of course! I can help you with that.

To find the angles PQS and SQR, we need to use the fact that rays PQ and QR are perpendicular. Let's break down the problem step by step.

Step 1: Identify the given information.
We are given that ray PQ and ray QR are perpendicular. Additionally, we know that point S lies in the interior of angle PQR. Finally, we are given the measures of angle PQS as 4+7a and angle SQR as 9+4a.

Step 2: Identify the relationship between the angles.
Since ray PQ and ray QR are perpendicular, this creates a right angle at point Q. Therefore, angle PQR is a right angle and measures 90 degrees.

Step 3: Write an equation using the given information.
We can use the fact that the sum of angles in a triangle is 180 degrees to write an equation:
angle PQS + angle SQR + angle PQR = 180 degrees

Step 4: Substitute the given angle measures into the equation.
We substitute angle PQS = 4+7a and angle SQR = 9+4a into the equation:
(4+7a) + (9+4a) + 90 = 180

Step 5: Solve the equation.
Combine like terms on the left side:
13 + 11a + 90 = 180
11a + 103 = 180
11a = 180 - 103
11a = 77
a = 77/11
a = 7

Step 6: Find the values of angle PQS and angle SQR.
Now, substitute the value of a = 7 into the expressions for the angles:
angle PQS = 4 + 7a = 4 + 7(7) = 4 + 49 = 53 degrees
angle SQR = 9 + 4a = 9 + 4(7) = 9 + 28 = 37 degrees

Therefore, angle PQS measures 53 degrees and angle SQR measures 37 degrees.