I"m stuck on this question

Question

I"m stuck on this question

"Rays PQ and QR ae perpendicular.Point S lies in the interior of angle PQR.If angle PQS!=4+7a and angle SQR=9+4a,find angle PQS and angle SQR.
PLEASE HELP!!!!

Answer 1

4 + 7 a + 9 + 4 a = 90 degrees

13 + 11 a = 90
11 a = 77
a = 7

PQS = 4 + 49
SQR = 9 + 28

Answer 2

Thank you so much!!!!!

Answer 3

To solve this problem, we can start by using the fact that rays PQ and QR are perpendicular. This means that the angles formed at point Q are right angles (90 degrees).

Let's label angle PQS as x, so we have:

angle PQS = x (1)

We are given that angle SQR is equal to 9 + 4a. Since the angles in a triangle add up to 180 degrees, we can set up the following equation:

x + 9 + 4a + 90 = 180

Simplifying this equation, we have:

x + 4a + 99 = 180

Now, let's solve for x:

x = 180 - 4a - 99
x = 81 - 4a

So, we have found that angle PQS is equal to 81 - 4a.

Substituting this value back into equation (1), we have:

81 - 4a = 4 + 7a

Now, we can solve for a:

11a = 77
a = 7

Now that we have found the value of a, we can substitute it back into the expressions for angle PQS and angle SQR:

angle PQS = 81 - 4a
angle PQS = 81 - 4(7)
angle PQS = 81 - 28
angle PQS = 53

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

Therefore, the angle PQS is 53 degrees, and the angle SQR is 37 degrees.

Answer 4

To solve this problem, we need to use the fact that the sum of angles in a triangle is 180 degrees.

Given that rays PQ and QR are perpendicular, we can infer that angle PQR is a right angle (equal to 90 degrees).

Let's denote angle PQS as x and angle SQR as y.

According to the problem statement, we are given two equations for the angles:
1. angle PQS is not equal to 4 + 7a
2. angle SQR is equal to 9 + 4a

Since angle PQR is a right angle, the sum of angles PQS and SQR must equal 90 degrees.

x + y = 90 -- equation (1)

In addition, we have the first equation given:
x ≠ 4 + 7a -- equation (2)

Now, we can solve the system of equations to find the values of x and y.

To calculate angle SQR, we are given that it is equal to 9 + 4a:
y = 9 + 4a -- equation (3)

Substituting equation (3) into equation (1) will allow us to find the value of angle PQS:

x + (9 + 4a) = 90
x = 90 - 9 - 4a
x = 81 - 4a -- equation (4)

So, angle PQS is equal to 81 - 4a.

To find angle SQR, we already know its value from equation (3):
y = 9 + 4a

Therefore, angle SQR is equal to 9 + 4a.

Now we have the values of angle PQS and angle SQR. Angle PQS is 81 - 4a, and angle SQR is 9 + 4a.