Rays PQ and QR are perpendicular. Point S lies in the interior of angle PQR. If angle PQS= 4+7a and angle SQR=9+4a, fing angle PQS and angle SQR.
Angle PQR is 90 degrees.
4+7a + 9+4a = 90
Solve for a, and substitute back into the expressions for the angles
To find the measurements of angle PQS and angle SQR, we can set up an equation based on the given information and solve for the values of "a" first.
Given:
- Rays PQ and QR are perpendicular.
- Point S lies in the interior of angle PQR.
- Angle PQS measures 4 + 7a.
- Angle SQR measures 9 + 4a.
We know that the sum of angles in a triangle is 180 degrees. Since angle PQR is a right angle (90 degrees), we can use this information to set up the equation:
angle PQS + angle SQR + angle PQR = 180
Substituting the given measurements:
(4 + 7a) + (9 + 4a) + 90 = 180
Combining like terms:
13 + 11a = 180
Next, we can solve this equation for "a":
11a = 180 - 13
11a = 167
a = 167/11
a ≈ 15.18
Now that we have found the value of "a," we can substitute it back into the given expressions for angles PQS and SQR to find their respective measurements:
angle PQS = 4 + 7a
angle PQS = 4 + 7(15.18)
angle PQS ≈ 4 + 106.26
angle PQS ≈ 110.26
angle SQR = 9 + 4a
angle SQR = 9 + 4(15.18)
angle SQR ≈ 9 + 60.72
angle SQR ≈ 69.72
Hence, angle PQS measures approximately 110.26 degrees, and angle SQR measures approximately 69.72 degrees.