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.
since they are perpendicular,
4+7a + 9+4a = 90
11a = 77
a = 7
so angle PQS = 4+7a = 4+49 = 53
and angle SQR = 9+4a =9+28 = 37
check: is 53+73 = 90 ? YES
To find the measure of angles PQS and SQR, we can use the property that the sum of angles in a triangle is 180 degrees.
Given that angles PQS and SQR are part of triangle PQR and that PQ and QR are perpendicular, we can use the fact that a straight angle measures 180 degrees to solve for the missing angles:
Sum of angles PQR + PQS + SQR = 180 degrees
Since PQ and QR are perpendicular, we know that angle PQR is a right angle and measures 90 degrees:
90 + (4 + 7a) + (9 + 4a) = 180
Simplifying the equation:
103 + 11a = 180
Subtracting 103 from both sides:
11a = 77
Dividing both sides by 11:
a = 7
Now, we can substitute the value of a back into angle PQS and angle SQR:
angle PQS = 4 + 7a = 4 + 7(7) = 4 + 49 = 53 degrees
angle SQR = 9 + 4a = 9 + 4(7) = 9 + 28 = 37 degrees
Therefore, the measure of angle PQS is 53 degrees and the measure of angle SQR is 37 degrees.
To find the angles PQS and SQR, we can use the fact that the sum of angles in a triangle is 180 degrees.
Since PQ and QR are perpendicular, angle PQR is a right angle, which means it measures 90 degrees.
Therefore, we can set up the following equation:
angle PQS + angle SQR + angle PQR = 180 degrees.
Substituting the given values, we have:
(4 + 7a) + (9 + 4a) + 90 = 180.
Combining like terms, we get:
13 + 11a + 90 = 180.
Simplifying further:
103 + 11a = 180.
Now, we can isolate the variable:
11a = 180 - 103.
11a = 77.
To solve for a, we divide both sides of the equation by 11:
a = 77 / 11.
Simplifying, we find that a = 7.
Now, substituting this value into the expressions for angle PQS and SQR:
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.