Rays PQ and QR are perpindicular. Point S lies on the interior of angle PQR. If m<PQS=4+7a and m<SQR = 9+4a, find m<PQS and m<SQR
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To find the measures of angles PQS and SQR, we can start by using the fact that rays PQ and QR are perpendicular.
Since PQ and QR are perpendicular, they form a right angle at point Q. This means that the sum of the measures of angles PQS and SQR will be equal to 90 degrees.
We are given that m<PQS = 4 + 7a and m<SQR = 9 + 4a.
To find the values of a and the measures of angles PQS and SQR, we need to set up an equation using the fact that the sum of the measures of angles PQS and SQR is 90 degrees.
So, we have:
m<PQS + m<SQR = 90
Substituting the given values, we get:
(4 + 7a) + (9 + 4a) = 90
Combining like terms and simplifying, we have:
13 + 11a = 90
Subtracting 13 from both sides, we get:
11a = 77
Dividing both sides by 11, we get:
a = 7
Now that we have the value of a, we can substitute it back into the original equations to find the measures of the angles.
First, substitute a = 7 into the equation for m<PQS:
m<PQS = 4 + 7(7)
= 4 + 49
= 53
So, m<PQS = 53.
Next, substitute a = 7 into the equation for m<SQR:
m<SQR = 9 + 4(7)
= 9 + 28
= 37
So, m<SQR = 37.