Rays PQ and QR are perpindicular. Point S lies i the interior of angle PQR. If the measure of the angle SQR=4+7a and the measure of angle SQR=9=4a, find the measure of angle of PQS and angle SQR.
angle PQS + angle PQR= 90
so, 4+7a+9+4a=90
7a+4a=90-4-9
11a=77
a=7
check:
PQS= 4+7a
4+7(7)=53
PQR= 9+4a
=9+4(a)=37
53+37=90
angle PQS + angle SQR= 90
so, 4+7a+9+4a=90
7a+4a=90-4-9
11a=77
a=7
check:
PQS= 4+7a
4+7(7)=53
SQR= 9+4a
=9+4(a)=37
53+37=90
To find the measures of angle PQS and angle SQR, we need to use the given information about the angles and the fact that rays PQ and QR are perpendicular.
First, let's restate the given information:
- The measure of angle SQR is 4 + 7a.
- The measure of angle PQS is 9 - 4a.
Now, since rays PQ and QR are perpendicular, we know that angle SQR and angle PQS are complementary angles (they add up to 90 degrees). In other words, SQR + PQS = 90 degrees.
Using this information, we can set up an equation:
(4 + 7a) + (9 - 4a) = 90.
Simplifying this equation, we get:
13 + 3a = 90.
Next, let's solve this equation for the value of 'a':
Subtract 13 from both sides: 3a = 77.
Divide both sides by 3: a = 77/3.
Now that we have the value of 'a', we can substitute it back into the expressions for the angles to find their measures:
Angle SQR = 4 + 7a = 4 + 7(77/3) = 4 + 539/3 = 4 + 179 2/3 = 183 2/3 degrees.
Angle PQS = 9 - 4a = 9 - 4(77/3) = 9 - 308/3 = 9 - 102 2/3 = 6 1/3 degrees.
Therefore, the measure of angle PQS is 6 1/3 degrees and the measure of angle SQR is 183 2/3 degrees.