The figure is an illustration of the relationship between the angles in a circle and arc
The measure of angle PSQ
From the figure, the arc PQ is subtended by the angle PAQ.
This means that:
PQ = ∠PAQ
Given that ∠PAQ = 130, it means that:
Arc PQ = 130
The measure of PSQ is then calculated using:
∠PSQ = 0.5 * Arc PQ ----- inscribed angle is half a subtended angle.
This gives
∠PSQ = 0.5 * 130
∠PSQ = 65
Hence, the measure of ∠PSQ is 65 degrees
The measure of arc QR
A semicircle measures 180 degrees.
This means that:
QR + PQ = 180
So, we have:
QR = 180 - PQ
Substitute 130 for PQ
QR = 180 - 130
QR = 50
Hence, the measure of QR is 50 degrees
The measure of arc RS
The measure of arc RS is then calculated using:
∠RPS = 0.5 * Arc RS ----- inscribed angle is half a subtended angle.
Where ∠RPS = 35
So, we have:
35 = 0.5 * Arc RS
Multiply both sides by 2
Arc RS = 70
Hence, the measure of RS is 70 degrees
The measure of angle AQS
In (a), we have:
∠PSQ = 65
This means that:
∠PSQ = ∠PSB = 65
So, we have:
∠PSB = 65
Next, calculate SBP using:
∠SBP + ∠BPS + ∠PSB = 180 ---- sum of angles in a triangle.
So, we have:
∠SBP + 35 + 65 = 180
∠SBP + 100 = 180
This gives
∠SBP = 80
The measure of AQS is then calculated using:
AQS = AQB = 180 - (180 - SBP) - (180 - PAQ)
This gives
AQS = 180 - (180 - 80) - (180 - 130)
Evaluate
AQS = 30
Hence, the measure of AQS is 30 degrees
The measure of arc PS
A semicircle measures 180 degrees.
This means that:
PS + RS = 180
This gives
PS = 180 - RS
Where RS = 70
So, we have:
PS = 180 - 70
Evaluate
PS = 110
Hence, the measure of arc PS is 110 degrees
Read more about circles and arcs at:
https://brainly.com/question/15096899
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