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The figure of Circle A shown below has diameter PR which intersects QS at point B and the measurements shown. Calculate the following measures:

m

Complete all calculations and work on a separate piece of paper, including any figures that you use. Once you have found all of the measures, upload the solution and your work here.

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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