Question

Use the figure below for #1 and #2.

1. PR is tangent to circle Q at R and PS is tangent to circle Q at S. Find m∠RQS

2. Use your knowledge of central and inscribed angles to find m∠RTS.​

Answer 1

Given: Circle Q with center O, tangents PR and PS drawn from external point P to circle Q.

To prove: Angle RPS is supplementary to angle AOB.

Construction: Draw radii OA and OB.

Proof:

Since PR and PS are tangents to circle Q, we have:

Angle OAP = 90 degrees (because a tangent is perpendicular to the radius at the point of contact)

Angle OBP = 90 degrees (because a tangent is perpendicular to the radius at the point of contact)

Therefore, we have:

Angle OAP + Angle OBP = 90 degrees + 90 degrees = 180 degrees

But angle AOB is the sum of angles OAP and OBP, so we have:

Angle AOB = 180 degrees

Therefore, we have:

Angle RPS + Angle AOB = 180 degrees

This means that angle RPS is supplementary to angle AOB.