Prove: If two circles are tangent internally at point P and chord PA
of the larger circle intersects the smaller circle at B, then the measure of arc PA is equal to the measure of arc PB.

Proved

Step-by-step explanation:

Given:

Two circles are tangent internally at point P and chord PA of the larger circle intersects the smaller circle at B.

Prove: measure of arc PA is equal to measure of arc PB

To better understand this, the diagrams have been attached.

From triangle PQB:

Line PQ  and line QB are radius of the smaller circle.

Reason:  their line are straight lines  drawn from the center of the circle to the circumference of the circle

Line PQ = Line QB

That is, side PQ = side QB

Since side PQ = side QB, it is an isosceles triangle.

Therefore < QPB = <PBQ

From triangle PRA

Line PR = line RA

Side PR = side RA

From triangle PRA:

Line PR  and line RA are radius of the bigger circle.

Reason:  their line are straight lines  drawn from the center of the circle to the circumference of the circle

Line PR = Line RA

Since side PR = side RA, it is an isosceles triangle.

Therefore < RPA = <PAR

Using similar triangles theorem:

Triangle PQB = Triangle PRA

<PBQ = <PAR

Since they are equal, the measure of the arc (angle subtended by the arc) are equal.

Therefore measure of arc PA = measure of arc PB

d

step-by-step explanation:

d