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

ca = ec

cpctc makes it so that corresponding parts of the triangle you proved to be equal are equal, the other options aren't part of the part of the triangles you proved to be equal.