Answer:
Proved
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
Hope this helps!