Answer:
∆APQ is isosceles, so sides AP, AQ are congruent
Explanation:
You want to show that AP≅AQ in the given figure.
Angles of Intersecting Chords Theorem
The Angles of Intersecting Chords Theorem tells you that the angle where chords cross is half the sum of the intercepted arcs. Here, that means ...
∠APY = (arc BX +arc AY)/2
∠AQX = (arc AX +arc CY)/2
Substituting equals
Given that arc AX ≅ arc BX, and arc AY ≅ arc CY, we can substitute for BX and CY to get ...
∠APY = (arc AX +arc AY)/2
∠AQX = (arc AX +arc AY)/2
Two angles equal to the same thing are equal (substitution property), so ...
∠APY ≅ ∠AQX
These are base angles in the isosceles triangle APQ, so their opposite sides, AQ and AP, respectively, are congruent.
AP ≅ AQ