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10-33.

In  at right, assume that . Prove that . Use the format of your choice.  Circle, with center, Y, with points, in order, O, K, E, P, line segments from, O, to, P, and from, E, to, k.

On circle, Draw line segments from, O, to, Y,  from, K, to, Y, from, E, to, Y, & from, P,  to, Y. Proof: Oval #1: O, Y, = K, Y, =, E, Y, =, P, Y, All Radii are =. (Definition of a circle)

Proof: Oval #2 added in row: arc, P, Q, =, arc, E, K, (Given).

Proof: Oval #3 added in row: angle, P, Y, O, = arc, P, O, (Intercepted arc). Oval #4 added in row: angle, E, Y, K, = arc, E, K, labeled, (Equal Central Angle).

Proof: Oval #5 below right three ovals: angle, P, Y, O, = angle, E, Y, K, (Transitive Property). An arrow from ovals 2, 3, and 4 point to oval 5.

Proof: Oval #6 below the first and second ovals has triangle, P, Y, O, = triangle, E, Y, K, by S, A, S congruency. An arrow from Oval #1 and oval #5 point to Oval #6.

Proof:Oval #7 below Oval #6: segment, P, O, = segment, E, K, (congruent triangle yields congruent parts).