- Emily Crum
Conjecture: Angle alpha (RPQ) is 1/2 Angle beta (ROQ) Proof: Line segments PO, QO, and RO are all radii and are therefore congruent. This makes Triangles POR, QOR, and ROQ are isosceles triagles because they have 2 congruent sides each. Because they are isosceles triangles, they each have congruent base angles. Thus angle OPR = angle ORP (let these angles be called delta), angle ORQ = angle OQR (let these be called lambda) and angle OPQ = angle OQP. Since the angles of a triangle add to 180 degrees, angle POR = 180- 2delta, and angle POQ = 180 - 2lambda. angle POR and Angle QOA are supplementary. Thus these 2 angles must add to 180 degrees. Thus 180 - 2 lambda + QOA =180, therefore QOA = 2lambda Angles POR and ROA are supplementary and must add to 180 degrees. Thus 180 - 2delta + ROA =180. Therefore ROA = 2 delta. Angle ROQ = QOA + ROA = 2 lambda + 2 delta = 2(lambda + delta). Angle RPQ = RPO + QPO = delta + lambda. Therefore Angle alpha is 1/2 Angle beta.