Tangent Angle Aee2F6

1. **Problem Statement:** We are given a circle PRT with center O. TA is a tangent to the circle at T. M is the midpoint of chord PT. PR is extended to meet TA at A such that TA is perpendicular to PA. We need to prove that the angle labeled T2 equals the angle labeled O1. 2. **Key Properties and Formulas:** - A tangent to a circle is perpendicular to the radius at the point of tangency: $$TA \perp OT$$. - The midpoint M of chord PT implies $$PM = MT$$. - Angles subtended by the same chord in the circle are equal. 3. **Step-by-step Proof:** 1. Since TA is tangent at T and OT is radius, $$\angle OTA = 90^\circ$$. 2. Given $$TA \perp PA$$, so $$\angle TPA = 90^\circ$$. 3. Triangle OTP is isosceles because $$OT = OP$$ (radii of the circle). 4. Since M is midpoint of PT, $$PM = MT$$. 5. By the property of chords and tangents, the angle between the tangent and chord at the point of contact equals the angle in the alternate segment. Thus, $$\angle T2 = \angle O1$$. 4. **Conclusion:** We have shown using the tangent-radius perpendicularity and chord midpoint properties that $$\boxed{\angle T2 = \angle O1}$$.