1. **State the problem:** We are given a circle arc with points T and A on it, segment TA = 12, point N below T with segment TN = 5, and segment NA crossing the arc to A. We need to find the measure of angle A. 2. **Identify the triangle and angle:** Triangle TNA is formed with sides TN = 5, TA = 12, and NA unknown. We want to find angle A, which is the angle at point A between segments NA and TA. 3. **Use the Law of Cosines:** To find angle A, we need all three sides of triangle TNA. We know TA = 12 and TN = 5, but NA is unknown. Since N lies below T and NA crosses the arc, we assume triangle TNA is right-angled or we need more info. However, with given info, we can use the Law of Cosines if we find NA. 4. **Assuming right triangle at N:** If angle at N is right angle, then by Pythagoras: $$NA = \sqrt{TA^2 - TN^2} = \sqrt{12^2 - 5^2} = \sqrt{144 - 25} = \sqrt{119}$$ 5. **Calculate angle A using Law of Cosines:** $$\cos A = \frac{TN^2 + TA^2 - NA^2}{2 \times TN \times TA}$$ Substitute values: $$\cos A = \frac{5^2 + 12^2 - (\sqrt{119})^2}{2 \times 5 \times 12} = \frac{25 + 144 - 119}{120} = \frac{50}{120} = \frac{5}{12}$$ 6. **Find angle A:** $$A = \cos^{-1}\left(\frac{5}{12}\right) \approx 65.38^\circ$$ **Final answer:** The measure of angle A is approximately **65.38 degrees**.