In the given figure, O is the center of the circle. If $$\angle BAC = 65^{\circ}$$ and $$\angle ACB = 84^{\circ}$$, find the value of $$x$$.

Let's break this down step-by-step!

1. Understanding the Angles: We are given a circle with center O. We know two angles within the triangle ABC: $$\angle BAC = 65^{\circ}$$ and $$\angle ACB = 84^{\circ}$$. We need to find the value of $$x$$, which represents $$\angle CA T$$, where AT is a tangent to the circle at point A.
2. Finding $$\angle ABC$$: The sum of angles in a triangle is always $$180^{\circ}$$. So, in triangle ABC:
   $$\angle ABC = 180^{\circ} - \angle BAC - \angle ACB$$
   $$\angle ABC = 180^{\circ} - 65^{\circ} - 84^{\circ}$$
   $$\angle ABC = 180^{\circ} - 149^{\circ}$$
   $$\angle ABC = 31^{\circ}$$
3. Tangent-Chord Theorem: Now, let's consider the angle $$x$$. The angle between a tangent (AT) and a chord (AC) through the point of contact (A) is equal to the angle in the alternate segment. The alternate segment for the angle $$\angle CAT$$ is the angle subtended by the chord AC at the circumference, which is $$\angle ABC$$.
4. Applying the Theorem: According to the Tangent-Chord Theorem, the angle between the tangent AT and the chord AC is equal to the angle subtended by the chord AC at the circumference in the alternate segment.
   So, $$x = \angle ABC$$.
5. The Solution: We found that $$\angle ABC = 31^{\circ}$$. Therefore, $$x = 31^{\circ}$$.

Ответ: $$x = 31^{\circ}$$