## Task A-2.3. (8 points)

In triangle $A B C$, the angles $\varangle C A B=35^{\circ} \text{ and } \varangle A B C=60^{\circ}$ are known. If $t$ is the tangent to the circumcircle of this triangle at vertex $C$, and $p$ is the line parallel to line $A B$ through vertex $C$, determine the angle between lines $p$ and $t$.
## Solution.

The measure of the angle formed by line $p$ and line $B C$ is equal to the measure of $\angle A B C$, which is $60^{\circ}$ (parallel sides).

![](https://cdn.mathpix.com/cropped/2024_05_30_59925bf190fc107f19b7g-07.jpg?height=617&width=562&top_left_y=1368&top_left_x=770)

Let $S$ be the center of the circumscribed circle. The angle $\angle B S C$ is $70^{\circ}$ because it is the central angle of the chord $\overline{B C}$, and the inscribed angle $\angle B A C$ is $35^{\circ}$. Now, $\angle S C B = 55^{\circ}$ (from the isosceles triangle $B C S$), and the angle between line $B C$ and the tangent $t$ is $90^{\circ} - 55^{\circ} = 35^{\circ}$, since the tangent is perpendicular to the radius $\overline{S C}$.

(We can also arrive at this fact using the theorem about the angle between a chord and a tangent: the angle between a chord of a circle and a tangent to that circle drawn from one of the endpoints of the chord is equal to the inscribed angle over that chord.)

Finally, the angle between lines $p$ and $t$ is $60^{\circ} - 35^{\circ} = 25^{\circ}$.

For 4 points: correct method with a calculation error.

For 8 points: correct result and method.