## Task 4 - 320714

In a triangle $A B C$, let $D$ be the midpoint of side $A B$. The segments $A D$ and $C D$ are of equal length, and the measure of the interior angle $\angle B C D$ in the sub-triangle $B C D$ is $35^{\circ}$.

Determine the sizes of the interior angles of triangle $A B C$ from these data!
Given that $A D=B D=C D$ and (1) $\angle B C D=35^{\circ}$. (2) Since triangle $B C D$ is isosceles due to (1), it follows from the base angle theorem and (2) that $\angle D B C=35^{\circ}$, i.e., (3) $\angle A B C=35^{\circ}$. (4)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0537.jpg?height=323&width=552&top_left_y=244&top_left_x=752)

From (2) and (3), it follows from the exterior angle theorem that $\angle A D C=70^{\circ}$. (5) Due to (1), triangle $A C D$ is also isosceles; thus, by the base angle theorem, $\angle C A D=\angle A C D$.

From this and (5), it follows from the interior angle theorem that

$$
\angle C A D=\angle A C D=\frac{1}{2}\left(180^{\circ}-70^{\circ}\right)=55^{\circ}
$$

Thus, on the one hand, $\angle C A B=55^{\circ}$, (7) and on the other hand, from (6) and (2)

$$
\angle A C B=\angle A C D+\angle B C D=55^{\circ}+35^{\circ}=90^{\circ}
$$

With (4), (7), (8), the required interior angle measures are determined.

Solutions of the I. Round 1992 adopted from $[5]$

### 4.34.2 II. Round 1992, Class 7