3. Given a square $A B C D$. Inside the square lies a point $U$ such that $B U$ and $A B$ are of equal length. The intersection of $B U$ and the diagonal $A C$ is $V$. Angle $D A U$ is equal to 28 degrees.

What is the angle at $V$ in triangle $B V C$?

![](https://cdn.mathpix.com/cropped/2024_04_17_227837a5972cf599de5fg-2.jpg?height=331&width=371&top_left_y=1699&top_left_x=1476)
B3. $101^{\circ}$

First, we look at triangle $A B U$. The angle at $A$ is equal to $90^{\circ}-28^{\circ}=62^{\circ}$. Since triangle $A B U$ is isosceles, the angle at $U$ is also equal to $62^{\circ}$. Because the sum of the angles in a triangle is $180^{\circ}$, the angle at $B$ must be equal to $180^{\circ}-62^{\circ}-62^{\circ}=56^{\circ}$.

Now we look at triangle $A B C$. Triangle $A B C$ is isosceles. The two equal angles at $A$ and $C$ plus the right angle at $B$ together are

![](https://cdn.mathpix.com/cropped/2024_04_17_87bdce5b1b523bcd9951g-3.jpg?height=346&width=386&top_left_y=1209&top_left_x=1429)
$180^{\circ}$. From this, it follows that the angle at $C$ is equal to $\frac{180^{\circ}-90^{\circ}}{2}=45^{\circ}$.

Finally, we look at triangle $B C V$. The angle at $C$ is equal to $45^{\circ}$. The angle at $B$ is equal to $90^{\circ}-56^{\circ}=34^{\circ}$. The angle at $V$ is then equal to $180^{\circ}-45^{\circ}-34^{\circ}=101^{\circ}$.