## Task 3 - 250713

Students Gerd and Uwe are discussing the following requirements for a convex quadrilateral $A B C D$ (see figure).

It should hold that $A B=B C=A D$, and if $M$ is the intersection of the two diagonals $A C$ and $B D$, then the angle $\angle B M A$ should be $70^{\circ}$ and the angle $\angle B C M$ should be $50^{\circ}$.

Gerd claims that these requirements uniquely determine the size of the angle $\angle D A M$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0484.jpg?height=437&width=394&top_left_y=313&top_left_x=1345)

Uwe argues that there are convex quadrilaterals that meet these requirements but have different sizes for the angle $\angle D A M$. Who is right?
In every convex quadrilateral $A B C D$ that meets the requirements, the following statements hold for the angle sizes

$$
\angle B A C=\angle B A M=\alpha, \quad \angle A B D=\angle A B M=\beta, \quad \angle A D B=\angle A D M=\gamma, \quad \angle D A M=\angle D A C=\phi
$$

Since $\angle B A C$ and $\angle B C A(=\angle B C M)$ are base angles in the isosceles triangle $A B C$, we have $\alpha=50^{\circ}$.

By the interior angle theorem, applied to triangle $A B M$, it follows that $\beta=180^{\circ}-70^{\circ}-\alpha=60^{\circ}$. Since $\angle A D B$ and $\angle A B D$ are base angles in the isosceles triangle $A B D$, we have $\gamma=\beta=60^{\circ}$.

By the exterior angle theorem, applied to triangle $A D M$, it follows that $\gamma+\phi=70^{\circ}$, thus $\phi=70^{\circ}-\gamma=10^{\circ}$. This angle size is therefore uniquely determined by the requirements; Gerd is right, Uwe is not.