## Task 4 - 290714

When reviewing the interior angle theorem for convex quadrilaterals, Anja and Klaus get into an argument: Klaus claims:

"If a convex quadrilateral $A B C D$ is divided by drawing the diagonal $A C$ into the two sub-triangles $A B C$ and $A D C$, then the sum of the interior angles of each of these sub-triangles is $180^{\circ}$. Consequently, the sum of the interior angles in the quadrilateral $A B C D$ must be $2 \cdot 180^{\circ}=360^{\circ}$."

Anja counters: "But if you draw the second diagonal $B D$ as well, you get four sub-triangles. The sum of the interior angles of $A B C D$ must therefore be $4 \cdot 180^{\circ}=720^{\circ}$." Investigate which of the two students is correct! Justify your decision!
Klaus is right. As the first figure shows, all interior angles of triangles $ABC$ and $ADC$ are also interior angles or parts of interior angles of quadrilateral $ABCD$.

Moreover, there are no interior angles or parts of interior angles of this quadrilateral that are not interior angles of the mentioned triangles.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0514.jpg?height=269&width=443&top_left_y=1733&top_left_x=812)

On the other hand, Anja is not right, as the second figure shows. Four of the interior angles of the triangles formed by the two diagonals lie within quadrilateral $ABCD$ and together form a full angle. Consequently, Anja should have subtracted $360^{\circ}$ from the obtained $720^{\circ}$, which would have resulted in the $360^{\circ}$ interior angle sum determined by Klaus.

Solutions of the 1st Round 1989 taken from [5]

### 4.31.2 2nd Round 1989, Class 7