A3. What is the sum of the sizes of the angles marked in the figure, if $\varphi=$ $77^{\circ}$ ?
(A) $283^{\circ}$
(B) $360^{\circ}$
(C) $385^{\circ}$
(D) $437^{\circ}$
(E) Cannot be determined.

![](https://cdn.mathpix.com/cropped/2024_06_07_66bdf7a61d9bfcb32ca4g-06.jpg?height=417&width=517&top_left_y=1296&top_left_x=1386)
A3. We additionally mark it as $\varphi$. Considering the fact that the sum of the interior angles at two vertices in a triangle is equal to the exterior angle at the third vertex, we sequentially conclude that the sum of the angles on the sketch is equal to the sum of the angles on the following sketches:
![](https://cdn.mathpix.com/cropped/2024_06_07_66bdf7a61d9bfcb32ca4g-14.jpg?height=236&width=1512&top_left_y=919&top_left_x=270)

The sum of the angles marked on the sketch is therefore $2 \cdot 180^{\circ}=360^{\circ}$, and the sum of the initially marked angles is $360^{\circ}-\varphi=360^{\circ}-77^{\circ}=283^{\circ}$.