A car drives a route through town as indicated alongside, crossing square M five times while doing so. How large is the total angle his car turned through when it has completed the route?
I just don't understand how the answer is supposed to be 1080. I just see 5 triangles, so I immediately assumed 900.
On
The Car travels through 5 equal triangles. Each of the triangles vertex angle (angle centered at M) is $\frac{360}{10} = 36^0$
Each time, the car traverses the entire triangle, it covers a total angle of $180^0$ and then starts traversing the next triangle, which is at an offset of $36^0$ from its current position. Note, as there are 5 equal triangles with vertex angle of $36^0$, and the triangles are equally spaced, they are each at an angle of $36^0$.
So For each triangle, the car traverses $180^0+36^0=216^0$.
As there are 5 triangles, total angle covered is $216^0 \cdot 5 = 1080^0$
Note that if the path were a triangle, the car would turn $360$ degrees, not $180.$ The amount the car turns is the exterior angle, not the internal angle at each point, which is $180$ degrees minus the interior angle. If the car went all the way around each triangle, it would turn $5 \cdot 360= 1800$ degrees. It did not make the turn at each of the angles in the square. Because the five segments are straight, the five angles inside the triangles add up to $180$ degrees, so the angle we didn't turn is $5 \cdot 180 - 180=720$ and the total angle turned is $1800-720=1080$ degrees.