# What is the angle the car turned?

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.

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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.

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inside the triangles? Also, if the line is straight, why can't we say 0 degrees? The car doesn't turn on a straight line? –  Ylyk Coitus Mar 20 '13 at 20:00
@YlykCoitus: do you see the argument for $360$ in a triangle? At each corner of an equilateral triangle you turn $120$ degrees, not $60$. In the five triangle problem, I am trying to figure out how much the car would have turned in the center if it went all the way around all the triangles, so I can subtract it. –  Ross Millikan Mar 20 '13 at 20:04
I really don't understand the slightest bit of this –  Ylyk Coitus Mar 20 '13 at 20:16

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 $216^0 \cdot 5 = 1080^0$

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This seems logical and easy to follow. I'm only concerned by one thing: how are you sure they are 5 isosceles triangles? –  Ylyk Coitus Mar 20 '13 at 20:21
@YlykCoitus: On a second though, it actually doesn't matters as long as the car returns back to $M$ to start on a different triangle. The reason I said isosceles is because as per your diagram, the sides are equal (notice two slashes next to the sides of the triangle) –  Abhijit Mar 20 '13 at 20:26
@YlykCoitus: I agree we are not given that the triangles match. My solution does not assume that. This one might help you understand the more general case. –  Ross Millikan Mar 20 '13 at 20:26