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# Angles

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What is the measure (in degrees) of the smallest interior angle of a triangle in which the exterior angle measures have the ratio $$3:5:7$$?

May 14, 2022

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Note that the exterior angles of any convex polygon always add up to 360 degrees.

Since an interior angle and its corresponding exterior angle always add up to 180 degrees, a larger exterior angle means a smaller interior angle. So we are actually looking for the largest exterior angle.

Using the ratio, we calculate that the largest exterior angle is $$360^\circ \times \dfrac{7}{3 + 5 + 7} = 168^\circ$$. You can subtract that from 180 degrees to find the smallest interior angle.

This is one step away from the answer. Please continue on your own, but I hope you have an idea of what I explained, because that would be useful if you are to solve a similar problem entirely on your own.

May 14, 2022