A triangle with two line segments. What’s the angle α?

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The angle α is 180°/7.

First do some angle chasing to come up with the angles as shown. The trick then is to glue a copy of the triangle to the original. This forms a parallelogram stacked on a rhombus. Now because ∠UPA equals ∠URA, we see that URPA is a cyclic quadrilateral.

Because chords UR and RP are congruent, we have that 180-4a=3a-90. The rest is basic algebra.

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