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Macedonian Mathematical Competitions

Geometry Difficulty 4.7 AIME Prove it North Macedonia

Two lines intersect in one point and form four angles, two acute and two obtuse. The sum of the two acute angles is half of the one obtuse angle. Calculate these angles?

Solution

Let the size of the two acute angles be $x$ (they are equal). Then each of the obtuse angles equals $180^{\circ} - x$ . From the condition in the problem we have that $2x = \frac{1}{2}(180^{\circ} - x)$ , i.e. $4x = 180^{\circ} - x$ , from where we obtain $x = 36^{\circ}$ . Hence each of the obtuse angles equals $180^{\circ} - 36^{\circ} = 144^{\circ}$ .

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