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Geometry

# Angles and Lines: Level 3 Challenges

In a simple decagon (10-sided polygon with no self-intersections), what is the maximum number of internal angles that could be acute?

For example, the above image shows 6 acute angles, marked in blue.

In the square above, $$E$$ and $$F$$ are the midpoints of $$\overline{BC}$$ and $$\overline{CD} ,$$ respectively. If $$G$$ is the intersection point of $$\overline{AC}$$ and $$\overline{BF}$$ and $$x=\angle AEG,$$ find $$x$$ (in degrees) in terms of $$\alpha.$$

In a quadrilateral $$ABCD$$, $$\angle ABD = \angle BCA = \angle CDB = \angle DAC = 30^\circ$$.

Find the smaller of the two angles (in degrees) between the diagonals AC and BD.


Note: The image might not be necessarily up to scale.

An irregular hexagon is inscribed in a circle, and I am interested in finding the measure of one specific interior angle of the hexagon.

If I am not allowed to measure it directly, what is the minimum number of other interior angles that I need to measure?

Determine the value of $$x$$.

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