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Angles and Lines

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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