Angles and Lines

Angles and Lines: Level 3 Challenges


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.

\(AB\) is parallel to \(CD\) and \(AB =2CD\).

If \(m\angle ABC = 45^o \) and \(m\angle CBD = 15^o,\) find \(m\angle BAD\) in degrees.

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 \(\alpha.\)

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?

Let \(P\) be an interior point of triangle \(ABC\).
Let \(Q\) and \(R\) be the reflections of \(P\) in \(AB\) and \(AC\), respectively.

If \(QAR\) is a straight line, find \(\angle BAC\) in degrees.

The image is not drawn to scale.


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