Geometry

# 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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