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## Angles and Lines

Forgive us for being obtuse, but this is a cute concept, and we think it’s right for you. See more

# Level 3

$$ABC$$ is a right triangle with $$\angle C=90^\circ$$. $$M$$ is the midpoint of $$AB,$$ and $$D$$ is a point such that $$MC = MD$$ with points $$C$$ and $$D$$ lying on opposite sides of $$AB,$$ as shown in the diagram.

If $$\angle DAB = 40 ^ \circ$$, what is $$\angle DCA$$?

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 figure above, 3 unit squares are placed side by side. Find the sum of the angle measurements $$a,$$ $$b,$$ and $$c,$$ in degrees.

Determine the value of $$x$$.

What is the average of all the angles (in degrees) on this line?

Note: Count only angles at the marked points, ignoring the locations where the line crosses itself. At each point pick the "internal" angle, the one that is less than $$180^\circ$$.

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