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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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