2008 AMC 10B Problems/Problem 24
Quadrilateral has , angle and angle . What is the measure of angle ?
Draw the angle bisectors of the angles and . These two bisectors obviously intersect. Let their intersection be . We will now prove that lies on the segment .
Note that the triangles and are congruent, as they share the side , and we have and .
Also note that for similar reasons the triangles and are congruent.
Now we can compute their inner angles. is the bisector of the angle , hence , and thus also . (Faster Solution picks up here) is the bisector of the angle , hence , and thus also .
It follows that . Thus the angle has , and hence does indeed lie on . Then obviously .
Faster Solution: Because we now know three angles, we can subtract to get , or .
Draw the diagonals and , and suppose that they intersect at . Then, and are both isosceles, so by angle-chasing, we find that , , and . Draw such that and so that is on , and draw such that and is on . It follows that and are both equilateral. Also, it is easy to see that and by construction, so that and . Thus, , so is isosceles. Since , then , and .
Again, draw the diagonals and , and suppose that they intersect at . We find by angle chasing the same way as in solution 2 that and . Applying the Law of Sines to and , it follows that , so is isosceles. We finish as we did in solution 2.
Start off with the same diagram as solution 1. Now draw which creates isosceles . We know that the angle bisector of an isosceles triangle splits it in half, we can extrapolate this further to see that it's is
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