2008 AMC 10B Problems/Problem 24
Quadrilateral has , angle and angle . What is the measure of angle ?
This solution requires the use of cyclic quadrilateral properties but could be a bit time-consuming during the contest.
To start off, draw a diagram like in solution one and label the points. Now draw and and call their intersection point . Note that triangle is an isosceles triangle so angles and are each degrees. Since equals , angle equals degrees, thus making angle equal to degrees. We can also find out that angle equals degrees.
Extend and and let their intersection be . Since angle plus angle equals degrees, quadrilateral is a cyclic quadrilateral.
Next, draw a line from point to point . Since angle and angle point to the same arc, angle is equal to degrees. Since is an isosceles triangle (based on angle properties) and is also an isosceles triangle, we can find that is also an isosceles triangle. Thus, each of the other angles is degrees. Finally, we have angle equals degrees.
~Minor edits by BakedPotato66
First, connect the diagonal , then, draw line such that it is congruent to and is parallel to . Because triangle is isosceles and angle is , the angles and are both . Because angle is , we get angle is . Next, noticing parallel lines and and transversal , we see that angle is also , and subtracting off angle gives that angle is .
Now, because we drew , triangle is equilateral. We can also conclude that meaning that triangle is isosceles, and angles and are equal.
Finally, we can set up our equation. Denote angle as . Then, because is a parallelogram, the angle is also . Then, is . Again because is a parallelogram, angle is . Subtracting angle gives that angle equals . Because angle equals angle , we get , solving into .
Side note: this solution was inspired by some basic angle chasing and finding some 60 degree angles, which made me want to create equilateral triangles.
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