2020 AIME I Problems/Problem 1
In with point lies strictly between and on side and point lies strictly between and on side such that The degree measure of is where and are relatively prime positive integers. Find
If we set to , we can find all other angles through these two properties: 1. Angles in a triangle sum to . 2. The base angles of an isosceles triangle are congruent.
Now we angle chase. , , , , , . Since as given by the problem, , so . Therefore, , and our desired angle is for an answer of .
See here for a video solution: https://youtu.be/4e8Hk04Ax_E
Let be in degrees. . By Exterior Angle Theorem on triangle , . By Exterior Angle Theorem on triangle , . This tells us and . Thus and we want to get an answer of .
Solution 3 (Official MAA)
Let . Because is isosceles, . Then Because and are also isosceles, Because is isosceles, is also , so , and it follows that . The requested sum is .
See here for a video solution:
Solution without words
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