Three quadrilaterals have these angles, taken in order A, B, C, D: (60°, 110°, 120°, 70°), (80°…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Three quadrilaterals have these angles, taken in order , , , : , and . How many of them can be cyclic?
Need a reminder? The key ideaShow
If a pair of opposite angles of a quadrilateral adds up to , the quadrilateral is cyclic. A trapezium is cyclic if and only if its legs are equal. A cyclic parallelogram is a rectangle. The centre of a circle through three points is at equal distances from all of them.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The first. and . It can be cyclic.
Step 2. The second. , which is not . It cannot be cyclic.
Step 3. The third. for both pairs. It can be cyclic: it is a rectangle.
Answer: .
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