2016 AMC 8

Complete problem set with solutions and individual problem pages

Problem 23 Hard

Two congruent circles centered at points A and B each pass through the other circle's center. The line containing both A and B is extended to intersect the circles at points C and D. The circles intersect at two points, one of which is E. What is the degree measure of \angle CED?

  • A.

    90

  • B.

    105

  • C.

    120

  • D.

    135

  • E.

    150

Answer:C

Observe that \triangle{EAB} is equilateral (all are radii of congruent circles). Therefore, m\angle{AEB}=m\angle{EAB}=m\angle{EBA} = 60^{\circ}. Since CD is a straight line, we conclude that m\angle{EBD} = 180^{\circ}-60^{\circ}=120^{\circ}. Since BE=BD (both are radii of the same circle), \triangle{BED} is isosceles, meaning that m\angle{BED}=m\angle{BDE}=30^{\circ}. Similarly, m\angle{AEC}=m\angle{ACE}=30^{\circ}.

Now, \angle{CED}=m\angle{AEC}+m\angle{AEB}+m\angle{BED} = 30^{\circ}+60^{\circ}+30^{\circ} = 120^{\circ}. Therefore, the answer is \boxed{\textbf{(C) }\ 120}.

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