1954 AMC 12 Problem 13

Question

A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, without intersecting the sides between vertices, their sum will be:

180^\circ

540^\circ

360^\circ

450^\circ

1080^\circ

Answer: B

Concepts:inscribed angle arc angle sum

Difficulty rating: 1630

Small hint:

Let the four consecutive arc measures be

a,b,c,d

Big hint:

An angle whose vertex lies in the arc of measure

a

intercepts the other three arcs, of total measure

360^\circ-a

Solution:

Let the four arcs have measures

a,b,c,d,

whose sum is

360^\circ.

The angle placed in the arc of measure

a

intercepts the other three arcs, so it measures

\frac{360^\circ-a}{2},

and similarly for the others. Their sum is

\begin{aligned} &\frac{4(360^\circ)-(a+b+c+d)}2\\ &\qquad=\frac{1080^\circ}{2}\\ &\qquad=540^\circ. \end{aligned}

Thus, the correct answer is B.