The angles of quadrilateral ABCD satisfy ∠ A=2∠ B=3∠ C=4∠ D. What is the degree measure of ∠ A, rounded to the nearest whole number?

2007 AMC 12B Problem 11

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11.

The angles of quadrilateral

ABCD

satisfy

\angle A=2\angle B=3\angle C=4\angle D.

What is the degree measure of

\angle A,

rounded to the nearest whole number?

125

144

153

173

180

Let

x=\angle A;

then

\angle B=\tfrac x2,

\angle C=\tfrac x3,

\angle D=\tfrac x4

The four angles of a quadrilateral sum to

360^\circ

Let

x=\angle A.

Then

\angle B=\tfrac x2,

\angle C=\tfrac x3,

and

\angle D=\tfrac x4.

The angle sum gives

x+\dfrac x2+\dfrac x3+\dfrac x4=\dfrac{25x}{12}=360,

x=\dfrac{12\cdot360}{25}=172.8\approx173.

Thus, the correct answer is D.