2007 AMC 10B Problem 15

Attempt Problem 15 of the 2007 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 10B solutions, or check the answer key.

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15.

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

Answer: D

Concepts:angle sum ratio and proportion

Difficulty rating: 1370

Small hint:

Write

\angle B,

\angle C,

and

\angle D

in terms of

\angle A

Big hint:

The four angles sum to

360^\circ

Solution:

Let

x=\angle A.

Then

\angle B=\dfrac{x}{2},

\angle C=\dfrac{x}{3},

and

\angle D=\dfrac{x}{4}.

The angles sum to

360^\circ,

so

x+\dfrac{x}{2}+\dfrac{x}{3}+\dfrac{x}{4}=\dfrac{25x}{12}=360.

Thus

x=\dfrac{12\cdot 360}{25}=172.8\approx 173.

Thus, the correct answer is D.