Problem 247: (2007 AMC 10b, 15; 12b 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? $\mathrm {(A)} 125\qquad$ $ \mathrm {(B)} 144\qquad$ $ \mathrm {(C)} 153\qquad$ $ \mathrm {(D)} 173\qquad$ $ \mathrm {(E)} 180$
Solution
Let $\angle A=2\angle B=3\angle C=4\angle D=k$, so $\angle B=\frac k2$, $\angle C=\frac k3$, $\angle D=\frac k4$. $k\left(1+\frac12+\frac13+\frac14\right)=360^\circ$, so $\frac{25}{12}k=360^\circ$ and $k=\dfrac{864}5=172.8^\circ$. Rounded, $\angle A\approx173^\circ$. Answer is (D) .
The angles of quadrilateral $ABCD$ satisfy $\angle A=2\angle B=3\angle C=5\angle D$. What is the degree measure of $\angle A$, rounded to the nearest whole number?
