Sample Question 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?

$\mathrm {(A)} 125\qquad \mathrm {(B)} 144\qquad \mathrm {(C)} 153\qquad \mathrm {(D)} 173\qquad \mathrm {(E)} 180$

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Answer Keys

Question 11: D

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Solutions

Question 11
Step 1: Note that the sum of the interior angles of any quadrilateral is $360^\circ$.

Step 2: Given that $\angle A=2\angle B=3\angle C=4\angle D$, we represent each angle in terms of $\angle A$ and substitute these back into the sum of angles equation. This gives us $360 = \angle A + \frac{1}{2}A + \frac{1}{3}A + \frac{1}{4}A$.

Step 3: To simplify the equation, we express the coefficients of $\angle A$ in the equation as fractions with the same denominator. The equation becomes $360 = \frac{12}{12}A + \frac{6}{12}A + \frac{4}{12}A + \frac{3}{12}A$, and further simplification gives us $360 = \frac{25}{12}A$.

Step 4: Solving for $\angle A$ we get $\angle A = 360 \cdot \frac{12}{25} = 172.8$.

Step 5: Rounding to the nearest whole number, the degree measure of angle A is approximately 173.

Thus, the answer is $\mathrm{(D) \ } 173$.