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Sample 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 \(\mathrm{(D) \ } 173\).
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?
\(\mathrm {(A)} 125\qquad \mathrm {(B)} 144\qquad \mathrm {(C)} 153\qquad \mathrm {(D)} 173\qquad \mathrm {(E)} 180\)
Answer Keys
Question 11: DSolutions
Question 11Step 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 \(\mathrm{(D) \ } 173\).