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The angles of a quadrilateral are in the ratio 2:5:7:10. Find the difference between the greatest and the smallest angles of the quadrilateral.
50°
120°
180°
60°
120°
Find the sum of ratio parts (2+5+7+10 = 24). Calculate the value of one unit by dividing 360 by 24 (360/24 = 15). Find the difference between the ratio parts of the greatest (10) and smallest (2) angle, which is 8, and multiply by 15.
The angles of a quadrilateral are in the ratio 2 : 5 : 7 : 10. The sum of the interior angles of a quadrilateral is 360°.
2x+5x+7x+10x=360°
Find the sum of ratio parts (2+5+7+10 = 24). Calculate the value of one unit by dividing 360 by 24 (360/24 = 15). Find the difference between the ratio parts of the greatest (10) and smallest (2) angle, which is 8, and multiply by 15.
Students often calculate the individual angles and then subtract, wasting time; instead, directly subtract the ratio parts (10 - 2 = 8) and multiply by the unit value.
Sum of ratios
Let the angles be 2x, 5x, 7x, and 10x. Since the sum of angles in a quadrilateral is 360°, we set up the equation.
2x+5x+7x+10x=360°
Solve for x
Combine the terms and solve for x.
24x=360°⟹x=24360=15°
Calculate the difference
The greatest angle is 10x and the smallest is 2x. The difference is 10x−2x=8x.
8×15°=120°
B is correct because the difference between the greatest angle (150°) and the smallest angle (30°) is 120°.
This ratio-sum method is applicable to any polygon where the sum of interior angles is (n−2)×180°.
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