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If the ratio of four angles of a quadrilateral is ∠A:\angle B:\angle C:\angle D::2:3:5:8, what is the value of ∠B?
60°
90°
50°
40°
60°
Sum of ratios is 2+3+5+8=18. Each part is 360°/18=20°. Since ∠B is 3 parts, ∠B = 3×20°=60°.
The ratio of angles of a quadrilateral is ∠A : ∠B : ∠C : ∠D = 2 : 3 : 5 : 8. The sum of interior angles of a quadrilateral is 360°.
2x+3x+5x+8x=360°
Sum of ratios is 2+3+5+8=18. Each part is 360°/18=20°. Since ∠B is 3 parts, ∠B = 3×20°=60°.
Confusing the sum of angles of a quadrilateral (360°) with that of a triangle (180°).
Define variables
Let the angles be 2x,3x,5x, and 8x based on the given ratio.
2x,3x,5x,8x
Apply quadrilateral angle sum property
The sum of interior angles of any quadrilateral is 360°.
2x+3x+5x+8x=360°
Solve for x
Combine like terms and divide by the sum.
18x=360°⟹x=20°
Calculate ∠B
Substitute x=20° into the expression for ∠B.
∠B=3x=3×20°=60°
A is correct because calculating the value of one part as 20° and multiplying by the ratio factor of ∠B (which is 3) gives 60°.
This concept of partitioning a total sum based on ratios is essential in mixture problems, profit sharing, and geometric division.
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