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Question
If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3 : 7 : 6 : 4, then ABCD is a ______.
Options
rhombus
parallelogram
trapezium
kite
Solution
If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3 : 7 : 6 : 4, then ABCD is a trapezium.
Explanation:
Given in the question, ratio of angles of quadrilateral ABCD is 3 : 7 : 6 : 4.
Let the angles of quadrilateral ABCD be 3x, 7x, 6x and 4x respectively.
So, 3x + 7x + 6x + 4x = 360° ...[Sum of the all angles of a quadrilateral is 360°]
20x = 360°
x = `(360^circ)/20`
x = 18°
So, angles of the quadrilateral are:
∠A = 3 × 18° = 54°
∠B = 7 × 18° = 126°
∠C = 6 × 18° = 108°
∠D = 4 × 18° = 72°
See the figure, ∠BCE = 180° – ∠BCD ...[Linear pair axiom]
∠BCE = 180° – 108° = 72°
∠BCE = ∠ADC = 72°
Now, BC || AD ...[The corresponding angles are equal]
The sum of co interior angles is:
∠A + ∠B = 126° + 54° = 180°
And ∠C + ∠D = 108° + 72° = 180°
Hence, ABCD is a trapezium.
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