Question

if the angles of a quadrilateral,taken in order,are in the ratio 1 : 2 : 3 : 4 ,prove that it is a trapezium.

Answer 1

Let the coefficient of the following ratios be x.
we know that sum of angles of quadrilateral equals 360°.

So,
1x+2x+3x+4x = 360.
10x = 360°.
x = 36°.

First angle = 36°.
Second angle = 72°.
Third angle = 108°.
Fourth angle = 144°.

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Answer 2

The measures of angles of a quadrilateral taken in order are 1:2:3:4. Them the quadrilateral is 
The angel are 
Angle A =360°/(1+2+3+4) =360°/10=36° 
Angle B =360°×2/(1+2+3+4) = 72° 
Angle C =360°×3/(1+2+3+4) = 108° 
Angle D =360°×4/(1+2+3+4) = 144° 
Since Angle A+ Angle D =36°+144° =180° 
Angle B +Angle C = 72°+ 108°=180° 
Hence AB is parallel to CD 
and the quadrilateral is a trapezium.