Question

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

Answer 1

let the constant of ratio be x
so x+2x+3x+4x=360(angle sum of quadrilateral)
=>10x=360
=>x=360/10=36°
so 2x=72°,3x=108° and 4x=144°
we find that the pairs of angles X and 4x
36+144=180°
and also 2x+3x=72+108=180°
as the angles add up to 180° and we know when two lines are parallel the cointerior angles are supplementary .we also know a trapezium is a quadrilateral where a pair of opposite sides are parallel. (proved)

Answer 2

All 4 angles in a quadrilateral add up to 360°

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Find x:

x + 2x + 3x + 4x = 360

10x = 360

x = 36

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Find each angle:

x = 36

2x = 36 x 2 = 72

3x = 36 x 3 = 108

4x = 36 x 4 = 144.

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To prove that quadrilateral is a trapezium, we need to prove that:

• It has a pair of parallel lines.

To prove that there is a pair of parallel line, we need to prove that:

• the interior angles of the two lines add up to 180°

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Find 1st pair of angles that make up 180°:

2x + 3x = 72 + 102 = 180°

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Find another pair of angles that make up 180°:

x + 4x = 36 + 144 = 180°

.

So we know that there is a pair of parallel lines in this quadrilateral

⇒ It is a trapezium.

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