Name: Rectangles and Squares Formed by Four Points
Uploaded: 2019-07-19
Duration: 6 min 3 s
Description: What are the conditions for a quadrilateral to be a rectangle? How does it differ from a square? Unlock your doubts and clear your basics from this video!

Question

If the angles of a quadrilateral (taken in order) are in the ratio

1 : 2 : 3 : 4

, find the measures of all the angles of the quadrilateral. Hence, show that the quadrilateral is a trapezium.

Accepted Answer

Given, the angles of a quadrilateral (taken in order) are in the ratio $1 : 2 : 3 : 4$ .

Let the angles of the quadrilateral be $x, 2 x, 3 x$ and $4 x$ respectively.

We know that the sum of the interior angles of a quadrilateral is $360 °$ .

$∴ x + 2 x + 3 x + 4 x = 360 °$

$\Rightarrow 10 x = 360 °$

$\Rightarrow x = \frac{360 °}{10}$

$\Rightarrow x = 36 °$

$∴ 2 x = 2 \times 36 ° = 72 °, 3 x = 3 \times 36 ° = 108 °$ and $4 x = 4 \times 36 ° = 144 °$

If $A B C D$ is the quadrilateral, then $∠ A = 36 °, ∠ B = 72 °, ∠ C = 108 °$ and $∠ D = 144 °$

In the quadrilateral $A B C D, ∠ A + ∠ D = 36 ° + 144 ° = 180 °$

We know that when a transversal intersects two parallel lines, the interior angles on the same side of the transversal are supplementary.

$∴ A B ∥ C D$

One pair of opposite sides in a trapezium are parallel.

Therefore, $A B C D$ is a trapezium since one pair of opposite sides are parallel.

Hence, proved.

If the angles of a quadrilateral (taken in order) are in the ratio $1 : 2 : 3 : 4$ , find the measures of all the angles of the quadrilateral. Hence, show that the quadrilateral is a trapezium.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

If the angles of a quadrilateral (taken in order) are in the ratio 1:2:3:4, find the measures of all the angles of the quadrilateral. Hence, show that the quadrilateral is a trapezium.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

If the angles of a quadrilateral (taken in order) are in the ratio $1 : 2 : 3 : 4$ , find the measures of all the angles of the quadrilateral. Hence, show that the quadrilateral is a trapezium.