Question

measure of angle of quadrilateral ABCD are in ratio of 1:2:3:4 . show that quadrilateral ABCD is trapezium​

Answer 1

ANSWER

⇒ Let the measure of angles be x,2x,3x and 4x.

We know that sum of measure of all four angles is 360

o

.

∴ x+2x+3x+4x=360

o

⇒ 10x=360

o

⇒ x=36

o

⇒ 2x=2×36

o

=72

o

⇒ 3x=3×36

o

=108

o

⇒ 4x=4×36

o

=144

o

⇒ The measures of angles of quadrilateral are 36

o

,72

o

,108

o

and 144

o

.

⇒ We can see measure of all 4 angles are different so, the given quadrilateral is trapezium.

Answer 2

To Find,

• Show that the quadrilateral ABCD is a trapezium

Solution,

Given that,

• ABCD is a quadrilateral
• The measure of all angles are in a ratio of 1 : 2 : 3 : 4

Figure,

• Refer the attachment. [ Image 1 ]

Let us assume the angles ( ∠A, ∠B, ∠C and ∠D ) as 1x, 2x, 3x and 4x.

As we know that,

Sum of all angles of quadrilateral is 360°,

∠A + ∠B + ∠C + ∠D = 360°

➠ 1x + 2x + 3x + 4x = 360

➠ 10x = 360

➠ x = 360 / 10

➠ x = 36

The value of x is 36.

The measure of all angles are :-

⋄ 1x

➠ 1 × 36

➠ 36°

⋄ 2x

➠ 2 × 36

➠ 72°

⋄ 3x

➠ 3 × 36

➠ 108°

⋄ 4x

➠ 4 × 36

➠ 144°

The angles ( ∠A, ∠B, ∠C and ∠D ) are 36°, 72°, 108° and 144°.

Type of quadrilateral :- Trapezium

Figure,

• Refer the attachment. [ Image 2 ]

Because, By the property of trapezium

If ABCD is a trapezium,

• Two pair of adjacent angles ( which form pairs of consecutive angles ) are supplementary ( 180° )

[ ∠A + ∠D = 180° and ∠B + ∠C = 180° ]

Proof :

We got,

• ∠A = 36°
• ∠B = 72°
• ∠C = 108°
• ∠D = 144°

Let's proof that, ∠A + ∠D = 180° and ∠B + ∠C = 180°

• ∠A + ∠D = 180°

➠ 36 + 144 = 180

➠ 180 = 180

Hence, Proved !

• ∠B + ∠C = 180°

➠ 72 + 108 = 180

➠ 180 = 180

Hence, Proved !