Question

(2) The measures of the angles of a quadrilateral taken
in order are as 6:7:11 : 12. Prove that it is a
trapezium.
​

Answer 1

Answer:

The given quadrilateral is a trapezium.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure, ABCD is a quadrilateral.

We have given the ratio of the measures of the angles of a quadrilateral.

We have to prove that the quadrilateral is a trapezium.

The ratio of the measures of the angles is 6 : 7 : 11 : 12.

Let the common multiple be x.

∴ m∠A = 6x

m∠B = 7x

m∠C = 11x

m∠D = 12x

Now, we know that,

The sum of measures of all angles of a quadrilateral is 360°.

∴ m∠A + m∠B + m∠C + m∠D = 360°

⇒ 6x + 7x + 11x + 12x = 360°

⇒ 13x + 11x + 12x = 360°

⇒ 10x + 3x + 10x + x + 10x + 2x = 360°

⇒ 10x + 10x + 10x + 3x + 2x + x = 360°

⇒ 30x + 5x + x = 360°

⇒ 30x + 6x = 360°

⇒ 36x = 360°

⇒ x = 360° ÷ 36

⇒ x = 10°

Now,

m∠A = 6x

⇒ m∠A = 6 * 10

⇒ m∠A = 60°

Now,

m∠B = 7x

⇒ m∠B = 7 * 10

⇒ m∠B = 70°

Now,

m∠C = 11x

⇒ m∠C = 11 * 10

⇒ m∠C = 110°

Now,

m∠D = 12x

⇒ m∠D = 12 * 10

⇒ m∠D = 120°

Now, in □ABCD,

m∠A + m∠B

⇒ 60° + 70°

⇒ 130°

∴ m∠A + m∠B = 130° ≠ 180°

Angles A and B are adjacent angles, but aren't supplementary.

∴ AD ∦ BC - - - [ Adjecnt angles test ]

Now,

m∠B + m∠C

⇒ 70° + 110°

⇒ 180°

∴ m∠B + m∠C = 180°

Angles B and C are adjacent angles and supplementary.

∴ AB ∥ CD - - - [ Adjacent angles test ]

Now, in □ABCD,

AB ∥ CD &

AD ∦ BC

∴ There is only one pair of parallel sides in the quadrilateral.

∴ □ABCD is a trapezium. - - - [ By definition ]

∴ The given quadrilateral is a trapezium.

Hence proved!