Question

measures of angles of square ABCD are in the ratio 4 : 5 : 7 : 8. show that squre ABCD is a trapezium.

Answer 1

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Answer 2

Given:

The measures of the angles of the square ABCD are in ratio 4:5:7:8.

To Find:

ABCD is a trapezium.

Solution:

Let the angles of the trapezium be 4x, 5x, 7x, 8x of square ABCD respectively.

Now, the sum of all the angles of the quadrilateral is 360°.

So, ∠A + ∠B + ∠C + ∠D = 360°

Putting the values as ∠A = 4x, ∠B = 5x, ∠C = 7x, ∠D = 8x

⇒ 4x + 5x + 7x + 8x = 360°

Adding all the values keeping x common.

⇒ 24x = 360°

Divide 360 by 24 to find the value of x.

⇒ x = 360/24

⇒ x = 15°

So, ∠A = 4x = 4×15 = 60°

∠B = 5x = 5×15 = 75°

∠C = 7x = 7×15 = 105°

∠D = 8x = 8×15 = 120°

Now, ∠A + ∠B = 75°+60°=180°

∠B+∠C = 75°+105° = 180°

[CD is parallel to BA and BC is not parallel to AD]

⇒ CD║BA ..(i)

BC∦ AD ..(ii)

Now, Using (i) and (ii)

so, ABCD is a trapezium.

Hence proved that ABCD is a trapezium.