### Theory:

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*Trapezium*A trapezium is a quadrilateral (four-sided figure) in which a pair of opposite sides are parallel and a pair of opposite sides are not parallel.

In the figure, the sides \(AB\) and \(CD\) are parallel, and \(AD\) and \(BC\) are non-parallel.

Important!

• The distance between the parallel sides of a trapezium is called the height. In the figure, \(CX\) is the height.

• Sum of all the angles in a trapezium is \(360^{\circ}\).

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*Isosceles trapezium*If the non-parallel sides of a trapezium are equal in length and form equal angles at one of its bases, then it is called an isosceles trapezium.

In the figure, the non-parallel sides \(AD\) and \(BC\) are equal, and \(\angle A\) and \(\angle B\) are equal.

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*Area of the trapezium*The area of any trapezium is calculated using the formula \(A = \frac{1}{2} \times h \times (a +b)\) square units.

Where \(h\) represents the height of the trapezium.

\(a\) and \(b\) are the lengths of the parallel sides.

Example:

If the height of the trapezium is \(12\) \(cm\) and the lengths of the parallel sides are \(5\) \(cm\) and \(6\) \(cm\) respectively then find its area.

Solution:

Given \(h\) \(=\) \(12\) \(cm\), \(a\) \(=\) \(5\) \(cm\) and \(b\) \(=\) \(6\) \(cm\).

Area of the trapezium, \(A\) \(=\) \(\frac{1}{2} \times h \times (a +b)\)

\(=\) \(\frac{1}{2} \times 12 \times (5 + 6)\)

\(=\) \(6 \times 11\)

\(=\) \(66\) \(cm^2\)

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*Ways to construct a trapezium*A trapezium can be constructed if one of the following four measurements are given.

**Three sides and one diagonal.**

**(i)****Three sides and one angle.**

**(ii)****Two sides and two angles.**

**(iii)****Four sides.**

**(iv)**