# Isosceles Trapezoid

An isosceles trapezoid is a two-dimensional closed figure with one pair of congruent non-parallel sides (legs) and two pairs of congruent base angles. In other words, an isosceles trapezoid is a trapezoid with congruent legs. Therefore, an isosceles trapezoid has a pair of parallel but unequal opposite sides (bases) and a pair of non-parallel but congruent opposite sides (legs). A non-isosceles trapezoid has a pair of parallel bases, but its four sides are non-congruent.

## Properties

1. Has one pair of parallel and unequal opposite sides (bases); AD II BC
2. Has one pair of congruent non-parallel sides (legs); AB ≅ CD
3. Lower base angles are congruent; ∠B ≅ ∠C
4. Upper base angles are congruent; ∠A ≅ ∠D
5. Diagonals are congruent; AC ≅ BD
6. Any lower base angle is supplementary (adds up to 180°) to any upper base angle; ∠A + ∠C = 180° and ∠B + ∠D = 180°
7. Has one line of symmetry connecting the parallel sides (bases) at their midpoints; PQ is the line of symmetry, and PQ ⊥ BC

## Isosceles Trapezoid Theorem

The isosceles trapezoid theorem states that if a quadrilateral with one pair of parallel sides is an isosceles trapezoid, its legs must be congruent.

## Formulas

### Area

Let us solve an example to understand the concept better.

Find the area of an isosceles trapezoid whose bases are 181 mm and 73 mm, and height is 75 mm.

Solution:

As we know,
Area (A) = ½ (a + b) × h, here a = 181 mm, b = 73 mm, h = 75 mm,
∴A = ½ (181 + 73) × 75
= 9525 mm2

### Perimeter

Let us solve an example to understand the concept better. DEFG is an isosceles trapezoid. Find the measure of its perimeter.

Solution:

As we know,
Perimeter (P) = a + b + 2c, here a = 30 cm, b = 18.3 cm, c = 9.5 cm
∴P = 30 + 18.3 + 2 × 9.5
= 67.3 cm

## FAQs

Q1. Is an isosceles trapezoid a parallelogram?

Ans. No. In a parallelogram, the opposite sides are parallel and equal. In contrast, a trapezoid has parallel bases but unparallel legs. It can be a parallelogram ONLY if the remaining opposite sides are parallel. However, that is not an ideal case when it comes to defining an isosceles trapezoid.

Q2. How many lines of symmetry does the isosceles trapezoid have?

Ans. One. The line of symmetry connects the midpoints of the bases of an isosceles trapezoid.