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A trapezium is a 2D shape and a quadrilateral in which of two pairs of opposite sides, only one pair is parallel. The opposite parallel sides are referred to as the base and the non-parallel sides are referred to as the legs of the trapezium.

Let’s understand how to construct a trapezium using a compass and ruler with steps and examples.

## Construction of Trapezium

A trapezium is a quadrilateral with one pair of opposite sides parallel and two pairs of adjacent angles supplementary.

There are three methods of constructing a trapezium depending on the parameter known for construction. You can construct a unique trapezium when

- four sides are given (parallel sides are marked)
- three sides and one diagonal is given
- two parallel sides, one non-parallel side, and one angle is given
- one parallel and one non-parallel and two angles are given

### Construction of Trapezium When Four Sides Are Given

The following are the steps to construct a unique trapezium when four sides are given.

**Step 1:** Draw line segment $\text{AB}$ equal to the first parallel side (longer one)

**Step 2:** Mark a point $\text{E}$ on $\text{AB}$ such that $\text{AE}$ equal to second parallel side (shorter one)

**Step 3:** Draw an arc with $\text{E}$ as the centre and radius equal to the third side (non-parallel side)

**Step 4:** Draw another arc with $\text{B}$ as the centre and radius equal to the fourth side (non-parallel side), cutting the previous arc at $\text{C}$

**Step 5:** Draw an arc with $\text{C}$ as the centre and radius equal to the second non-parallel side

**Step 6:** Draw another arc with $\text{A}$ as the centre and radius equal to the second non-parallel side, cutting the previous arc at $\text{D}$

**Step 7:** Join points $\text{B}$ and $\text{C}$, points $\text{C}$ and $\text{D}$ and points $\text{D}$ and $\text{A}$

$\text{ABCD}$ is the required trapezium.

### Construction of Trapezium When Three Sides and One Diagonal Are Given

The following are the steps to construct a unique trapezium when three sides and one diagonal are given.

**Step 1: **Draw a line segment $\text{AB}$ equal to the first side

**Step 2:** With $\text{A}$ and $\text{B}$ as centers draw arcs of radii equal to the diagonal and the second side of a trapezium cutting at $\text{C}$

**Step 3:** Draw $\text{XC}$ parallel to $\text{BA}$

**Step 4:** With $\text{C}$ as centre and radius equal to the third side draw an arc cutting $\text{CX}$ at $\text{D}$

**Step 5:** Join $\text{AD}$.

$\text{ABCD}$ is the required trapezium.

### Construction of Isosceles Trapezium When Two Parallel Sides, One Non-Parallel Side, and One Angle are Given

The following are the steps to construct a unique trapezium when two parallel sides, one non-parallel side, and one angle is given.

**Step 1:** Draw a line segment $\text{PQ} equal to the length of one parallel side of a trapezium

**Step 2:** At $\text{Q}$ on $\text{PQ}$ construct an $\angle \text{PQX}$ equal to the measure of a given angle

**Step 3:** With $\text{Q}$ as centre and radius equal to the length of an equal side draw an arc. This cuts $\text{QX}$ at $\text{R}

**Step 4:** Construct $\text{RY}$ parallel to $\text{QP}$

**Step 5:** With $\text{P}$ as centre and radius equal to the length of the non-parallel side draw an arc cutting $\text{RY}$ at $\text{S}$

**Step 6:** Join $\text{PS}$

$\text{PQRS}$ is the required trapezium.

### Construction of Trapezium When One Parallel and One Non-Parallel and Two Angles are Given

The following are the steps to construct a unique trapezium when one parallel and one non-parallel and two angles are given.

**Step 1:** Draw a line segment $\text{AB}$ equal to the length of a parallel side

**Step 2:** On $\text{AB}$ at $\text{A}$ construct a given $\angle \text{BAX}$

**Step3: **On $\text{AB}$ at $\text{B}$ construct another given $\angle \text{ABY}$

**Step 4:** With $\text{B}$ as centre and radius equal to the length of the non-parallel side draw an arc cutting $\text{BY}$ at $\text{C}$

**Step 5:** Draw $\text{CZ}$ paralllel to $\text{AB}$. This cuts $\text{AX}$ at $\text{D}$

$\text{ABCD}$ is the required trapezium.

### Practice Problems

- What is a trapezium?
- Construct a trapezium in which $\text{AB} = 6 \text{ cm}$, $\text{BC} = \text{CD} = 4 \text{ cm}$ and $\text{DA} = 5 \text{ cm}$. Also, $\text{AB} || \text{CD}$.
- Construct a trapezium $\text{ABCD}$ in which $\text{AB} || \text{DC}$, $\text{AB} = 10 \text{cm}$, $\text{BC} = 5 \text{cm}$, $\text{AC} = 8 \text{cm}$ and $\text{CD} = 6 \text{ cm}$.
- Construct a trapezium $\text{PQRS}$ in which $\text{PQ}$ is parallel to $\text{SR}$, $\text{PQ} = 8 \text{ cm}$, $\angle \text{PQR} =70^{\circ}$, $\text{QR} = 6 \text{ cm}$ and $\text{PS} = 6 \text {cm}$
- Construct a trapezium $\text{ABCD}$ in which $\text{AB} || \text{DC}$, $\text{AB} = 7 \text{ cm}$, $\text{BC} = 6 \text{ cm}$, $\angle \text{BAD} = 80^{\circ}$ and $\angle \text{ABC} =70^{\circ}$.

## FAQs

### What are the 4 properties of a trapezium?

The four important properties of a trapezium are

a) One pair of opposite sides are parallel.

b) Two pairs of adjacent angles are supplementary.

c) The non-parallel sides in the trapezium are unequal except in the isosceles trapezium.

d) The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half of the sum of parallel sides.

### How many measurements are required to construct a trapezium?

To construct a unique trapezium, a minimum of four measurements are required. The most common methods of constructing a trapezium are when

a) four sides are given (parallel sides are marked)

b) three sides and one diagonal is given

c) two parallel sides, one non-parallel side, and one angle are given

d) one parallel and one non-parallel and two angles are given

## Conclusion

A trapezium is a quadrilateral with one pair of opposite sides parallel and two pairs of adjacent angles supplementary. You can construct a unique trapezium when four sides are given (parallel sides are marked), or three sides and one diagonal is given, or two parallel sides, one non-parallel side and one angle are given or one parallel and one non-parallel and two angles are given.

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