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A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices, and four angles. It is formed by joining four non-collinear points. The sum of four angles in a quadrilateral is $360^{\circ}$.

Let’s understand the construction of quadrilaterals with steps and examples.

## Construction of Quadrilaterals

A quadrilateral is a closed 2D geometrical shape with four sides and four angles. The sum of four angles of a quadrilateral is $360^{\circ}$. There are three methods of constructing a quadrilateral depending on the parameter known for construction. You can construct a unique quadrilateral when

- four sides and one diagonal is given
- three angles and two included sides are given
- three sides and two included angles are given

**Note:** To construct a unique quadrilateral, at least five parameters are required.

### Construction of Quadrilaterals When Four Sides and One Diagonal is Given

Let’s consider an example of a quadrilateral to understand the construction of quadrilaterals when four sides and one diagonal is given.

Construct a quadrilateral $\text{ABCD}$ with $\text{BC} = 4.5 \text{ cm}$, $\text{DA} = 5.5 \text{ cm}$, $\text{CD} = 5 \text{ cm}$ and the diagonal $\text{AC} = 5.5 \text{ cm}$, diagonal $\text{BD} = 7 \text{ cm}$.

The following are the steps to construct a unique quadrilateral with given four sides and one diagonal.

**Step 1:** Construct a $\triangle \text{ACD}$ using $\text{SSS}$ construction criteria.

Read the construction of triangles to understand the steps.

**Step 2:** Taking $\text{D}$ as a centre and radius equal to the length of diagonal $\text{BD} = 7 \text{ cm}$ draw an arc opposite side of the diagonal $\text{AC}$

**Step 3:** Taking $\text{C}$ as a centre and radius equal to the length of side $\text{BC} = 4.5 \text{ cm}$ draw an arc intersecting the arc drawn in Step 2 at $\text{B}$

**Step 4:** Join the points $\text{A}$, $\text{B}$ and $\text{C}$, $\text{B}$

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

### Construction of Quadrilaterals When Three Sides and Two Included Angles are Given

Let’s consider an example of a quadrilateral to understand the construction of quadrilaterals when three sides and two included angles are given.

Construct a quadrilateral $\text{PQRS}$, with sides $\text{PQ} = 3.5 \text{ cm}$, $\text{QR} = 4.5 \text{ cm}$, $\text{RS} = 4 \text{ cm}$ and $\angle {\text{Q}} = 110^{\circ}$, $\angle {\text{R}} = 80^{\circ}$.

The following are the steps to construct a unique quadrilateral with given three angles and two included sides.** ****Step 1:** Draw the side $\text{QR} = 4.5 \text{ cm}$

**Step 2:** At point $\text{Q}$ with base line as $\text{QR}$, draw an angle of $110^{\circ}$

**Step 3:** At point $\text{R}$ with base line as $\text{QR}$, draw an angle of $80^{\circ}$

**Step 4:** With $\text{Q}$ centre and radius equal to side $\text{PQ} = 3.5 \text{ cm}$, draw an arc intersecting the line at $\text{Q}$ at the point $\text{P}$.

**Step 5:** With $\text{R}$ centre and radius equal to side $\text{RS} = 4 \text{ cm}$, draw an arc intersecting the line at $\text{R}$ at the point $\text{S}$.

**Step 6:** Join the points $\text{P}$ and $\text{S}$

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

**Construction of Quadrilaterals When Three Angles and Two Included Sides are Given**

Let’s consider an example of a quadrilateral to understand the construction of quadrilaterals when three angles and two included sides are given.

Construct a quadrilateral $\text{ABCD}$, with sides $\text{AB} = 4 \text{ cm}$, $\text{BC} = 5 \text{ cm}$, $\text{DA} = 3 \text{ cm}$ and $\angle \text{{A}} = 100^{\circ}$, $\angle \text{{B}} = 120^{\circ}$.

The following are the steps to construct a unique quadrilateral with given three angles and two included sides.

**Step 1:** Draw the side $\text{AB} = 5 \text{ cm}$

**Step 2:** At the point $\text{A}$, draw an angle of $120^{\circ}$, with $\text{AB}$ as a base

** Step 3: **At the point $\text{B}$, draw an angle of $110^{\circ}$, with $\text{BA}$ as a base

**Step 4:** With $\text{B}$ as a centre and radius equal to $3 \text{cm}$, draw an arc intersecting the line at $\text{B}$, at $\text{C}$

**Step 5:** At the point $\text{C}$, draw an angle of $90^{\circ}$, with $\text{BC}$ as a base intersecting at $\text{D}$

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

## Practice Problems

- Construct a quadrilateral $\text{LMNO}$ in which $\text{LM} = 4.2 \text{ cm}$, $\text{MN} = 6 \text{ cm}$, $\text{NO} = 5.2 \text{ cm}$, $\text{OL} = 5 \text{ cm}$ and $\text{LN} = 8 \text{ cm}$.
- Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 3.5 \text{ cm}$, $\text{BC} = 3.8 \text{ cm}$, $\text{CD} = \text{DA} = 4.5 \text{ cm}$ and diagonal $\text{BD} = 5.6 \text{ cm}$.
- Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 3.6 \text{ cm}$, $\text{BC} = 3.3 \text{ cm}$, $\text{AD} = 2.7 \text{ cm}$, diagonal $\text{AC} = 4.6 \text{ cm}$ and diagonal $\text{BD} = 4 \text{ cm}$.
- Construct a quadrilateral $\text{LMNO}$ in which $\text{LN} = \text{LO} = 6 \text{ cm}$, $\text{MN} = 7.5 \text{ cm}$ , $\text{MO} = 10 \text{ cm}$ and $\text{NO} = 5 \text{ cm}$.
- Construct a quadrilateral $\text{PQRS}$ in which $\text{PQ} = 5.4 \text{ cm}$, $\text{BC} = 6 \text{ cm}$, $\text{QR} = 4.6 \text{ cm}$, $\text{RS} = 4.3 \text{ cm}$, $\text{SP} = 3.5 \text{cm}$ and diagonal $\text{BD} = 5.6 \text{ cm}$.
- Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = \text{BC} = 3.5 \text{ cm}$, $\text{AD} = \text{CD} = 5.2 \text{ cm}$ and $\angle \text{ABC} = 120^{\circ}$.
- Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 2.9 \text{ cm}$, $\text{BC} = 3.2 \text{ cm}$, $\text{CD} = 2.7 \text{ cm}$, $\text{DA} = 3.4 \text{ cm}$ and $\angle \text{A} = 70^{\circ}$.
- Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 3.5 \text{ cm}$, $\text{BC} = 5 \text{ cm}$, $\text{CD} = 4.6 \text{ cm}$, $\angle \text{B} = 125^{\circ}$ and $\angle \text{C} = 60^{\circ}$.
- Construct a quadrilateral $\text{ABCD}$ in which $\text{AB} = 5.6 \text{ cm}$, $\text{BC} = 4 \text{ cm}$, $\angle \text{A} = 50^{\circ}$, $\angle \text{B} = 105^{\circ}$ and $\angle \text{D} = 80^{\circ}$.

## FAQs

### How many minimum measurements are required to construct a unique quadrilateral?

The minimum number of measurements required to construct a unique quadrilateral is five.

### What are the different methods of construction of quadrilaterals?

There are three different methods of constructing a unique quadrilateral. These are when

a) four sides and one diagonal is given

b) three angles and two included sides are given

c) three sides and two included angles are given

## Conclusion

A quadrilateral is a closed 2D geometrical shape with four sides and four angles. The sum of four angles of a quadrilateral is $360^{\circ}$. To construct a unique quadrilateral, at least five parameters are required. You can construct a unique quadrilateral when four sides and one diagonal is given, or three angles and two included sides are given or three sides and two included angles are given.

## Recommended Reading

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