__maths__>__Construction / Practical Geometry (basics)__>__Construction of Quadrilaterals__### Construction of Parallelograms

In this page, a short and to-the-point overview of constructing parallelograms is provided. It is outlined as follows.

• Properties of parallelograms is explained

• The number of independent parameters in a parallelogram is `3`

• For a given parameter, construction of parallelograms is approached as combination of triangles (sss, sas, asa, rhs, sal) and using the properties of parallelograms.

*click on the content to continue..*

What is a parallelogram?

- a quadrilateral with two pairs of parallel sides
- a quadrilateral with two pairs of parallel sides
- not a quadrilateral

The answer is "a quadrilateral with two pair of parallel sides"

Quadrilateral is defined by `5` parameters. In a parallelogram, the following properties provide dependency of parameters

• opposite sides are parallel and that makes them equal

• opposite angles are equal

• adjacent angles are supplementary

• diagonals bisect

• two angles on diagonals are supplementary These properties cause two parameters to be dependent on other parameters and so, *a parallelogram is defined by `3` parameters*.

To construct a parallelogram, `2` (`bar(AB)`, `bar(BC)`) sides and a diagonal (`bar(AC)`) are given. This is illustrated in the figure. Which of the following helps to construct the specified parallelogram?

- Consider this as two SSS triangles `ABC` and `ACD`
- Consider this as two SSS triangles `ABC` and `ACD`
- Consider this as an SSS triangle `ABC` and another SAS triangle `ABD`

The answer is "Consider this as two SSS triangles `ABC` and `ACD`"

To construct a parallelogram, `2` sides (`bar(AB)`, `bar(BC)`) and an angle (`/_B`) are given. This is illustrated in the figure. Which of the following helps to construct the specified parallelogram?

- Consider this as two SSS triangles `ABC` and `ABD`
- Consider this as an SAS triangle `ABC` and another SSS triangle `ACD`
- Consider this as an SAS triangle `ABC` and another SSS triangle `ACD`

The answer is "Consider this as a SAS triangles `ABC` and another SSS triangle `ACD`".

Note: Once the first SAS triangle `ABC` is completed, the `bar(AC)` is fixed. Using that SSS triangle `ACD` is constructed.

To construct a parallelogram, a diagonal (`bar(AC)`), a side (`bar(AB)`), and an obtuse angle (`/_B`) are given. This is illustrated in the figure. Which of the following helps to construct the specified parallelogram?

- Consider this as two SSS triangles `ABC` and `ACD`
- Consider this as an SSA triangle `ABC` and an SSS triangle `ACD`
- Consider this as an SSA triangle `ABC` and an SSS triangle `ACD`

The answer is "Consider this as an SSA triangles `ABC` and an SSS triangle `ACD`".

Note: Once the first SAS triangle `ABC` is completed, that triangle can be copied to a SSS triangle `ACD`.

To construct a parallelogram, a side (`bar(AB)`), and two diagonals (`bar(AC)`, `bar(BD)`) are given. This is illustrated in the figure. Which of the following helps to construct the specified parallelogram?

- With a side and two diagonals, a parallelogram cannot be constructed
- Consider this as an SSS triangle `AOB`. Then construct points `C` and `D`
- Consider this as an SSS triangle `AOB`. Then construct points `C` and `D`

The answer is "Consider this as an SSS triangles `AOB`. Then construct points `C` and `D`".

Note: The diagonals bisect, and `AOB` is constructed with half-diagonals. The `vec(AO)` and `vec(BO)` are extended. The half diagonals are marked from point `O` to construct vertices `C` and `D`

To construct a parallelogram, two diagonals (`bar(AC)`, `bar(BD)`) and the angle between diagonals (`/_AOB`) are given. This is illustrated in the figure. Which of the following helps to construct the specified parallelogram?

- With two diagonals and angle between them, a parallelogram cannot be constructed
- Consider this as two SAS triangles `DOC` and `AOB`
- Consider this as two SAS triangles `DOC` and `AOB`

The answer is "Consider this as two SAS triangles `DOC` and `AOB`".

Note: Draw line `AOC` where points `A` and `C` are marked with half diagonal from point `O`. At the given angle line `BOD` is drawn and points `B` and `D` are marked.

**Construction of Parallelograms** :

Properties of Parallelograms

• opposite sides are parallel and equal

• opposite angles are equal

• adjacent angles are supplementary

• diagonals bisect

• two angles on diagonals are supplementary The formulations of questions

• `2` sides and `1` diagonal

• `2` sides and `1` angle

• `1` side, 1 diagonal and `1` angle

• `1` side and `2` diagonals

• `2` diagonals and `1` angle between diagonals *use properties to figure out dependent parameters and look for triangles*

*slide-show version coming soon*