Construction of Parallelograms

overview

In this page, constructing parallelograms is explained. 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.

recap

A parallelogram 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, $2$ ($\overline{AB}$, $\overline{BC}$) sides and a diagonal ($\overline{AC}$) are given. This is illustrated in the figure. To construct, Consider this as two SSS triangles $ABC$ and $ACD$.

To construct a parallelogram, $2$ sides ($\overline{AB}$, $\overline{BC}$) and an angle ($\angle B$) are given. This is illustrated in the figure.

To construct the specified parallelogram, "Consider this as a SAS triangles $ABC$ and another SSS triangle $ACD$".

Note: Once the first SAS triangle $ABC$ is completed, the $\overline{AC}$ is fixed. Using that SSS triangle $ACD$ is constructed.

To construct a parallelogram, a diagonal ($\overline{AC}$), a side ($\overline{AB}$), and an obtuse angle ($\angle B$) are given. This is illustrated in the figure. To construct the specified parallelogram "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 ($\overline{AB}$), and two diagonals ($\overline{AC}$, $\overline{BD}$) are given. This is illustrated in the figure.

To construct the specified parallelogram, "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 $\overrightarrow{AO}$ and $\overrightarrow{BO}$ are extended. The half diagonals are marked from point $O$ to construct vertices $C$ and $D$

To construct a parallelogram, two diagonals ($\overline{AC}$, $\overline{BD}$) and the angle between diagonals ($\angle AOB$) are given. This is illustrated in the figure.

To construct the specified parallelogram, "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.

summary

**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*

Outline

The outline of material to learn "Construction / Practical Geometry at 6-8th Grade level" is as follows.
Note: * click here for detailed outline of "constructions / practical geometry".*

• Four Fundamenatl elements

→ __Geometrical Instruments__

→ __Practical Geometry Fundamentals__

• Basic Shapes

→ __Copying Line and Circle__

• Basic Consustruction

→ __Construction of Perpendicular Bisector__

→ __Construction of Standard Angles__

→ __Construction of Triangles__

• Quadrilateral Forms

→ __Understanding Quadrilaterals__

→ __Construction of Quadrilaterals__

→ __Construction of Parallelograms__

→ __Construction of Rhombus__

→ __Construction of Trapezium__

→ __Construction of Kite__

→ __Construction of Rectangle__

→ __Construction of Square__