Construction / Practical Geometry (basics) : Construction of Parallelograms

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Construction of Parallelograms

what you'll learn...

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 $3$ $3$

• 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 $5$ $5$ 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 $3$ $3$ parameters.

parallelogram construction 2 sides and a diagonal

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

parallelogram construction 2 sides and an angle

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

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

Note: Once the first SAS triangle $A B C$ $ABC$ is completed, the $\bar{A C}$ $bar(AC)$ is fixed. Using that SSS triangle $A C D$ $ACD$ is constructed.

parallelogram construction a diagonal, a sides, and an angle

To construct a parallelogram, a diagonal ( $\bar{A C}$ $bar(AC)$ ), a side ( $\bar{A B}$ $bar(AB)$ ), and an obtuse angle ( $∠ B$ $/_B$ ) are given. This is illustrated in the figure. To construct the specified parallelogram "Consider this as an SSA triangles $A B C$ $ABC$ and an SSS triangle $A C D$ $ACD$ ".

Note: Once the first SAS triangle $A B C$ $ABC$ is completed, that triangle can be copied to a SSS triangle $A C D$ $ACD$ .

parallelogram construction a sides, and 2 diagonals

To construct a parallelogram, a side ( $\bar{A B}$ $bar(AB)$ ), and two diagonals ( $\bar{A C}$ $bar(AC)$ , $\bar{B D}$ $bar(BD)$ ) are given. This is illustrated in the figure.

To construct the specified parallelogram, "Consider this as an SSS triangles $A O B$ $AOB$ . Then construct points $C$ $C$ and $D$ $D$ ".

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

parallelogram construction 2 diagonals and angle

To construct a parallelogram, two diagonals ( $\bar{A C}$ $bar(AC)$ , $\bar{B D}$ $bar(BD)$ ) and the angle between diagonals ( $∠ A O B$ $/_AOB$ ) are given. This is illustrated in the figure.

To construct the specified parallelogram, "Consider this as two SAS triangles $D O C$ $DOC$ and $A O B$ $AOB$ ".

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

• $2$ $2$ sides and $1$ $1$ diagonal

• $2$ $2$ sides and $1$ $1$ angle

• $1$ $1$ side, 1 diagonal and $1$ $1$ angle

• $1$ $1$ side and $2$ $2$ diagonals

• $2$ $2$ diagonals and $1$ $1$ angle between diagonals

use properties to figure out dependent parameters and look for triangles

Outline