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Geometry: Congruent Polygons: Quadrilaterals: Parallelograms, Squares, Rectangle, Rhombus, Trapezoid





 



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One basic way of telling if two or more polygons are congruent is by checking to see if all their corresponding sides and angles are congruent. This is similar to the triangle except carried further to polygons with four or more sides. The converse is also true so that if two or more polygons have all of their corresponding parts congruent then the polygons are congruent.

If all the corresponding parts of two polygons are congruent then the polygons are congruent.

The definition for congruent polygons is as follows:

Congruent Polygons
If there is correspondence between the vertices of the polygons with their corresponding angles and sides being congruent, then the polygons are congruent.


Polygons with corresponding congruent sides and angles are congruent.
CorrespondenceCongruence
VerticesLine Segments
Angles
Line Segments
Angles
A ↔ EABDEAB congruent to DE
B ↔ FBCEFBC congruent to EF
C ↔ GCDFGCD congruent to FG
D ↔ HADEFAD congruent to EF
angleABC ↔ angleEFGangleABC congruent to angleEFG
angleBCD ↔ angleFGHangleBCD congruent to angleFGH
angleACD ↔ angleEHGangleADC congruent to angleEHG
angleDAB ↔ angleHEFangleDAB congruent to angleHEF

The triangle is the simplest polygon. The isosceles triangle, however, has certain congruence properties stated by the following.

Isosceles Triangle Congruence
If two sides of a triangle are congruent then the two angles that are opposite to these sides are also congruent.


This theorem can also be applied to equilateral triangles as a corollary.

If a triangle is equilateral then it is also equiangular. In other words, if all the sides are equal then all the angles are equal in measure and therefore congruent.

Equilateral triangles with equal corresponding sides also have equal corresponding angles and are congruent.

This can also be applied to regular polygons that have equal sides. If all the sides of a regular polygon are congruent then the angles must be congruent. The converse is also true.

Quadrilaterals

Quadrilaterals are polygons with four sides and four angles. There are many different types of quadrilaterals that have certain characteristics that define them. For example, a square is equal in length on all four sides. The usual kinds of quadrilaterals described in geometry are the parallelogram, trapezoid, rhombus, kite, rectangle and square. They can be defined by their congruent parts as well as their other features.

Parallelograms

A parallelogram is a quadrilateral in which its opposite sides are parallel but can have any angle to adjacent sides.

A parallelogram has parallel opposite sides.

Some of the theorems associated with the parallelogramís congruent parts are the following:

Parallelogram Opposite Angles
The opposite angles in a parallelogram are congruent.


The opposite angles of a parallelogram are congruent.
Parallelogram Opposite Sides
The opposite sides of a parallelogram are congruent.


The opposite sides of a parallelogram are congruent.

In addition to the congruency theorems there are others:

Parallelogram Consecutive Angles
Consecutive angles in a parallelogram are supplementary.


Adjacent angles in a parallelogram are supplementary.
Parallelogram Diagonals
The diagonals of a parallelogram bisect each other.


Parallelogram diagonals bisect each other.

In the case of congruency the converse is also true. This can be used along with the following theorem to test for a parallelogram.

Quadrilateral To Parallelogram
If one of the pairs of opposite sides in a quadrilateral are both parallel and congruent then the quadrilateral is a parallelogram.


Kite

A kite is a special kind of quadrilateral that gets its name from the shape of a kite that flies in the wind.

A kite is a special type of quadrilateral.

The following is the definition for the kite.

Kite
A kite is a quadrilateral that has two pairs of congruent sides that are adjacent.


A kite has two pairs of adjacent sides that are congruent.

The kite also has one pair of its opposite angles being congruent.

There is only one pair of opposite angles in a kite that are congruent.

Rectangles

One of the most familiar quadrilaterals is the rectangle which is also a parallelogram since its opposite sides are parallel and congruent. The rectangle, however, only has right angles. Some of the special theorems associated with rectangles are as follows:

Rectangle
A rectangle is a parallelogram that has a right angle.


Rectangle Diagonals
A rectangle is a parallelogram that has congruent diagonals.


Rectangle Corollary
All the angles of a rectangle are right angles.


All the angles of a rectangle are congrent since they are right angles.

Square

The square is a special case of the rectangle where all the sides are equal to one another. The square can be defined as follows:

Square
The square is a rectangle with adjacent sides being congruent.


A square is a rectangle with congruent adjacent sides.
Square Corollary
All the sides of a square are congruent.


A square has all sides congruent.

Rhombus

The rhombus is a special type of parallelogram that looks like a square tilted on its side.

A rhombus is a parallelogram with all congruent sides, like a square.

The plural for rhombus is rhombi. Just like the square, all the sides of a rhombus are congruent. A square, however, only contains right angles.

A rhombus does not have to include right angles.

Some of the definitions and theorems associated with a rhombus are as follows:

Rhombus
A rhombus is a parallelogram with two congruent adjacent sides.
Rhombus Diagonals
The diagonals of a rhombus are perpendicular.
Rhombus Bisectors
Each pair of opposite angles in a rhombus are bisected by the diagonals of the rhombus.
The angles of a rhombus are bisected by the diagonals.
Rhombus Sides
All the sides of a rhombus are congruent.
While a rhombus has all congruent sides, only opposite angles are congruent.
Perpendicular Rhombus Diagonals
All the diagonals of a rhombus are perpendicular.


Trapezoids

A trapezoid is a quadrilateral that looks like a triangle with the top removed.

A trapezoid is the result when a triangle is cut into two parts by a line parallel to one of its sides.

It consists of two parallel line segments on opposite sides called bases. The other two line segments are called legs and are not parallel like a parallelogram. A trapezoid can be formally defined as follows:

Trapezoid
A quadrilateral with exactly one pair of parallel sides.


Isosceles Trapezoids

Isosceles trapezoids are trapezoids with congruent legs. Some of the properties of isosceles trapezoids are as follows:

Isosceles Trapezoid Base Angles
The pair of base angles for each base of the trapezoid are congruent.


Isosceles Trapezoid Diagonals
The diagonals of an isosceles trapezoid are congruent.


An isosceles trapezoid has congruent diagonals.

Median

The median of a trapezoid bisects the legs.

The legs of a trapezoid are bisected by the median.

Altitude

The altitude of a trapezoid extends from one base to the other and is perpendicular to both bases.

The trapezoid altitude is the minimum distance between its parallel sides.
 




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