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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.
The definition for congruent polygons is as follows:
The triangle is the simplest polygon. The isosceles triangle, however, has certain congruence properties stated by the following.
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.
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 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.
A parallelogram is a quadrilateral in which its opposite sides are parallel but can have any angle to adjacent sides.
Some of the theorems associated with the parallelogramís congruent parts are the following:
In addition to the congruency theorems there are others:
In the case of congruency the converse is also true. This can be used along with the following theorem to test for a parallelogram.
A kite is a special kind of quadrilateral that gets its name from the shape of a kite that flies in the wind.
The following is the definition for the kite.
The kite also has one pair of its opposite angles being congruent.
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:
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:
The rhombus is a special type of parallelogram that looks like a square tilted on its side.
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.
Some of the definitions and theorems associated with a rhombus are as follows:
A trapezoid is a quadrilateral that looks like a triangle with the top removed.
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:
Isosceles trapezoids are trapezoids with congruent legs. Some of the properties of isosceles trapezoids are as follows:
The median of a trapezoid bisects the legs.
The altitude of a trapezoid extends from one base to the other and is perpendicular to both bases.
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