Types of quadrilaterals are polygons with four sides (or edges) and four vertices (or corners) that can have different shapes.

## Triangles: An Introduction

Triangles are plane (i.e., flat, two-dimensional) figures with three straight sides (labeled a, b, and c in the figure shown on the left) and three angles. Triangles are formed by three straight lines that join three vertices (labeled A, B, and C). The symbol for a triangle is Δ. So the triangle shown is referred to as ΔABC. The small square at vertex C is used to denote an angle of 90 degrees or a right angle. Triangles have many applications in physics, architecture, and many other disciplines. They are so important within geometry, that they have a specialized field of study termed “trigonometry”, which will be considered in Chapter 9. The longest side, c, of a right triangle is called the hypotenuse.

## Classification of a Triangle: By Angle

An angle (shown below) is a figure formed by non-parallel segments that intersect each other at a given point. The word angle comes from the Latin angulus which means corner. As you may already know, the word triangle angle describes a 3-sided polygon comprised of 3 angles. An angle is the arc formed by two straight segments of a line, called the sides of the angle. The sides of the angle meet at the vertex or intersection point of both lines. A protractor (shown on the right) is a tool or instrument that is used to measure an angle in units of degrees, minutes, and seconds. Here is one such protractor that can measure angles from 0 to 360 degrees (usually written at 360°) or a full circle.

Degrees can be further divided into units of minutes of arc and seconds of arc. There are 60 minutes of arc in 1 degree, and 60 seconds of arc in one minute. Thus, 5 degrees 30 minutes 18 seconds (usually written as 5° 30’ 18”) is equivalent to 5.505°―since 18”=18/60=0.3’ and 0.3’=0.3/60=0.005°, and 30’=30/60=0.5°). The Greek letter theta (ϴ) is commonly used to denote the measure of an angle, such as

Angles can be classified according to the measure of their angle. There are three specific types of angles: acute, right, and obtuse as shown in the table below.

Type of Angle | Description | Example |

acute | Angle of less than 90° | |

right | 90° angle is formed when two lines are perpendicular to each other. | |

obtuse | Angle of greater than 90° |

There are two special cases of angles shown below over which a protractor has been placed. Notice in this 1st case that the red line, forming one side of the angle, is parallel (or coincident lines—on top of one another) to the blue line, forming the other side of the angle. These sides form an obtuse angle of 180° on the protractor.

In the 2nd case, notice that the red line, forming on side of the angle, is again parallel and directly on the blue line, forming the other side of the angle. The angle formed here on the protractor is an acute angle of 0°. Alternatively, the outside angle (measured from the red line, counter-clockwise around to the blue line) could be considered an obtuse angle of 360°.

Triangles can be classified by their interior angles as right (with at least one 90° angle) or oblique (without any 90° angle).

Triangle | Description | Example |

acute (oblique) | Contains interior angles that are all less than 90°. | |

right | Contains a 90° angle (or right angle). | |

obtuse (oblique) | Contains one obtuse angle, i.e. one angle greater than 90°. |

## Classification of a Triangle: By Congruent Sides

Congruent sides are simply sides that have the same length or measure. In addition to classifying triangles like we did previously by considering their angles, we can also classify triangles based on the number of congruent sides of which they are comprised. The table below shows three different classifications of a triangle based on the congruency (comparison of the lengths) of its sides.

Triangle | Description | Example |

equilateral | All three sides have equal lengths and the three interior angles are acute and 60°. An equilateral triangle has three congruent sides. Notice the “tick” marks placed on each side. These denote that all the sides have the same length. | |

isosceles | Two of the three sides of the triangle have equal lengths and the angles facing those sides are equal. An isosceles triangle has two congruent sides. Notice only two of the sides have tick marks. | |

scalene | The three sides have different lengths and the angles have different measurements. A scalene triangle has no congruent sides. |

## Quadrilaterals: An Introduction

As we saw previously, quadrilaterals are polygons with four sides (or edges) and four vertices (or corners) that can have different shapes. The interior angles of a quadrilateral add up to 360 degrees. The word “quadrilateral” comes from two Latin words—quadri meaning four and latus meaning sides. Let’s identify six different types of quadrilaterals.

Quadrilateral | Shape | Description |

square (regular quadrilateral) | A regular quadrilateral having all four sides of equal length (equilateral), and all four angles are right angles (90°)– opposite sides are parallel (a square is a parallelogram), the diagonals perpendicularly bisect each other (a square is a rhombus), and are of equal length (a square is a rectangle). | |

rectangle (irregular quadrilateral) | Opposite sides are parallel and of equal length. Also, all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. When sides are of equal length, this special condition is called a square. | |

parallelogram (irregular quadrilateral) | A quadrilateral with two pairs of parallel sides—opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid (similar to rhombus, but with different length sides). | |

rhombus (irregular quadrilateral) | A quadrilateral with four sides of equal length—opposite sides are parallel and opposite angles are equal, or the diagonals bisect each other at right angles (i.e., the diagonals are perpendicular). Every rhombus is a parallelogram, but not every parallelogram is a rhombus. | |

trapezoid (irregular quadrilateral) | A trapezoid has a pair of opposite sides that are parallel. If the sides that are not parallel are of equal length and both angles from a parallel side are equal, this is a special case called an isosoles trapezoid. | |

irregular quadrilateral | Only the square is a regular quadrilateral, so all other quadrilaterals are irregular. |