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.

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).

## 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.