Angles

Angles are used by almost everyone. Engineers and architects use them to craft designs for buildings, roads, and various other projects. Football players and tennis stars use them to enhance their performance in sports. Carpenters use angles to make tables and chairs. Even artists use angles to sketch their drawings and frame elaborate paintings.

Any time two lines intersect at a point, an angle is formed. The point of the intersection is called the vertex, which can be labeled with a letter such as B. The two lines that make up the angle are called the legs, and they extend out to the other two end points, which can be labeled A and C. This means the angle has three points and can be referred to as angle ABC. The interior of the angle is the space inside the "jaws" that extend outward to infinity. The exterior of the angle is everything outside those jaws.

There are six types of angles used in geometry, trigonometry and calculus: right angles, acute angles, obtuse angles, straight angles, reflex angles, and full angles. Angles can also be categorized in relation to each other as adjacent, complementary, or supplementary angles.

Overview

The history of angle measurement goes back to 1500 BC and the Babylonians. At this time, measurements were made by the sun's shadow as it shone down on a vertical rod known as a gnomon. A gnomon would cast a shadow against various markings on a stone tablet that acted like a sundial. It could be used to measure the degrees of an angle. Another tool, discovered by scientists excavating Pompeii, was the "groma," which had four stones hanging from cords to form four right angles that could be used to make various measurements. A surveying tool called the optical square is the present-day version of the groma.

The concept of angles becomes more complex when considering the way in which they are measured. A protractor is often the quickest way of measuring, but there are also many formulas and calculations that can be used to determine the size, or degree, of angles. The size of an angle is sometimes given as "angle degrees" or in notation form as " ."

Drawing Acute, Obtuse, and Right Angles

Angles are drawn by using three simple concepts; the point, the line segment, and the ray. The first point is the vertex of the angle, usually labeled B. Two line segments are drawn from point B to point A and from point B to point C, so that the final angle looks like an opening mouth of a crocodile. Add two small arrow signs at the end of each line segment and you will have two rays pointing outward from the vertex and creating an infinite angle.

To distinguish an acute, obtuse, or right angle, imagine the mouth of the crocodile. If the jaws are opened exactly 90°, with its nose (point A) pointing straight up, it is called a right angle. If the jaws are opened less than 90°, this is an acute angle. When the mouth of the crocodile is more than 90°, stretching its jaws so that the nose points backwards toward its tail, this is an obtuse angle.

Measuring Angles

In trigonometry, angles can be very complex and have measurements even larger than 360°. They are named as ABC or with a single lowercase letter, such as a so that the measurement (m) is expressed as . A Greek letter, such as α (alpha) or θ (theta), is also sometimes used to name the angle, as in inches. These angles can be either positive or negative, as in inches, and they can be placed on a grid with an x- and y-axis to determine the measurements around a circle. The common formula used to determine these measurements is where r = radian, s = length of the arc, and a = angle in degrees. See Figure 1.

At this higher level, angles are measured in radians found by placing the angle inside the circle and measuring from the center to the outer arc. The leg of the angle is the radian, and it can be laid along the outside of the circle. When the arc of the angle is the same as the radian it is expressed as .

Adjacent Angles

When two angles are sitting next to each other with a common side, they are called adjacent. These angles will share the same vertex, and the interior of the angles will never overlap. For example, and share the common side AC and common vertex point A. These are adjacent angles also form a larger angle , which is the sum of the two adjacent angles.

Adjacent angles can be found in polygons where there are many angles sitting next to each other. In this case, the angles are obtuse. With six points on this polygon there are six vertexes and therefore six sets of adjacent angles. See Figure 2.

Exploring Angle Pairs

When one line crosses over two other parallel lines, it is called a transversal. A transversal creates many different angles that are often similar. An example of a transversal is shown in Figure 3.

The angles that are similar are referred to as angle pairs. Because there are many types of transversals, there are also many types of angle pairs, including vertical angles, corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. Angle pairs can be used to test whether two of the lines are parallel.

Complementary and Supplementary Angles

Two angles together can be categorized as complementary or supplementary. Angles are complementary when their combined measurements equal 90°. Figure 4 shows two angles that are adjacent, because they share a ray and a vertex, and also complementary.

These angles are complementary because the measure of angle x plus the measure of angle y equal 90° but they don’t have to be adjacent to be complementary. The angles shown in Figure 5 are also complementary because their sum equals 90° but they are not adjacent.

Angles are categorized as supplementary when their measurements total 180°. Figure 6 shows two angles that are supplementary.

Like complementary angles, supplementary angles do not have to be adjacent as long as the sum of their angles equals 180°. See Figure 7.

Angle Addition Postulate and Decomposing Angles

The angle addition postulate states that when a point lies in the interior of an angle such as ABC, this point can be used to create two angles whose total is equal to the measurement of the original larger angle. This is shown in Figure 8.

Decomposing angles is best understood by simply understanding the word decompose, that is, to break down into smaller parts. When angles are decomposed, their measurements can be found because the larger angle has already been measured. An example is a set of smaller angles, such as those inside a circle. These will always total up to 360°. A circle split down the middle will always create two angles of 180° each. If the measure of all but one angle inside a circle are known, the known angles can be added up and the total subtracted from 360°, leaving the measure of the remaining angle. Figure 9 shows how complex these angle measurements may look while still totaling up to 360°.

Angle Bisector Theorem

The angle bisector theorem says that a line AD that perfectly bisects an angle can help to determine the lengths of other segments around the triangle. The formula

can be used to make various calculations and determine the lengths of different line segments. The angle bisector theorem is often used when at least one length of the bisector or segment on the triangle are known. In Figure 10, three sides are known where AB = 10, CD = 3, and BD is 4; therefore AC = 7.5.

Bibliography

McKellar, Danica. Girls Get Curves. New York: Penguin, 2012.

Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015.

Wolfe, Harold E. Introduction to Non-Euclidean Geometry. Mineola, NY: Dover, 2012.