Lines and Angles Part 1: What are Lines and Angles?

Photo by Elizaveta Dankevich on Unsplash

A pool player lining up the perfect shot to sink the 8-ball relies on a similar set of skills to the ones a sailor in the 1500s would use to navigate the open seas: an understanding of the properties of lines and angles. Professional pool players have developed an intuitive understanding of the angles at which billiard balls scatter. Ancient sailors used various instruments, like the astrolabe, to triangulate their position based on the location of the sun, moon, and stars. Lines and angles are used in all kinds of fields, including architecture, fashion, and astronomy.

If you connect any two points, you will form a line. In the Cartesian coordinate plane (the one you’re familiar with from math class), any point is represented by 2 or 3 coordinates (depending on whether you are in 2 or 3 dimensions): (x, y) or (x, y, z). These coordinates represent the shortest distance (line) from a point to each respective axis. For example, the point (1, 2) represents the intersection of perpendicular lines drawn from the 1 on the x-axis and the 2 on the y-axis.

When two or more lines intersect, they form an angle. The measure of an angle is the amount of rotation that separates two lines, line segments, or rays about the point of intersection.  

Here are some of the most common types and properties of lines and angles.

Before we go on, it is important to clarify that these properties come from a set of 5 axioms, or accepted truths, first put forth by the Greek mathematician Euclid (hence the name Euclidean Geometry). Check them out here! You might be surprised by how simple they are.

The following definitions and properties all come directly from these 5 axioms, and so, the properties only apply to Euclidean Geometry. If you study more advanced math, you will come across other geometries (space that is shaped like a sphere or hyperbola) that behave quite differently from Euclidean space.  

Lines

A line is a one-dimensional object that extends infinitely in two directions with a constant slope (in math, a line has length but no width). It is defined as the set of values (x,y) that satisfy a linear equation: y = m * x + b where m and b are constants (a value that doesn’t change). Lines are notated \( \overleftrightarrow{A}\). There are 180 degrees in a line.

Line Segment

A line segment is part of a line. Unlike a line, a line segment has two distinct endpoints. \(\overline{AB}\) below is a line segment.

Ray

A ray is a line with one endpoint. In the example below, the ray begins at point A and continues infinitely. Rays are denoted: \(\overrightarrow{A}\)

Linear Function

A linear function is a function of the form y = m * x + b. This is called slope-intercept form. y is the output value, x is the input value, m is the slope of the line, and b is the y-intercept (the place where the line crosses the y-axis). The example below shows the linear functions y = 3 and y = 5x + 7.

Slope

A line’s slope is the difference between y values divided by the difference between x values of any two points on that line. A positive slope means the line is moving upwards from left to right, a negative slope means the line is moving downwards from left to right, a slope of 0 means the line is horizontal, and an undefined slope means the line is vertical.

In the diagram below, the blue line has a positive slope (3), the orange line has a negative slope (-3), and the green line has a slope of 0.

To find the slope of a line if given two points, use the following formula:

This represents the difference in y values (rise) over the difference in x values (run) of two points on the line.

Parallel

Two lines are parallel if they have the same slope and different intercepts. Parallel lines never intersect.

Perpendicular

Two lines are perpendicular if their slopes are negative reciprocals of each other. Perpendicular lines intersect at a right angle.

Tangent

A tangent line is a line that intersects a curve at a single point. If a line is tangent to a circle, it is perpendicular to its radius and diameter where it intersects the circle.

Points are Collinear

A set of points are collinear if they lie on a single line segment, line, or ray. In the example below, points W, X, Y, and Z are collinear.

Midpoint

The midpoint of a line segment is the point equidistant from both endpoints. The midpoint formula of a line segment averages the x and y coordinates of the endpoints as follows:

Distance

Distance is the length of the line connecting two points. To find the length of a vertical line segment, just find the difference between the y values of the endpoints. To find the length of a horizontal line segment, find the difference between the x values of the endpoints.

For any line segment on a diagonal, you can find the distance between two points by using the distance formula, which is a version of the Pythagorean Theorem \((a^2 + b^2 = c^2)\). Essentially, you are making the line by finding the length of the hypotenuse of a right triangle.

Transversal

A transversal is a line that crosses two or more other lines at distinct points. In the example below, the green line is the transversal that crosses the pair of parallel lines.

Angles

Acute

Acute angles are \(> 0 ^\circ\) but \(< 90 ^\circ\). For example, a \( 35 ^\circ\) angle is acute. In the diagram below, the highlighted angles marked \(a^\circ\) are both less than \( 90 ^\circ\).

Right

Right angles = \(90 ^\circ\). The angles created where the two lines intersect below are right angles. Right angles are marked by a box.

 

Obtuse

Obtuse angles are \( > 90 ^\circ\) but \( < 180 ^\circ\) . For example, a \( 135 ^\circ\) angle is obtuse. In the diagram below, the highlighted angles marked b are both greater than \( 90 ^\circ\) but less than \( 180 ^\circ\).

Bisect

A line, line segment, or ray bisects an angle if it divides it into 2 equal angles. In the example below, the  \( 90 ^\circ\) angle is divided into two equal angles marked a, which are both \( 45 ^\circ\).

Interior

Interior angles are angles in the inside of any shape or figure. In the example below, the \(60 ^\circ\) angles are interior angles.

Exterior

Exterior angles are angles on the outside of any shape or figure. In the example below, the \(120 ^\circ\) angles are exterior angles.

Corresponding

Corresponding angles are two or more angles that are at the same relative position with respect to each angle’s vertex. On the test, you’re likely to use this property when you see similar shapes (proportional sides and identical angle measures). You will also see it when two parallel lines are cut by a transversal. In the example below, the angles marked A are corresponding, and the angles marked B are corresponding.

 

Complementary

Two angles are complementary if they add to \(90 ^\circ\). In the example below, two angles that are each \(45 ^\circ\) add to \(90 ^\circ\), but the angles don’t have to be equal to be complementary. For example, two angles that are \(27 ^\circ\) and \(63 ^\circ\) would also be complementary.

Supplementary

Two angles are supplementary if they add to \(180 ^\circ\). In the examples below, the highlighted angles marked a and b are supplementary (the unhighlighted a and b angles are also supplementary).

Vertical

Vertical angles are the pair of angles opposite the intersection of two lines or line segments. In the example below, the angles marked a are a pair of vertical angles, and the angles marked b are another pair of vertical angles.

Alternate interior

When a transversal crosses two lines, the pairs of angles on opposite sides of the transversal and inside the two lines are opposite interior angles. If the two lines are parallel, then alternate interior angles are equal. In the example below, the highlighted angles marked a are equal (they are a pair of alternate interior angles), and the highlighted angles marked b are equal (they are also a pair of alternate interior angles).

Same-side interior

When a transversal crosses two lines, the pairs of angles on the same side of the transversal and inside the two lines are same-side interior angles. If the two lines are parallel, then alternate interior angles are supplementary. In the example below, the highlighted angles marked a and b are supplementary.

Same-side exterior

When a transversal crosses two lines, the pairs of angles on the same side of the transversal and outside the two lines are same-side exterior angles. If the two lines are parallel, then same-side exterior angles are supplementary. In the example below, the highlighted angles marked a and b are supplementary.

Alternate exterior

When a transversal crosses two lines, the pairs of angles on the opposite side of the transversal and outside the two lines are alternate exterior angles. If the two lines are parallel, then the opposite exterior angles are equal. In the example below, the highlighted angles marked a and b are supplementary.