Properties of Angles

Learning Objectives

• Identify parallel and perpendicular lines.
• Find measures of angles.
• Identify complementary and supplementary angles.

Imagine two separate and distinct lines on a plane. There are two possibilities for these lines: they will either intersect at one point, or they will never intersect. When two lines intersect, four angles are formed. Understanding how these angles relate to each other can help you figure out how to measure them, even if you only have information about the size of one angle.

Parallel linesTwo or more lines that lie in the same plane but which never intersect. are two or more lines that never intersect. Likewise, parallel line segments are two line segments that never intersect even if the line segments were turned into lines that continued forever. Examples of parallel line segments are all around you, in the two sides of this page and in the shelves of a bookcase. When you see lines or structures that seem to run in the same direction, never cross one another, and are always the same distance apart, there’s a good chance that they are parallel.

Perpendicular linesTwo lines that lie in the same plane and intersect at a `90^@` angle. are two lines that intersect at a `90^@` (right) angle. And perpendicular line segments also intersect at a `90^@` (right) angle. You can see examples of perpendicular lines everywhere as well: on graph paper, in the crossing pattern of roads at an intersection, to the colored lines of a plaid shirt. In our daily lives, you may be happy to call two lines perpendicular if they merely seem to be at right angles to one another. When studying geometry, however, you need to make sure that two lines intersect at a `90^@` angle before declaring them to be perpendicular.

The image below shows some parallel and perpendicular lines. The geometric symbol for parallel is `"||"`, so you can show that `overset(harr)(AB)text(||)overset(harr)(CD)`. Parallel lines are also often indicated by the marking >> on each line (or just a single `>` on each line). Perpendicular lines are indicated by the symbol `_|_`, so you can write `overset(harr)(WX)_|_overset(harr)(YZ)`.

If two lines are parallel, then any line that is perpendicular to one line will also be perpendicular to the other line. Similarly, if two lines are both perpendicular to the same line, then those two lines are parallel to each other. Let’s take a look at one example and identify some of these types of lines.

Understanding how parallel and perpendicular lines relate can help you figure out the measurements of some unknown angles. To start, all you need to remember is that perpendicular lines intersect at a `90^@` angle, and that a straight angle measures `180^@`.

The measure of an angle such as `angleA` is written as `mangleA`. Look at the example below. How can you find the measurements of the unmarked angles?

In this example, you may have noticed that angles `angleHJI`, `angleIJF`, and `angleHJM` are all right angles. (If you were asked to find the measurement of `angleFJM`, you would find that angle to be `90^@`, too.) This is what happens when two lines are perpendicular: the four angles created by the intersection are all right angles.

Not all intersections happen at right angles, though. In the example below, notice how you can use the same technique as shown above (using straight angles) to find the measurement of a missing angle.

In the example above, `mangleBAC` and `mangleDAC`  add up to `180^@`. Two angles whose measures add up to `180^@` are called supplementary anglesTwo angles whose measurements add up to `180^@`. . There’s also a term for two angles whose measurements add up to `90^@`; they are called complementary anglesTwo angles whose measurements add up to `90^@`. .

One way to remember the difference between the two terms is that “corner” and “complementary” each begin with c (a `90^@` angle looks like a corner), while straight and “supplementary” each begin with s (a straight angle measures `180^@`).

If you can identify supplementary or complementary angles within a problem, finding missing angle measurements is often simply a matter of adding or subtracting.

Parallel lines do not intersect, while perpendicular lines cross at a `90^@` angle. Two angles whose measurements add up to `180^@` are said to be supplementary, and two angles whose measurements add up to `90^@` are said to be complementary. For most pairs of intersecting lines, all you need is the measurement of one angle to find the measurements of all other angles formed by the intersection.