Properties of Angles

 

Learning Objective(s)

·         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 lines 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 lines are two lines that intersect at a 90º (right) angle. And perpendicular line segments also intersect at a 90º (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 90º 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 . 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 .

 

 

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.

 

 

 

 

Which statement most accurately represents the image below?     A) B) C) D)   Show/Hide Answer A) Incorrect. This image shows the lines  and , not  and . Both  and  are marked with >> on each line, and those markings mean they are parallel. The correct answer is .   B) Incorrect.  does intersect , but the intersection does not form a right angle. This means that they cannot be perpendicular. The correct answer is .   C) Correct. Both  and  are marked with >> on each line, and those markings mean they are parallel.   D) Incorrect.  and  intersect, so they cannot be parallel. Both  and  are marked with >> on each line, and those markings mean they are parallel. The correct answer is .

 

 

 

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 90º angle, and that a straight angle measures 180º.

 

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

 

 

 

 

In this example, you may have noticed that angles , and  are all right angles. (If you were asked to find the measurement of , you would find that angle to be 90º, 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.

 

 

 

 

Find the measurement of .     A) 43º B) 137º C) 147º D) 317º   Show/Hide Answer A) 43º Incorrect. You found the measurement of , not . Since  measures 43º, the measure of  must be 180º – 43º = 137º. The correct answer is 137º.   B) 137º Correct.  is a straight angle measuring 180º. Since  measures 43º, the measure of  must be 180º – 43º = 137º.   C) 147º Incorrect. It looks like you subtracted incorrectly. Since  measures 43º,  must be 180º – 43º = 137º. The correct answer is 137º.   D) 317º Incorrect. Remember that a straight angle measures 180º, not 360º. Since  measures 43º, the measure of  must be 180º – 43º = 137º. The correct answer is 137º.

 

 

 

In the example above,  and  add up to 180º. Two angles whose measures add up to 180º are called supplementary angles. There’s also a term for two angles whose measurements add up to 90º, they are called complementary angles.

 

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

 

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

 

 

 

 

 

 

 

 

 

 

Which pair of angles is complementary?     A) B) C) D)   Show/Hide Answer A) Incorrect. The measures of complementary angles add up to 90º. It looks like the measures of these angles may add up to 90º, but there is no way to be sure, so you cannot say that they are complementary. The correct answer is .   B) Incorrect.  are supplementary angles (not complementary angles) because together they comprise the straight angle . The correct answer is .   C) Incorrect.  are supplementary angles (not complementary angles) because together they comprise the straight angle . The correct answer is .   D) Correct. The measurements of two complementary angles will add up to 90º.  is a right angle, so  must be a right angle as well. , so  are complementary.

 

 

 

Parallel lines do not intersect, while perpendicular lines cross at a 90º angle. Two angles whose measurements add up to 180º are said to be supplementary, and two angles whose measurements add up to 90º 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.