Math Core

Every shape you have ever drawn is built from a few simple parts: points, lines, segments and rays. Once you know their names, you can describe any figure exactly, and you can talk about the corners where the parts meet, called angles.

The building blocks

A point marks an exact location. It has no size at all. We draw it as a dot and name it with a capital letter, like point $A$ .

A line is perfectly straight and goes on forever in both directions. The arrows on each end show that it never stops. You name a line with any two points on it: line $BC$ , written $\overleftrightarrow{BC}$ .

A line segment is the part of a line between two endpoints. It has a beginning and an end, so you could measure it with a ruler. Segment $DE$ is written $\overline{DE}$ .

A ray has one endpoint and goes on forever in the other direction, like a beam of light from a flashlight. Ray $FG$ is written $\overrightarrow{FG}$ .

Left to right: point A, line BC, segment DE and ray FG.

Common mistake

The endpoint always comes first in a ray's name. Ray $FG$ starts at $F$ and passes through $G$ . Ray $GF$ would start at $G$ and point the other way, so it is a different ray.

Angles

When two rays share the same endpoint, they make an angle.

Definition

Angle

An angle is formed by two rays with a common endpoint. The common endpoint is the vertex of the angle. The two rays are its sides.

To name an angle, use three points with the vertex in the middle. The angle below is $\angle ABC$ or $\angle CBA$ . When only one angle is at a vertex, you can also call it $\angle B$ .

Angle ABC. The vertex is B. The sides are ray BA and ray BC.

Kinds of angles

You can sort angles by comparing them to a right angle, the square corner you see on a sheet of paper.

A right angle makes a square corner. We mark it with a small square.
An acute angle is smaller than a right angle.
An obtuse angle is bigger than a right angle, but its sides do not make a straight line.
A straight angle has sides that point in opposite directions, so together they make a straight line.

Left to right: an acute angle, a right angle, an obtuse angle and a straight angle.

Sorting angles

Compare the angle to a square corner. Smaller is acute, exactly a square corner is right, and bigger (but not a straight line) is obtuse.

Tip

The size of an angle depends on how far apart the sides open, not on how long the sides are drawn. Rays go on forever anyway!

Worked example: Name the figure

A figure is straight, starts at point $P$ , and goes on forever through point $Q$ . What is it, and what is its name?

It has one endpoint, $P$ , and goes on forever in one direction, so it is a ray. The endpoint comes first: ray $PQ$ , or $\overrightarrow{PQ}$ .

Worked example: Angles on a clock

What kind of angle do the hands of a clock make at 3:00? At 2:00? At 4:00?

At 3:00 the hands make a square corner: a right angle.
At 2:00 the minute hand is closer to the hour hand, so the angle is smaller than a right angle: acute.
At 4:00 the hands are farther apart than a square corner: obtuse.

Worked example: Which letter goes in the middle?

Rays $TS$ and $TU$ make an angle. Which of these names is correct: $\angle STU$ or $\angle TSU$ ?

The two rays share the endpoint $T$ , so $T$ is the vertex. The vertex goes in the middle, so the name is $\angle STU$ . ( $\angle UTS$ is also correct.)

Practice

Practice 1

Which figure goes on forever in both directions?

Practice 2

How many endpoints does a line segment have?

Type a number.

Practice 3

What is the name of this figure?

Practice 4

What is a correct name for this angle?

Practice 5

What kind of angle do the clock hands make at 9:00?

Practice 6

Which angle is obtuse?

Practice 7

How many endpoints does a ray have?

Type a number.

Practice 8

The capital letter T is made of two segments: a flat one across the top and a straight-up-and-down one below its middle. How many right angles are formed where the two segments meet?

Type a number.