Middle Years

Lesson

An angle is formed between two lines, rays, or segments whenever they intersect. We can think of an angle as a turn from one object to the other.

The most important angle in geometry is called a right angle, and represents a quarter of a turn around a circle. When two objects form a right angle, we say they are perpendicular. We draw a right angle using a small square rather than a circular arc:

We draw all other angles with a circular arc. An angle that is smaller than a right angle is called an acute angle. Here are two:

Two right angles together form a straight angle:

Four right angles is the same as two straight angles, making a full revolution:

An angle that is larger than a right angle but smaller than a straight angle is called an obtuse angle. Here are two:

We met the last kind of angle in the previous lesson - a reflex angle is larger than a straight angle, but smaller than a full revolution. Here are two:

Angle types

Angles are a measure of turning. All angles can be compared to a right angle, representing a quarter turn.

Select the obtuse angle:

- ABCD

We divide a full revolution up into $360$360 small turns called degrees, and write the unit using a small circle, like this: $360^\circ$360°.

Since $90$90 is one quarter of $360$360, we know that a right angle is exactly $90^\circ$90°. This circle has markings every $45^\circ$45°:

We can measure angles more precisely using a protractor, or an applet like this one:

This lets us associate numbers with the angle types we learned about above.

Angle size

A full revolution is made up of $360$360 degrees, a single degree is written $1^\circ$1°.

Angle type | Angle size |
---|---|

Acute angle | Larger than $0^\circ$0°, smaller than $90^\circ$90°. |

Right angle | $90^\circ$90° |

Obtuse angle | Larger than $90^\circ$90°, smaller than $180^\circ$180°. |

Straight angle | $180^\circ$180° |

Reflex angle | Larger than $180^\circ$180°, smaller than $360^\circ$360°. |

Full revolution | $360^\circ$360° |

Select the angle that is closest to $120^\circ$120°:

- ABCD

Whenever two angles share a defining line, ray, or segment, and do not overlap, we say they are adjacent angles. Here are some examples:

Whenever two segments, lines, or rays intersect at a point, two pairs of equal angles are formed. Each angle in the pair is on the opposite side of the intersection point, and they are called vertically opposite angles.

If two angles form a right angle, we say they are complementary. We then know that they add to $90^\circ$90°.

If two angles form a straight angle, we say they are supplementary. We then know that they add to $180^\circ$180°.

Whenever we know that two (or more) angles form a right angle, a straight angle, or a full revolution, we can write an equation that expresses this relationship.

Solve for the value of $x$`x` in the diagram below:

Do: We write the equation:

$x+147+116=360$`x`+147+116=360

We then use subtraction to make $x$`x` the subject:

$x=360-147-116$`x`=360−147−116

We then do the subtraction to find $x$`x`:

$x=97$`x`=97

The angles in the diagram below are complementary. What is the value of $x$`x`?