7. Geometry

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:

- ABCDABCD

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°:

- ABCDABCD

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve equations for an unknown angle in a figure.