# Angles

### Concepts

Understand

- The concept of an angle.
- That angles are measured in degrees and drawn using a protractor.
- That there are three different types of angles (acute, right angle, and obtuse)
- Angle properties such as co-interior angles always sum up to 180 degrees.

### Theory

Where two straight lines meet, an angle is formed. Often the meeting point is called the vertex. If there is more than one vertex, then we call them vertices. In a triangle there are:

- three angles.
- three vertices.

The protractor is used to measure and draw angles, but it’s confusing to understand initially.

If you want to draw a 60 degree angle, you can align up with the 60 degree marker. But wait, why does it also say 120 degrees? And there are two pairs of 60/120 - what’s up with that? Read on, and once you have understood the different types of angles, it should be clear.

###### Acute angles

Always less than 90 degrees. If you want to open a door slightly and peek in, then this is the angle the door makes with the wall.

###### Obtuse angles

More than 90 degrees. That’s how much the door should ideally open when you burst into a room.

###### Right angle

Exactly 90 degrees, if you’ve seen a square you know what a right angle looks like. Intersections are often at 90 degree angles too. On paper they are depicted with a small notch.

###### Reflex angle

Greater than 180 degrees but less than 360 degrees.

###### Angles in a polygon

Shape | Sum of all angles | Notes |
---|---|---|

Triangle | 180 | This is important. The sum is NOT 360. |

Quadrilateral | 360 | Think of rectangle, each angle is 90. |

Pentagon | 540 |

But how?

- Use a protractor and verify.
- A pentagon is made of three triangles, each has 180. 180 times 3 = 540
- Introduce the concept of a equation
- Sum of all angles = (n-2) * 180

###### Angles in a triangle

A very common problem is to find the size of one angle in a triangle if the other two angles are known. Remember that the sum of all three angles of a triangle can NEVER add up to more than 180 degrees. Maybe in some other multiverse it might, but it’s not possible in this one.

###### Isosceles Triangle

The isosceles triangle is a special kind of triangle that has two opposite sides of equal length; it has two equal angles too.

### Misconceptions

- Sum of all all angles in a triangle is 360. No, 360 degrees is a full circle.