# Geometry » Angles

## Contents

1. Angle types2. Measuring angles

3. Calculating angles

4. Vertically opposite angles

5. F-angles and Z-angles

## 1. Angle types

An angle always consists of a vertex and two arms. These arms make a certain angle with each other. The smaller the angle, the 'sharper' the intersection between the lines. The angle is a measurement/quantity that tells you what kind of intersection there is between two lines. The unit is ° (degrees). When you extend the arms, the angle will not change.

### Right angle

A right angle is always 90°. The two lines are perpendicular to each other.

`A` below is a right angle. There are two different indicators for a right angle.

Those are and .

### Acute angle

An angle between 0° and 90° is an acute angle. `B` below is an acute angle.

### Obtuse angle

An angle between 90° and 180° is an obtuse angle. `C` below is an obtuse angle.

### Straight angle

A straight angle is always 180°. `D` below is a straight angle.

### Full rotation

A full rotation is an angle of 360°. `E` below is a full rotation. A full rotation can also be called a full turn.

### Reflex angle

A reflex angle is an angle between 180° and 360°. `F` and `G` below are examples of reflex angles.

## 2. Measuring angles

### With the compass rose

Put the dot in the centre of the compass rose exactly in the vertex. Put the 0 (or N) on one of the arms. Always make sure you can read off from the compass rose clockwise. In the example above you can read off that the angle is 46°.

Use this notation: `A` = 56°

### With the geo-triangle

Put the geo-triangle on the angle with the longest side *alongside* one of the arms. Make sure the vertex is at the 0 of the longest side of the geo-triangle. Now you can read off two values at the other arm. In the example above these are 56° and 124°. You can see that the angle is acute so the correct answer is 56°.

Another way is the fact that you start measuring at the arm where the longest side of the geo-triangle is alongside. You start counting there at 0°, continue with 10°, 20° etc. You then know which protractor you need and you can read off the correct value.

Use this notation: `A` = 56°

## 3. Calculating with angles

### Straight angles

A straight angle is always 180°. The angles that form a straight angle together should therefore be 180° together.

*Example*

`A`_{1} = 180° – 70° = 110°

### Triangle

The three angles in a triangle add up to 180°.

*Example*

In a triangle it is known that there is an angle of 68° and an angle of 34°.

What size is the third angle?

Answer: 180° – 68° – 34° = 78°

See triangles for angle properties in certain triangles.

### Quadrilateral

In every quadrilateral the four angles add up to 360°.

*Example*

In a quadrilateral it is known that there are two angles of 37° and one angle of 130°.

What size is the fourth angle?

Answer: 360° – 37° –37° – 130° = 156°

See quadrilaterals for angle properties in certain quadrilaterals.

### Polygon / `n`-gon

In a polygon with `n` angles, the total number of degrees can be calculated using:

`total number of degrees` = (`n` – 2) × 180°.

*Example*

What size is every angle in a regular 15-gon (pentadecagon)?

Answer: Total number of degrees = (15 – 2) × 180° = 2340°.

There are fifteen vertices so every angle is 2340° : 15 = 156°.

## 4. Vertically opposite angles

If two straight lines intersect it creates vertically opposite angles.

Vertically opposite angles are equal in size.

*Example*

`A`_{2} = `A`_{4} = 120°

`A`_{1} = `A`_{3} = 180° – 120° = 60°

## 5. F-angles and Z-angles

With *F-angles* and *Z-angles* we mean angles that are the same because two parallel lines are intersecting these vertices.

Because of *Z-angles*: `A` = `B` and `C` = `D`.

Because of *F-angles*: `E` = `F`

*Example*

In triangle `ABC` it is given that `DE` is a mid-parallel (so `DE` // `AB`).

It is also known that `A`_{2} = 38°, `D`_{2} = 104° and `E`_{1} = 50°.

Calculate all the other angles:

B | = E_{1} = 50° (F-angles) |

A_{2} | = E_{2} = 38° (Z-angles) |

E_{3} | = 180° – 38° – 50° = 92° |

A_{1} | = 180° – 38° – 104° = 38° |

D_{1} | = 180° – 104° = 76° |

C | = 180° – 76° – 50° = 54° |