**Angles** are a part of a plane, a flat endless surface, enclosed by two rays. They are measured in **degrees** which are denoted by $^{\circ}$. Minutes and seconds are smaller units of measure, $1^{\circ}$ equals $60’$ (minutes) and $3600”$ (seconds), furthermore one minute equals 60 seconds, $ 1’= 60”$.

An angle has three parts: a vertex and two rays, sometimes reffered to as arms.

We say that angles are **congruent **if they have the same measure.

Angles are divided into six groups based on their measure – acute, right, obtuse, straight, reflex and a full rotation angles.

**Acute angles** have a measure less than $\ 90^{\circ}$.

Angles with a measure exactly $\ 90^{\circ}$ are called **right angles.**

**Obtuse angles** have a measure greater than $\ 90^{\circ}$ but less than $\ 180^{\circ}$.

Angles with a measure exactly $\ 180^{\circ}$ are called **straight angles**. Their arms make a straight line.

**Reflex angle** has more than $\ 180^{\circ}$ and less than $\ 360^{\circ}$.

**Full rotation** always has exactly $\ 360^{\circ}$.

Angles can also be measured in radians, where $90^{\circ}$ equals $\pi/2$, $180^{\circ}$ equals $\pi$ , and $360^{\circ}$ equals $ 2\pi$.

Pair of angles:

**adjacent angles****a linear pair****vertical angles****complementary angles**if two angles sum to $90^{\circ}$**suplementary angles**if two angles sum to $ 180^{\circ}$**explementary angles**if two angles sum to $ 360^{\circ}$

**Adjacent angles**

Adjacent angles are two angles that share a common vertex, a common side, but they do not overlap (they don’t have common interior).

$\alpha$ and $\beta$ are adjacent angles. They have a common vertex $A$ and a common side $AB$.

$\gamma$ and $\alpha$ are **NOT** adjacent angles because $\gamma$ overlaps $\alpha$ (same for $\gamma$ and $\beta$)

**REMEMBER: **for $2$ angles to be adjacent they have to have common both vertex and a side, angles who share a common vertex but don’t have common a side are not adjacent!

On this last image, $\alpha$ and $\beta$ are **not** adjacent angles, because they only have a common vertex.

#### Linear pair

A linear pair are two adjacent angles whose non-common sides form on opposite ray (the non common sides lie on the same line).

Angles $\alpha$ and $\beta$ form a linear pair.

We can see that $\alpha + \beta =116.57^{\circ}+63.43^{\circ} = 180^{\circ}$, meaning that linear pair always forms a straight angle which is $180^{\circ}$.

If two congruent angles form a linear pair, the angles are right angles.

#### Vertical angles

Vertical angles are a pair of opposite angles formed by intersecting lines.

On the image we can see two pairs of vertical angles: $\alpha$ and $\alpha^{‘}$, $\beta$ and $\beta^{‘}$. Vertical angles are always equal in measure.

However, vertical angles are **not** adjacent: $\alpha$ and $\beta$ are not vertical angles, they are a linear pair.

#### Complementary angles

Two angles that add up to $90^{\circ}$.

To be complementary, angles don’t have to be adjacent. As we can see, $$\alpha + \alpha^{‘} = 55^{\circ} + 35^{\circ} = 90^{\circ} $$and they are complementary just like $\beta$ and $\beta^{‘}$.

**Supplementary angles**

Two angles whose measures sum to $180^{\circ}$.

To be supplementary, angles don’t have to be adjacent. As we can see, $$\alpha + \alpha^{‘} = 45^{\circ} + 135^{\circ} = 180^{\circ} $$and they are supplementary just like $\beta$ and $\beta^{‘}$.

A linear pair is always supplementary .

**Complementary or supplementary? How to remember which is which?**

- Complementary starts with “
**C**” meaning “Corner” or “$90^{\circ}$ angle”! - Supplementary starts with “
**S**” meaning “Straight” or “$180^{\circ}$ angle”!

**Explementary angles**

Angles that add up to $ 360^{\circ}$

We can see that $67^{\circ}+95^{\circ}+40^{\circ}+123^{\circ}+35^{\circ} = 360^{\circ}$.

Also, $\alpha + \alpha^{‘} = 230^{\circ} + 130^{\circ} = 360^{\circ} $

**Converting radians into degrees**

Full rotation has $\ 360^{\circ}$ which equals $ 2\pi$ radians. To convert $ 2\pi$ radians into $\ 360^{\circ}$ we multiply $2\pi$ by $180$ and divide it with $\pi$.

This is generalized with a formula:

Degrees = radians $\cdot \displaystyle{ \frac{180}{\pi}}$

For example, convert $\displaystyle{\frac{2\pi}{3}}$ into degrees.

$\displaystyle{\frac{2 \pi}{3} \cdot \frac{180}{\pi} = \frac{360 \pi}{3 \pi} = 120^{\circ}}$

### Converting degrees into radians

When converting degrees into radians, for example $360^{\circ}$ into $2 \pi$, we divide $360^{\circ}$ with $180^{\circ}$ and multiply it with $\pi$.

This is generalized with a formula:

Radians = degrees $\cdot \displaystyle{\frac{\pi}{180^{\circ}}}$

For example, convert $ 45^{\circ}$ into radians.

$ 45^{\circ} \cdot \displaystyle{\frac{\pi}{180^{\circ}} = \frac{\pi}{4}}$

## Constructing angles

**Constructing a $60^{\circ}$ angle**

Let’s say you have one ray and its endpoint $A$ and another point on it, point $B$. We want to construct an angle with measure of $60^{\circ}$ in point $A$. This angle is enclosed by side $AB$ and another one we need to construct. Draw a circle around the point $A$, it does not matter which radius you choose. Mark intersection of the circle with side $AB$ with $C$. Now draw an arc, with center in $C$, with the same radius you used to draw a circle. Draw a straight line that goes through point $A$ and intersection of circle and arc, point $D$. Side $AD$ is our second arm of an angle. $\angle CAD = 60^{\circ}$.

**Constructing a $120^{\circ}$ angle**

Constructing angle whose measure equals to $ 120^{\circ}$ is very easy since we know how to construct an angle with measure $ 60^{\circ}$. $ 120^{\circ} = 2 \cdot 60^{\circ}$, and that means that we’ll simply construct two angles of $ 60^{\circ}$ and add them together. Put the needle in your compass in point $D$ -intersection of your circle with previous angle and draw another arc with the same radius.

**Constructing a $30^{\circ}$ angle**

The first thing we can notice is that $ 30^{\circ} = 60^{\circ} : 2$. Because of that, to get $30^{\circ}$ angle we first need to construct the $60^{\circ}$ angle and then bisect it.

We can construct other angles by combining these three.

If you want an angle whose measure equals to 15ᵒ you’ll simply bisect angle of $30^{\circ}$.

If you want $90^{\circ}$ you’ll have to make twice $60^{\circ}$ angle, and bisect the other one to get $30^{\circ}$ and you’ll have $30^{\circ} + 60^{\circ} = 90^{\circ}$.

## Angles worksheets

**Determine a type of angle** (238.2 KiB, 701 hits)

**Angle relationship** (381.1 KiB, 587 hits)

**Measure an acute angles** (61.1 KiB, 676 hits)

**Measure an obtuse angles** (57.5 KiB, 669 hits)

**Angle measurement** (334.9 KiB, 619 hits)

**Draw an acute angle** (50.6 KiB, 822 hits)

**Draw an obtuse angle** (78.1 KiB, 479 hits)

**Draw different type of angles** (251.6 KiB, 554 hits)

**Converting degrees into radians** (121.2 KiB, 906 hits)

**Converting radians into degrees** (135.3 KiB, 827 hits)

**Coterminal angles** (136.9 KiB, 1,144 hits)