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, 1,008 hits)
Angle relationship (381.1 KiB, 764 hits)
Measure an acute angles (61.1 KiB, 925 hits)
Measure an obtuse angles (57.5 KiB, 931 hits)
Angle measurement (334.9 KiB, 809 hits)
Draw an acute angle (50.6 KiB, 1,070 hits)
Draw an obtuse angle (78.1 KiB, 737 hits)
Draw different type of angles (251.6 KiB, 739 hits)
Converting degrees into radians (121.2 KiB, 1,099 hits)
Converting radians into degrees (135.3 KiB, 1,091 hits)
Coterminal angles (136.9 KiB, 3,379 hits)
