In geometry, a **line** is an infinitely thin, infinitely long collection of points extending in two opposite directions. A closed interval corresponding to a finite portion of an infinite line is called a **line segment**. A line segment has two endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. **Ray** is a line which has one endpoint and stretches infintely in one direction.

**Angle** is 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.

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 $\ 90^{\circ}$.

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$.

**Converting radians into degrees**

Full rotation has $\ 360^{\circ}$ which equals $ 2\pi$ radians. To convert $ 2\pi$ radians into $\ 360^{\circ}$, $2\pi$ is multiplied by 180 and divided with $\pi$.

This is generalized with a formula:

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

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

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

### Converting degrees into radians

When converting degrees into radians, for example $360^{\circ}$ into $2 \pi$, $360^{\circ}$ is divided with $180^{\circ}$ and multiplied with $\pi$.

This is generalized with a formula:

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

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

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

## Constructing angles

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

Let’s say you have your side $c$ and points $A$ and $B$, and you want to construct an angle with measure of $60^{\circ}$ in point A. This angle is enclosed by sides $c$ and $b$, so you’ll construct your angle in point $A$. Draw a circle around the point $A$, it does not matter which radius you chose. Mark intersection of the circle with side $c$ with $D$. Now draw a circle, with center in $D$, with the same radius you used to draw previous circle. Draw a straight line that goes through point $A$ and intersection of two circles you drew. You got a line which contains side $b$. And you got your angle.

**Constructing a $120^{\circ}$ degree 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 F-intersection of your circle with previous angle and draw another arc with the same radius.

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

For constructing an angle with the measure of $ 30^{\circ}$ you’ll have to know how to bisect an angle. Since $ 30^{\circ} = 60^{\circ} : 2$, we’ll firstly draw angle with $ 60^{\circ}$. Mark the intersections of arms of an angle and circle with $H$ and $C$.

We’ll put the needle of a compass in $C$ and make an arc somewhere between $H$ and $C$ and do the same with $H$, then connect their intersection with our starting point and we got $ 30^{\circ}$.

Other angles are constructed by combining these. 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 $30^{\circ} + 60^{\circ} = 90^{\circ}$.

## Line segments and angles worksheets

__Line segments__

**Constructing line segments** (173.2 KiB, 432 hits)

**Constructing angles** (118.4 KiB, 536 hits)

**Construction of angle bisectors** (438.2 KiB, 505 hits)

**Measuring lines in centimeters** (105.0 KiB, 530 hits)

**Measuring lines in millimeters** (73.8 KiB, 542 hits)

__Angles__

**Determine a type of angle** (238.2 KiB, 552 hits)

**Angle relationship** (381.1 KiB, 489 hits)

**Measure an acute angles** (61.1 KiB, 542 hits)

**Measure an obtuse angles** (57.5 KiB, 530 hits)

**Angle measurement** (334.9 KiB, 493 hits)

**Draw an acute angle** (50.6 KiB, 688 hits)

**Draw an obtuse angle** (78.1 KiB, 412 hits)

**Draw different type of angles** (251.6 KiB, 421 hits)

**Converting degrees into radians** (121.2 KiB, 670 hits)

**Converting radians into degrees** (135.3 KiB, 632 hits)

**Coterminal angles** (136.9 KiB, 732 hits)