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Classifying and constructing angles by their measurement


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.

angle parts

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 $\ 90^{\circ}$.

acute angles

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

right angles

Obtuse angles have a measure greater than $\ 90^{\circ}$.

obtuse angles

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

straight angles

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

reflex angles

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$.

angles measured in radians

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.

60 degrees angle construction



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.

120 degrees angle construction

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 $C$.

30 degrees angle construction

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}$.

30 degrees angle construction second part

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, 444 hits)

  Constructing angles (118.4 KiB, 559 hits)

  Construction of angle bisectors (438.2 KiB, 527 hits)

  Measuring lines in centimeters (105.0 KiB, 561 hits)

  Measuring lines in millimeters (73.8 KiB, 563 hits)


  Determine a type of angle (238.2 KiB, 578 hits)

  Angle relationship (381.1 KiB, 507 hits)

  Measure an acute angles (61.1 KiB, 560 hits)

  Measure an obtuse angles (57.5 KiB, 554 hits)

  Angle measurement (334.9 KiB, 515 hits)

  Draw an acute angle (50.6 KiB, 715 hits)

  Draw an obtuse angle (78.1 KiB, 422 hits)

  Draw different type of angles (251.6 KiB, 449 hits)

  Converting degrees into radians (121.2 KiB, 706 hits)

  Converting radians into degrees (135.3 KiB, 666 hits)

  Coterminal angles (136.9 KiB, 779 hits)