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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.

Imagine a line – it has no beginning and no end.

Now somewhere, anywhere, put a point on it, and chop off one part – what you have now is called a ray. It has one endpoint and stretches infintely in one direction.


If you do the same on the some other part of the line you get a line segment.


A plane is a flat endless surface. An angle is a part of a plane enclosed by two rays. They are measured in degrees which are denoted by ^{\circ}.

\ 1^{\circ} equals \ 60' – minutes, and \ 3600'' – seconds, furthermore one minute equals 60 seconds, 1'= 60''.

An agnle 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} or 2\pi radians. To convert 2\pi radians into \ 360^{\circ} you have to multiply it by 180 and divide with \pi

By dividing with two, \ 180^{\circ} has \pi radians. If we multiply \pi with \ 180^{\circ} and divide it with \pi, we get \ 180^{\circ}.

We can generalize that with a formula:
Degrees = radians * \frac{180}{\pi} . Let’s see how it works.

Convert \frac{2\pi}{3} into degrees.

Degrees = \frac{2 \pi}{3} * \frac{180}{\pi} = \frac{360 \pi}{3 \pi} = 120^{\circ}


Converting degrees into radians

Opposite, when converting radians degrees into radians, for example 360^{\circ} into 2 \pi, you have to divide with 180^{\circ} and multiply with \pi.

Radians = degrees * \frac{\pi}{180^{\circ}}.

Convert 45^{\circ} into radians.

Radians = 45^{\circ} * \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 your 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.

60 degrees angle construction

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 * 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 30ᵒ.

If you want 90ᵒ you’ll have to make twice 60ᵒ angle, and bisect the other one to get 30ᵒ and you’ll have 30ᵒ + 60ᵒ = 90ᵒ.


Line segments and angles worksheets

Line segments

  Constructing line segments (173.2 KiB, 317 hits)

  Constructing angles (118.4 KiB, 352 hits)

  Construction of angle bisectors (438.2 KiB, 327 hits)

  Measuring lines in centimeters (105.0 KiB, 392 hits)

  Measuring lines in millimeters (73.8 KiB, 412 hits)


  Determine a type of angle (238.2 KiB, 382 hits)

  Angle relationship (381.1 KiB, 328 hits)

  Measure an acute angles (61.1 KiB, 354 hits)

  Measure an obtuse angles (57.5 KiB, 363 hits)

  Angle measurement (334.9 KiB, 314 hits)

  Draw an acute angle (50.6 KiB, 509 hits)

  Draw an obtuse angle (78.1 KiB, 352 hits)

  Draw different type of angles (251.6 KiB, 257 hits)

  Converting degrees into radians (121.2 KiB, 401 hits)

  Converting radians into degrees (135.3 KiB, 383 hits)

  Coterminal angles (136.9 KiB, 435 hits)