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Classifying and constructing angles by their measurement

Line segments is a part of the line limited by two remote points.

Imagine a line. A line has no beginning and no end.

infinite line

Now somewhere, anywhere, put a point on it, and chop off one part. Now you got a semi straight line, or a ray.


infinite line with one point

Now your line is limited on one side, and if you limit it to the other side you get a segment:

two point line


We all know what a plane is, a flat endless surface.

An angle is a part of a plane enclosed by two rays. They are measured in degrees (symbol: ᵒ).

\ 1^{\circ} = \ 60' (minutes) = \ 3600'' (seconds)
\ 1'= 60''

Every angle is made out of three parts vertex and 2 arms (rays).

angle parts

Angles are divided into six groups based on their measure.

First ones are the smallest – acute angles. Their measure is less than \ 90^{\circ}.

acute angles

Second is the right angle. Right angle is always \ 90^{\circ}.

right angles

Third is obtuse angle, the one that has more than \ 90^{\circ} but less than \ 180^{\circ}.

obtuse angles

Straight angle is the one whose two arms make a straight line. He always has exactly \ 180^{\circ}.

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 degrees is \pi/2, 180 degrees \pi , and 360 degrees is 2\pi.

angles measured in radians

Converting radians into degrees

We learned that full rotation has \ 360^{\circ} or 2\pi radians. To convert 2\pi into \ 360^{\circ} you have to multiply it by 180 and divide with \pi

By dividing that with two, we get that \ 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


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 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


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, 292 hits)

  Constructing angles (118.4 KiB, 301 hits)

  Construction of angle bisectors (438.2 KiB, 274 hits)

  Measuring lines in centimeters (105.0 KiB, 349 hits)

  Measuring lines in millimeters (73.8 KiB, 372 hits)


  Determine a type of angle (238.2 KiB, 318 hits)

  Angle relationship (381.1 KiB, 286 hits)

  Measure an acute angles (61.1 KiB, 308 hits)

  Measure an obtuse angles (57.5 KiB, 319 hits)

  Angle measurement (334.9 KiB, 260 hits)

  Draw an acute angle (50.6 KiB, 472 hits)

  Draw an obtuse angle (78.1 KiB, 346 hits)

  Draw different type of angles (251.6 KiB, 214 hits)

  Converting degrees into radians (121.2 KiB, 343 hits)

  Converting radians into degrees (135.3 KiB, 333 hits)

  Coterminal angles (136.9 KiB, 371 hits)