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Classifying triangles by sides


A triangle is a part of a plane bounded by three line segments. Triangles have three angles and three sides. Vertices of a triangles are denoted by capital letters and their sides by lowercase letters from their opposite vertex.

The sum of all angles in a triangle is always 180ᵒ.

We divide triangles considering how many sides have the same length and analog to that how many angles of the same measure.

A triangle that has a no equal sides and no equal angles is called scalene triangle.

scalene triangle

A triangle that has two equal sides and two equal angles is called isosceles triangle.

isosceles triangleSide f is called the base of isosceles triangle, because other two sides are the same that are called arms. Equal sides are always the ones that are enclosed by the base and arms.

A triangle that has all three equal sides and equal angles is called equilateral triangle.

equilateral triangle

Because the sum of all angles in a triangle is equal to 180^{\circ}, and equilateral triangle has three equal angles, each amounts to 60^{\circ}.

Also triangles are divided by their angles.

Acute triangle is a triangle that has all angles less than 90^{\circ}.

acute triangle

Right triangle has a right angle.


right triangle

Obtuse triangle has an angle more than 90^{\circ}.

obtuse triangle

Constructing triangles with given size of sides

To construct any geometric figure means to precisely draw their shape using only a compass, ruler and a pencil.

Example 1: constructing scalene triangle with sides 4 cm, 6 cm and 5 cm.

a = 4 cm
b = 6 cm
c = 5m

First draw a sketch, it doesn’t matter if it’s accurate or anything, what matters is your disposition of points and sides.

scalene triangle sketch

Now, we move on to the construction part. First we’ll draw side c. We’ll mark left point with A, and right with B. And then we take the compass, put the needle in zero, and open it to 6cm. Put the needle of compass into point A and draw a circle.

divider measurs centimeters


Next, again put the needle of the compass in zero on your ruler and open it to 4 cm, put the needle in point B and draw a circle. Their intersection represents point C. All that is left is connect the points and mark point C.

constructing scalene triangle

Example 2: Construct isosceles triangle whose base has length of 5 cm and arms 4 cm.

Firstly, draw a sketch. Since in the task it is not default which side is the base and which are arms, we’ll take that side c is our base.

isosceles triangle sketch

First draw the base. Then take your compass and draw a circle with radius 4, around point A and point B.

constructing isosceles triangle

Example 3: Draw an equilateral triangle whose side length equals to 4cm.

First, you draw line c, and then from points A and B draw a circle with 4cm radius.

constructing equilateral triangle

The Exterior Angle Theorem

The exterior angle is an angle between any side of a triangle and a line extended from its adjacent side.

Exterior angles: \alpha', \beta', \gamma'

Interior angles: \alpha, \beta, \gamma


exterior angle theorem

The sum of all measures of exterior angles equals to 360ᵒ.

The sum of measures of exterior and its attached interior angle equals to 180ᵒ.

The Exterior Angle Theorem says the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle.

\alpha' = \beta + \gamma

\beta' = \alpha + \gamma

\gamma' = \alpha + \beta

Example: Calculate all exterior and interior angles in following triangle.


exterior and interior angle example

\alpha = 115.89^{\circ}

\beta' = 151.23^{\circ}

\beta = ?, \gamma = ?, \alpha' = ?, \gamma' = ?

We know that \beta' + \beta = 180^{\circ}. If we include what we know:

\beta + 115.89^{\circ} = 180^{\circ}

\beta = 180^{\circ} - 115.89^{\circ} = 28.77^{\circ}

Now, by exterior angle theorem:

\gamma' = \alpha + \beta = 151.23^{\circ} + 28.77^{\circ} = 144.66^{\circ}

\gamma = 180^{\circ} - \gamma' = 180^{\circ} - 144.66^{\circ} = 35.34^{\circ}

\alpha' = \beta + \gamma = 28.77^{\circ} + 35.34^{\circ} = 64.11^{\circ}

And if you calculated everything correctly, the sum of all measures of interior angles should be \ 180^{\circ}, and of all exterior angles \ 360^{\circ}.

\alpha + \beta + \gamma = 115.89^{\circ} + 28.77^{\circ} + 35.34^{\circ} = 180^{\circ}

\alpha' + \beta' + \gamma' = 64.11^{\circ} + 151.23^{\circ} + 144.66^{\circ} = 360^{\circ}

Triangles Worksheets

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