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Triangle Classification


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Triangle Classification

Triangle Classification

A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle has three vertices (the points where the segments meet), three sides (the segments), and three interior angles (formed at each vertex). All of the following shapes are triangles.

You might have also learned that the sum of the interior angles in a triangle is \begin{align*}180^\circ\end{align*}. Later we will prove this, but for now you can use this fact to find missing angles. Angles can be classified by their size: acute, obtuse or right. In any triangle, two of the angles will always be acute. The third angle can be acute, obtuse, or right. We classify each triangle by this angle.

Right Triangle: When a triangle has one right angle.

Obtuse Triangle: When a triangle has one obtuse angle.



We can also classify triangles by its sides.

Scalene Triangle: When all sides of a triangle are all different lengths.


Equilateral Triangle: When all sides of a triangle are congruent.

Note that by the above definitions, an equilateral triangle is also an isosceles triangle.

Watch this video starting at around 2:30.

Recognizing Triangles 

Which of the figures below are not triangles?

\begin{align*}B\end{align*} is not a triangle because it has one curved side. \begin{align*}D\end{align*} is not a closed shape, so it is not a triangle either.

Describing Triangles 

Which term best describes \begin{align*}\triangle RST\end{align*} below?

This triangle has one labeled obtuse angle of \begin{align*}92^\circ\end{align*}. Triangles can only have one obtuse angle, so it is an obtuse triangle.

Classifying Triangles 

Classify the triangle by its sides and angles.



Example 1

How many triangles are in the diagram below?

Start by counting the smallest triangles, 16. Now count the triangle that are formed by four of the smaller triangles.

There are a total of seven triangles of this size, including the inverted one in the center of the diagram. Next, count the triangles that are formed by nine of the smaller triangles. There are three of this size. And finally, there is one triangle formed by the 16 smaller triangles. Adding these numbers together we get \begin{align*}16 + 7 + 3 + 1 = 27\end{align*}.

Example 2

Classify the triangle by its sides and angles.

This triangle has a right angle and no sides are marked as congruent. So, it is a right scalene triangle. 

Example 3

Classify the triangle by its sides and angles.

This triangle has an angle bigger than \begin{align*}90^\circ\end{align*} and two sides that are marked congruent. So, it is an obtuse isosceles triangle.


For questions 1-5, classify each triangle by its sides and by its angles.

  4. What must be true about the angles of your equilateral triangle from #8?

For 10-14, determine if the statement is ALWAYS true, SOMETIMES true, or NEVER true.

  1. Obtuse triangles are isosceles.
  2. A right triangle is acute.
  3. An equilateral triangle is equiangular.
  4. An isosceles triangle is equilaterals.
  5. Equiangular triangles are scalene.


For 15-16, construct the indicated figures.

  2. Construct a \begin{align*}60^\circ\end{align*} angle. (Hint: Think about an equilateral triangle.)

Review (Answers)

To view the Review answers, open this PDF file and look for section 1.11. 

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Acute Triangle

An acute triangle has three angles that each measure less than 90 degrees.

Interior angles

Interior angles are the angles inside a figure.

Isosceles Triangle

An isosceles triangle is a triangle in which exactly two sides are the same length.

Obtuse Triangle

An obtuse triangle is a triangle with one angle that is greater than 90 degrees.

Right Angle

A right angle is an angle equal to 90 degrees.

Scalene Triangle

A scalene triangle is a triangle in which all three sides are different lengths.




A polygon is equiangular if all angles are the same measure.

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