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# Triangle Classification

## Categories of triangles based on angle measurements or the number of congruent sides.

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Triangle Classification

What if you were given the angle measures or side lengths of several triangles and were asked to group them based on their properties? After completing this Concept, you'll be able to classify a triangle as right, obtuse, acute, equiangular, scalene, isosceles, and/or equilateral.

### Watch This

Watch this video starting at around 2:30.

### Guidance

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 180\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.

Acute Triangle: When all three angles in the triangle are acute.

Equiangular Triangle: When all the angles in a triangle are congruent.

We can also classify triangles by its sides.

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

Isosceles Triangle: When at least two sides of a triangle are congruent.

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

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

#### Example A

Which of the figures below are not triangles?

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

#### Example B

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.

#### Example C

Classify the triangle by its sides and angles.

We are told there are two congruent sides, so it is an isosceles triangle. By its angles, they all look acute, so it is an acute triangle. Typically, we say this is an acute isosceles triangle.

Watch this video for help with the Examples above.

### Vocabulary

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). A right triangle is a triangle with one right angle. An obtuse triangle is a triangle with one obtuse angle. An acute triangle is a triangle where all three angles are acute. An equiangular triangle is a triangle with all congruent angles. A scalene triangle is a triangle where all three sides are different lengths. An isosceles triangle is a triangle with at least two congruent sides. An equilateral triangle is a triangle with three congruent sides.

### Guided Practice

1. How many triangles are in the diagram below?

2. Classify the triangle by its sides and angles.

3. Classify the triangle by its sides and angles.

1. Start by counting the smallest triangles, 16. Now count the triangles 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*}.

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

3. 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.

### Practice

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

1. Can you draw a triangle with a right angle and an obtuse angle? Why or why not?
2. In an isosceles triangle, can the angles opposite the congruent sides be obtuse?
3. Construction Construct an equilateral triangle with sides of 3 cm. Start by drawing a horizontal segment of 3 cm and measure this side with your compass from both endpoints.
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.

In geometry it is important to know the difference between a sketch, a drawing and a construction. A sketch is usually drawn free-hand and marked with the appropriate congruence markings or labeled with measurement. It may or may not be drawn to scale. A drawing is made using a ruler, protractor or compass and should be made to scale. A construction is made using only a compass and ruler and should be made to scale.

For 15-16, construct the indicated figures.

1. Construct a right triangle with side lengths 3 cm, 4 cm and 5 cm.
2. Construct a \begin{align*}60^\circ\end{align*} angle. (Hint: Think about an equilateral triangle.)

### Vocabulary Language: English

Acute Triangle

Acute Triangle

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

Interior angles

Interior angles are the angles inside a figure.
Isosceles Triangle

Isosceles Triangle

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

Obtuse Triangle

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

Right Angle

A right angle is an angle equal to 90 degrees.
Scalene Triangle

Scalene Triangle

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

Equilateral

A polygon is equilateral if all of its sides are the same length.
Equiangular

Equiangular

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