### 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.

**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.

*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.

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.

### Examples

#### 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.

### Review

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

- Can you draw a triangle with a right angle and an obtuse angle? Why or why not?
- In an isosceles triangle, can the angles opposite the congruent sides be obtuse?
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.*Construction*- 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.

- Obtuse triangles are isosceles.
- A right triangle is acute.
- An equilateral triangle is equiangular.
- An isosceles triangle is equilaterals.
- 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.

- Construct a right triangle with side lengths 3 cm, 4 cm and 5 cm.
- 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.