Introduction

Triangles can be classified by sides or by angles.

First, let's consider the classifications by sides. We have three groups of triangles:

  • Equilateral triangles, where all sides have the same length.

  • Isosceles triangles, where only two sides are congruent. The congruent sides are called the legs of the triangle, and the third side is called the base.

  • Scalene triangles, where all sides have different measures.



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Classifying Triangles Based on Angles

Now, let's consider classifying triangles by angles. We have three groups of triangles.

  • In an acute triangle, all three angles are acute.

  • A triangle with a right angle is called a right triangle.

  • A triangle with an obtuse angle is called an obtuse triangle.



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Example: Classifying a Triangle Based on Its Sides

Classify the triangle shown below.



EXPLANATION

In the triangle we see two congruent sides, \overline{AB} \cong \overline{AC}, but the third side is not congruent: \overline{AB} \not\cong \overline{BC}.

Therefore, since only two sides are congruent, \triangle ABC is an isosceles triangle with base \overline{BC}.

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Practice: Classifying a Triangle Based on Its Sides

Classify the triangle above.

a
 $\triangle{ABC}$ is a right triangle
b
 $\triangle{ABC}$ is a left triangle
c
 $\triangle{ABC}$ is an isosceles triangle
d
 $\triangle{ABC}$ is an equilateral triangle
e
 $\triangle{ABC}$ is a scalene triangle
Practice: Classifying a Triangle Based on Its Sides

Which of the following statements best describes the triangle above?

a
 $\triangle ABC$ is an equilateral triangle
b
 $\triangle ABC$ is a scalene triangle
c
 $\triangle ABC$ is an isosceles triangle with base $\overline{BC}$
d
 $\triangle ABC$ is an isosceles triangle with base $\overline{AB}$
e
 $\triangle ABC$ is an isosceles triangle with base $\overline{AC}$
Example: Classifying a Triangle Based on Its Angles

Given a triangle \triangle ABC , we know that m\angle {B} = 20^{\circ} and m\angle {C} = 3 m\angle {A}. Based on the angles, which type of triangle is \triangle ABC?

EXPLANATION

Let x be the measure of \angle{A}. Then m\angle {C}=3x.

Since the sum of the interior angles is 180^{\circ} , we have \begin{align*} m\angle{A}+m\angle{B}+m\angle{C} &= 180^\circ \\[2pt] x + 20^{\circ} + 3x &= 180^{\circ} \\[2pt] 4x + 20^{\circ} &= 180^{\circ} \\[2pt] 4x &= 160^{\circ} \\[2pt] x &= 40^{\circ}. \end{align*}

Therefore, m\angle{A}=40^{\circ} and m\angle {C}=3 \cdot 40^\circ = 120^{\circ}.

So, ABC is an obtuse triangle because it has an obtuse angle ( m\angle {C} > 90^{\circ} ).

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Practice: Classifying a Triangle Based on Its Angles

Given a triangle $\triangle ABC$, we know that $m\angle{B}$ is $10^\circ$ less than $m\angle{A}$ and $40^\circ$ more than $m\angle{C}.$ Based on the angles, which of the following statements best describes the triangle $\triangle ABC?$

a
 $m\angle{B} = 90^\circ,$ $\triangle ABC$ is a right triangle
b
 $m\angle{B} = 80^\circ,$ $\triangle ABC$ is an obtuse triangle
c
 $m\angle{B} = 70^\circ,$ $\triangle ABC$ is an acute triangle
d
 $m\angle{B} = 30^\circ,$ $\triangle ABC$ is a right triangle
e
 $m\angle{B} = 110^\circ,$ $\triangle ABC$ is an obtuse triangle
Practice: Classifying a Triangle Based on Its Angles

For a triangle $\triangle ABC$, its known that $m\angle{B} = 16^{\circ}$ and $m\angle{C} = 3 m\angle{A}.$ Based on the angles, which of the following statements best describes the triangle?

a
 $m\angle{C}=90^{\circ},$ $\triangle ABC$ is a right triangle
b
 $m\angle{C}=89^{\circ},$ $\triangle ABC$ is an acute triangle
c
 $m\angle{C}=123^{\circ},$ $\triangle ABC$ is an obtuse triangle
d
 $m\angle{C}=115^{\circ},$ $\triangle ABC$ is an obtuse triangle
e
 $m\angle{C}=82^{\circ},$ $\triangle ABC$ is an acute triangle
Relationships Between Sides and Interior Angles of a Triangle

Consider the following scalene triangle, where AB=5 , BC=7, and AC=6 units, respectively.



Looking at the angles, it's hard to tell which angle is the largest.

However, since we know the sides, there is a simple rule that we can use to determine the relative size of the angles:

Longer sides are opposite to larger angles and vice versa.

Applying this rule to \triangle{ABC} depicted above, the longest side is \overline{BC}. Therefore, the largest angle is the angle opposite to \overline{BC}, which is \color{red}\angle{A}.



Similarly,

  • the mid-sized side ( \overline{AC} ) is opposite to the mid-sized angle ( \angle B ), and

  • the shortest side ( \overline{AB} ) is opposite to the smallest angle ( \angle C ).

So, from least to greatest, the angles of the triangle are ordered as follows: m\angle{C} < m\angle{B} < m\angle{A}

Note: This rule works in reverse too. If you know the biggest angle of a triangle, the longest side is always opposite to this angle.

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Example: Ordering the Sides of a Triangle by Size Given the Interior Angles

Order the sides of the triangle \triangle ABC above from the shortest to the longest.



EXPLANATION

First, let's find the missing angle \angle A. Since the interior angles of a triangle sum to 180^\circ, we have \begin{align*} m\angle A +80^\circ +75^\circ & = 180^\circ \\[3pt] m\angle A &= 180^\circ -155^\circ\\[3pt] m\angle A &= 25^\circ. \end{align*}

So, we have m\angle A < m\angle C < m\angle B.

Since \angle B is the largest angle, its opposite side \overline{CA} must be the longest side of the triangle.

Since \angle A is the smallest angle, its opposite side \overline{BC} is the shortest side of the triangle.

So, we have

BC < BA < CA

and therefore the required order is \overline{BC}, \overline{BA}, \overline{CA}.

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Practice: Ordering the Sides of a Triangle by Size Given the Interior Angles

Order the sides of the triangle $\triangle ABC$ above from the shortest to the longest.

a
 $\overline{BC}, \overline{AB}, \overline{AC}$
b
 $\overline{AB}, \overline{BC}, \overline{AC}$
c
 $\overline{AB}, \overline{AC}, \overline{BC}$
d
 $\overline{AC}, \overline{AB}, \overline{BC}$
e
 $\overline{BC}, \overline{AC}, \overline{AB}$
Practice: Ordering the Sides of a Triangle by Size Given the Interior Angles

Order the sides of the triangle $\triangle ABC$ above from the shortest to the longest.

a
 $\overline{AC}, \overline{AB}, \overline{BC}$
b
 $\overline{AC}, \overline{BC}, \overline{AB}$
c
 $\overline{AB}, \overline{AC}, \overline{BC}$
d
 $\overline{BC}, \overline{AC}, \overline{AB}$
e
 $\overline{AB}, \overline{BC}, \overline{AC}$
Example: Ordering the Angles of a Triangle by Size Given the Sides

Order the angles of the triangle \triangle ABC from smallest to largest.

EXPLANATION

The shortest side is \overline{BC}, so the smallest interior angle is \angle{A}.

The longest side is \overline{AB}, which means that the opposite angle, \angle C , must be the largest one.

Therefore, the required order must be \angle A,\, \angle B,\, \angle C.

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Practice: Ordering the Angles of a Triangle by Size Given the Sides

Order the angles of the triangle $\triangle ABC$ from smallest to largest.

a
 $\angle C,\, \angle B,\, \angle A$
b
 $\angle A,\, \angle C,\, \angle B$
c
 $\angle B,\, \angle C,\, \angle A$
d
 $\angle C,\, \angle A,\, \angle B$
e
 $\angle A,\, \angle B,\, \angle C$
Practice: Ordering the Angles of a Triangle by Size Given the Sides

Order the angles of the triangle $\triangle ABC$ from the smallest to the largest.

a
 $ \angle C,\, \angle B,\, \angle A\,$
b
 $ \angle B,\, \angle C,\, \angle A\,$
c
 $ \angle A,\, \angle C,\, \angle B\,$
d
 $ \angle A,\, \angle B,\, \angle C\,$
e
 $ \angle C,\, \angle A,\, \angle B\,$
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