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4. Triangles
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eCourses /
Pre-Algebra /
Chapter 9
4. 

Triangles

Triangles are foundational geometric shapes that appear in various mathematical and practical contexts. Understanding their properties and classifications enables deeper insight into geometry. Different types of triangles, such as acute, right, and obtuse, are categorized based on angles, while equilateral, isosceles, and scalene triangles are distinguished by their sides. The Triangle Inequality Theorem plays a critical role in determining possible side lengths and ensuring a valid triangle. Exploring these concepts enhances problem-solving skills and supports applications in fields like construction, design, and engineering. Mastery of triangle properties also serves as a stepping stone to more advanced topics in trigonometry and geometry.
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11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Triangles
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Two segments with a common endpoint form an angle. If the free endpoints of each segment are connected, then three angles are formed, one at each endpoint. The plane figure formed by these three angles and segments is called a triangle. In this lesson, the definition and different classifications of triangles are studied.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Studying the Angle Type

Consider three segments and that form three angles. Click and drag point to change the shape of the figure.
A triangle with a free vertex
What types of angle are formed at vertex
Challenge

Matchstick Riddles

Dylan and Zosia love to solve riddles and decided to have a competition to see who can solve them faster. In one riddle, they have to form a triangle with three matchsticks that are and centimeters long.

Three matchsticks

Dylan says that such a triangle cannot be formed, while Zosia says it can. Who is right?

Discussion

Figures With Three Angles

Consider an angle such that its two sides are segments.

Angle ABC

If the free endpoints of each side and are connected by a segment, a figure with three sides and three angles is formed. This kind of figure is called a triangle.

Concept

Triangle

A triangle is a polygon with three angles and three sides. The symbol is used to algebraically denote a triangle. Next to this symbol, the three vertices of the triangle are written in any order. For example, and both denote the triangle whose vertices are and

Interactive triangle with movable vertices
Discussion

Triangles With Three Acute Angles

Triangles can be classified according to the measures of their interior angles into four different types. The first type includes those triangles in which the three angles are acute.

Concept

Acute Triangle

An acute triangle is a triangle where all angles are acute. In other words, the three angle measures are less than

Acute triangle vs. Non acute triangle
Discussion

Triangles With a Right Angle

The second type of triangle groups all those triangles that have one right angle.

Concept

Right Triangle

A right triangle is a triangle that has one right angle. The side opposite the right angle is always the longest and is known as the hypotenuse. The other sides are commonly called legs. Notice that in a right triangle, the legs are perpendicular to each other.

Right triangle vs. Not a right triangle
Discussion

Triangles With an Obtuse Angle

The next type of triangle includes those triangles that have one obtuse angle.

Concept

Obtuse Triangle

An obtuse triangle is a triangle that has one obtuse angle. In other words, one of the angles measures more than

obtuse triangle vs. Not an obtuse triangle
Discussion

Triangles With Three Congruent Angles

The last classification of triangles according to the measure of their angles consists of those triangles that have three angles of the same measure.

Concept

Equiangular Triangle

An equiangular triangle is a triangle in which the three angles have the same measure. In other words, the three angles are congruent. In fact, the three angles measure each.
Interactive triangle with a movable vertex
Notice that an equiangular triangle is also an acute triangle. In an equiangular triangle, the three sides have the same measure.
Pop Quiz

Classifying Triangles

Classify the given triangle according to its angle measures as an acute, right, or obtuse triangle.

Random triangles
Discussion

Segments With the Same Length

The same way two angles with the same measure are called congruent angles, there is a special term for two segments that have the same length.

Concept

Congruent Segments

Two segments are congruent segments if both have the same length. In a diagram, congruent segments are usually indicated by the same number of ticks. The symbol is used to express algebraically that two segments are congruent.
Two congruent segments AB and CD
Therefore, if then
Discussion

Triangles With Three Congruent Sides

In addition to their angle measures, triangles can also be classified by comparing the lengths of their three sides. There are three different types. The first includes those triangles in which all three sides have the same length.

Concept

Equilateral Triangle

An equilateral triangle is a triangle in which the three sides have the same length. In other words, the three sides are congruent.
Interactive triangle with a movable vertex
In an equilateral triangle, the three angles have the same measure. Therefore, equilateral triangles are also equiangular triangles.
Discussion

Triangles With Only Two Congruent Sides

The second way of classifying a triangle according to its side lengths includes all the triangles in which only two sides have the same length.

Concept

Isosceles Triangle

An isosceles triangle is a triangle that has exactly two sides of the same length. In other words, it has two congruent sides. The congruent sides are called legs while the third side is called the base. The angle between the legs is called the vertex angle and the other two angles are called base angles.
Interactive triangle with a movable vertex
In an isosceles triangle, the base angles are congruent angles — they have the same measure.
Discussion

Triangles With Three Different Side Lengths

The last classification of triangles based on their side lengths is when the three sides have different lengths.

Concept

Scalene Triangle

A scalene triangle is a triangle in which all three sides have different lengths. In other words, it has no congruent sides.
Interactive triangle with a movable vertex
Additionally, in a scalene triangle, the three angles have different measures. In other words, scalene triangles have no congruent angles.
Pop Quiz

Classifying Triangles From Their Side Lengths

Classify the given triangle according to its side lengths as an equilateral, isosceles, or scalene triangle.

Random triangles
Discussion

Triangle Inequality Theorem

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
A triangle with movable vertices. The side lengths are printed. Three inequalities.
Therefore, given a triangle three inequalities hold true.
Example

Challenging Lunch

At lunch, Ali challenged Dylan and Zosia to build a triangle with the objects they had on hand. He offered a free dessert to whoever did it successfully.

Three friends eating

Since Zosia brought Chinese food and a soft drink, she decided to use the two chopsticks and the straw to build her triangle. The straw is centimeters long and the chopsticks are each centimeters long.

Chinese food, straw, and chopsticks

For his part, Dylan decided to use a pen, a short pencil, and a small crayon from his backpack. The pen, pencil, and crayon are and centimeters long, respectively.

Backpack, pencil, pen, and crayon
Who will get the free dessert?

Hint

Check if the lengths of the objects satisfy the Triangle Inequality Theorem.

Solution

In order for three segments to form a triangle, their lengths must satisfy the Triangle Inequality Theorem. In other words, the sum of the lengths of any two segments must be greater than the length of the third segment. Consider Zosia's and Dylan's attempts separately.

Zosia

Start by writing the lengths of the objects that Zosia chose.

Object Length (cm)
Straw
Chopstick
Chopstick

Now, calculate the sums of the lengths of the possible pairs of objects and check if they are greater than the length of the third object.

Pair of Objects Sum of Lengths Object Comparison
Straw and Chopstick Chopstick
Straw and Chopstick Chopstick
Chopstick and Chopstick Straw

As shown, the sum of the lengths of any two objects is always greater than the length of the third object. Therefore, Zosia can form a triangle with the objects she chose, so she will get a free dessert!

Isosceles triangle formed with a straw and two chopsticks

Notice that the triangle has two sides of the same length, which means that Zozia built an isosceles triangle.

Dylan

Dylan can build a triangle with the objects he chose only if their lengths meet the Triangle Inequality Theorem. Follow the same process to see if he can earn a free dessert. Begin by writing the lengths of the objects.

Object Length (cm)
Pen
Pencil
Crayon

Next, calculate the sums of the lengths of the possible pairs of objects and verify if they are greater than the length of the third object.

Pair of Objects Sum of Lengths Object Comparison
Pen and Pencil Crayon
Pen and Crayon Pencil
Pencil and Crayon Pen

The sum of the lengths of the pencil and the crayon in the last row of the table is not greater than the length of the pen, which means that the object's lengths do not meet the Triangle Inequality Theorem. Dylan cannot build a triangle with the objects he chose, so he will not get a free dessert. Bummer!

pencil, pen, and crayon
Pop Quiz

Building a Triangle

Determine whether the given segments can form a triangle.
Random Segments
Recall that three segments can form a triangle if their lengths satisfy the Triangle Inequality Theorem.

Extra

 Tip

There is no need to verify all three inequalities. Just check whether the sum of the lengths of the two shorter segments is greater than the length of the longer segment.

Closure

Solving the Riddle

At the beginning of the lesson, Dylan and Zosia were challenged to form a triangle using three matchsticks. However, they came to opposite conclusions. Dylan claimed that no triangle could be formed with the given matches, while Zosia said that a triangle could be formed.

Three matchsticks

The riddle can be solved with the information given in this lesson. The three matches can form a triangle if they satisfy the Triangle Inequality Theorem. In other words, check whether the sum of the lengths of any pair of matches is greater than the length of the third match.

Pair of Matches Sum of Lengths Third Match Comparison
Match & Match
Match & Match
Match & Match

As shown, the sum of the lengths of the two shorter matches is not greater than the length of the longest match. Therefore, the given matchsticks cannot form a triangle. Dylan was right!

Conversation between two friends
Before moving forward, keep in mind that a triangle can be classified simultaneously by angle measures and side lengths.
Interactive triangle with a movable vertex
There are a lot more facts about triangles to study! This information will be explored later in the course.
Level 1
Level 2
Level 3
Triangles
Exercises

Classify the given triangle based on its side lengths.

Billiard Triangle
Hanger

Let's begin by recalling the definitions of scalene, isosceles, and equilateral triangles. That way we will be able to classify the billiard triangle.

Definition
Scalene Triangle A triangle in which all three sides have different lengths.
Isosceles Triangle A triangle that has exactly two sides of the same length.
Equilateral Triangle A triangle in which the three sides have the same length.

In the given billiard triangle, we can see that the three sides have the same length, 28.5 centimeters.

Therefore, the billiard triangle is an equilateral triangle.


We can see that the triangle formed in the interior of the hanger has two sides that are 8.3 inches long and one side that is 15 inches long. This means that the triangle has exactly two sides of the same length.

The triangle is an isosceles triangle, according to the table in Part A.

Classify the given triangle based on its angle measures.

Remote and play button
Select all that apply!
Triangular Ruler
Select all that apply!

Let's start by writing the definitions of acute, right, obtuse, and equiangular triangles so that we can determine what type of triangle the play button is.

Definition
Acute Triangle A triangle where all angles are acute.
Right Triangle A triangle that has one right angle.
Obtuse Triangle A triangle that has one obtuse angle.
Equiangular Triangle A triangle in which the three angles have the same measure.

On the given remote, we can see that the three angles of the play button have the same measure, 60^(∘).

Therefore, the triangle is acute and equiangular.


The triangle formed in the interior of the ruler has a 90^(∘) angle — a right angle.

Because of this, according to the table in Part A, the triangle is a right triangle.

Fill in the blank in each of the following statements with the correct word.

In triangle the three angles are acute and have different measures. This means that is a(n) triangle.

In triangle the measure of is Therefore, is a(n) triangle.

We are given some information about △ PQR and its angles, so let's classify the triangle according to its angle measures. Remember, there are four different classifications.

Definition
Acute Triangle A triangle where all angles are acute.
Right Triangle A triangle that has one right angle.
Obtuse Triangle A triangle that has one obtuse angle.
Equiangular Triangle A triangle in which the three angles have the same measure.

We are told that the measures of all three angles are acute, which means that the triangle is not a right triangle or an obtuse triangle. The three angles are acute in an equiangular triangle, we are left with two options. cc acute & ? [0.75em] right & * [0.75em] obtuse & * [0.75em] equiangular &? We are told that the three angle measures are different, which means that △ PQR is not equiangular. By process of elimination, △ PQR is an acute triangle.

In triangle PQR, the three angles are acute and have different measures. This means that △ PQR is an acute triangle.

Notice that we did not consider the options isosceles and equilateral because these are classifications based on the side lengths, which we do not know.


All we know about this triangle is one angle measure, but this might be enough to classify it! Let's take a look. m∠ C = 95^(∘) We can see that ∠ C is an obtuse angle because its measure is greater than 90^(∘). This means that we can conclude that △ ABC is an obtuse triangle according to the table in Part A.

In triangle ABC, the measure of ∠ C is 95^(∘). Therefore, △ ABC is an obtuse triangle.


Fill in the blank in each of the following statements with the correct word.

If two sides of a triangle have the same length and it is different from the length of the third side, then the triangle is a(n) triangle.

If a triangle has no congruent sides, then the triangle is a(n) triangle.

All we know about the given triangle is certain information about its sides, so let's classify it according to the side lengths. Remember, there are three different classifications.

Definition
Scalene Triangle A triangle in which all three sides have different lengths.
Isosceles Triangle A triangle that has exactly two sides of the same length.
Equilateral Triangle A triangle in which the three sides have the same length.

We are told that the triangle has two sides of the same length and that the third side is not the same length. Therefore, the triangle has exactly two sides that are the same length. This means that the triangle is isosceles.

If two sides of a triangle have the same length and it is different from the length of the third side, then the triangle is a(n) isosceles triangle.


As in Part A, we have some information about the side lengths of a triangle. Let's classify the triangle according to what we know. This time the triangle has no congruent sides. No congruent sides ⇕ Sides with different lengths This means that all three sides have different lengths. As such, and using the table in Part A, the triangle is scalene.

If a triangle has no congruent sides, then the triangle is a scalene triangle.


Classify according to the given characteristics.

We are given the three side lengths of triangle ABC. Two sides are 5 centimeters long and the other side is 3 centimeters long. Let's draw the triangle to see what we are working with.

As we can see, △ ABC has exactly two sides of the same length, so it is an isosceles triangle.


This time we were not given the actual side lengths, but we can still analyze each piece of the given information. AB ≠ BC This statement tells us that the sides AB and BC have different lengths. We can come to similar conclusions from the other two statements.

Statement Conclusion
AB ≠ BC Sides AB and BC have different lengths.
BC ≠ AC Sides BC and AC have different lengths.
AC ≠ AB Sides AC and AB have different lengths.

If we combine the three conclusions, we get that the three sides of △ ABC have different lengths. In conclusion, △ ABC is a scalene triangle. Let's draw an example of what △ ABC could look like.


Classify according to the given characteristics.

We are given the three angle measures of triangle PQR. In light of this, let's recall the triangle classifications according to the angle measures.

Definition
Acute Triangle A triangle where all angles are acute.
Right Triangle A triangle that has one right angle.
Obtuse Triangle A triangle that has one obtuse angle.
Equiangular Triangle A triangle in which the three angles have the same measure.

Next, let's classify the angles of △ PQR according to their measures.

Angle Measure Angle Type
∠ P 30^(∘) Acute
∠ Q 50^(∘) Acute
∠ R 100^(∘) Obtuse

One of the angles is obtuse, so △ PQR is an obtuse triangle. Let's draw an example of what △ PQR may look like.


This time we can see that two of the angles of △ PQR have the same measure. We can also see that the three angle measures are all less than 90^(∘).

Angle Measure Angle Type
∠ P 59^(∘) Acute
∠ Q 62^(∘) Acute
∠ R 59^(∘) Acute

According to the right-hand column of the table, the three angles are acute, so we can conclude that △ PQR is an acute triangle. Now let's see an example of what △ PQR could look like.


Consider the following three triangles.

Three triangles
 Which of the triangles is a right triangle?
Which of the triangles is an acute triangle?
Which of the triangles is an isosceles triangle?
Which of the triangles is a scalene triangle?

We are looking for a right triangle. Recall that a right triangle has one right angle — that is, one of the angles of the triangle has a measure of 90^(∘). Let's scan the given three triangles with this in mind.

At first glance we do not see the measure 90^(∘) explicitly written. We might think that none of the triangles is a right triangle, but notice that ∠ K is denoted with a square angle marker. This means that ∠ K is a right angle. As such, △ JKL is a right triangle.


This time we are looking for an acute triangle. Remember, in an acute triangle, all three angles are acute. This means that we are looking for a triangle in which the three angle measures are less than 90^(∘).

If we consider the angle measures, we can make the following three conclusions.

  • In △ PQR, the measure of ∠ P greater than 90^(∘).
  • In △ ABC, the measures of all three angles are less than 90^(∘).
  • In △ JKL, the measure of ∠ K is 90^(∘).

Therefore, only △ ABC is an acute triangle.


We are looking for an isosceles triangle now. Remember, isosceles triangles have exactly two sides of the same length. Let's take a look at the given triangles again and pay attention to the side lengths.

After checking the triangles, we can come to the following three conclusions.

  • In △ PQR, the three sides have different lengths.
  • In △ ABC, the three sides have the same length.
  • In △ JKL, exactly two sides have the same length.

In light of these facts, only △ JKL is an isosceles triangle!


Finally, we are looking for a scalene triangle, which is a triangle in which all three sides have different lengths. Let's remember the observations we made in Part C.

  • In △ PQR, the three sides have different lengths.
  • In △ ABC, the three sides have the same length.
  • In △ JKL, exactly two sides have the same length.

According to this information, only △ PQR is a scalene triangle!

Triangles

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