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Pre-Algebra View details
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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Lesson Settings & Tools
18 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Triangles
Slide of 18
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

Select the most appropriate classification for the given triangle.

Triangle ABC
Triangle

Let's begin by classifying the given triangle according to its side lengths.

We can see that two sides have the same length, 4 units long, while the third side is 5 units long. Therefore, the triangle is an isosceles triangle. Next, let's consider to the angle measures of △ ABC.

The three angle measures are less than 90^(∘), which means they are all acute. Because of this, △ ABC is also an acute triangle. The most appropriate classification combines the two classifications we made. △ ABC is an acute triangle. △ ABC is an isosceles triangle. ⇓ △ ABC is an acute isosceles triangle.


Let's start by classifying △ PQR according to the angle measures.

We can see that angles Q and R are acute. Although the measure of ∠ P is not written using numbers, the angle is denoted with a square angle marker. This means that ∠ P is a right angle. Therefore, △ PQR is a right triangle. Let's now focus on the side lengths.

The three sides have different lengths, which means that △ PQR is a scalene triangle. Finally, let's combine the two classifications. △ PQR is a right triangle. △ PQR is a scalene triangle. ⇓ △ PQR is a right scalene triangle.

Fill in the blank with the most appropriate word or phrase.

Let be a triangle in which one angle has a measure greater than If and then is a(n) triangle.

In one of the angles has a measure of and the shorter sides have different lengths. Because of this, is a(n) triangle.

Let's begin by classifying △ ABC according to its angle measures. We are given that one angle has a measure greater than 100^(∘). This means that the triangle has one obtuse angle. One angle is obtuse. ⇓ △ ABC is an obtuse triangle. Let's now classify △ ABC based on its side lengths. We know that AB≅AC, which means that these two segments have the same length. On the other hand, we know that AC≆BC, which means these segments have different lengths. AB = AC AC ≠ BC ⇒ AB ≠ BC From these statements, we can say that only two sides have the same length. Only two sides have the same length. ⇓ △ ABC is an isosceles triangle. The most appropriate phrase to fill in the blank is the one combining the two classifications of △ ABC.

Let △ ABC be a triangle in which one angle has a measure greater than 100^(∘). If AB≅ AC and AC≆BC, then △ ABC is an obtuse isosceles triangle.


Let's classify △ JKL according to the measure of its angles. We are given that one angle has a measure of 90^(∘), so △ JKL has one right angle. One angle is a right angle. ⇓ △ JKL is a right triangle. Next, let's analyze the side lengths of △ JKL. Since the triangle is a right triangle, we know that one side is longer than the other two. This longer side is the hypotenuse. Additionally, we know that the shorter sides have different lengths, which means that the three sides all have different lengths. The sides have different lengths. ⇓ △ JKL is a scalene triangle. Finally, let's combine the two classifications to get the most appropriate phrase that fills in the blank.

In △ JKL, one of the angles has a measure of 90^(∘) and the shorter sides have different lengths. Because of this, △ JKL is a right scalene triangle.


Let and be three segments with the following lengths.
Can these segments form a triangle?
Let be a triangle for which and Which of the following cannot be length of

The Triangle Inequality Theorem says that in a triangle, the sum of the lengths of any two sides is greater than the length of the third side. The given three segments can form a triangle if their lengths satisfy such condition. AB = 7 BC = 5 AC = 3.5 Let's calculate the sums of the lengths of the possible pairs of segments and check if they are greater than the lengths of the third segment.

Pair of Segments Sum of Lengths Third Segment Comparison
AB & BC 7+5=12 AC 12 > 3.5 ✓
BC & AC 5+3.5=8.5 AB 8.5 > 7 ✓
AB & AC 7+3.5=10.5 BC 10.5 > 5 ✓

The lengths of the given segments meet the three inequalities. Therefore, the segments can form a triangle.


We are given the lengths of two sides of △ PQR. PQ = 5 QR = 8 PR = ? The points P, Q, and R already form a triangle. This means that the length of PR must be a number such that the three triangle inequalities are satisfied. Let x be the length of PR. Let's make a table like the one we made in Part A.

Pair of Segments Sum of Lengths Third Segment Comparison
PQ & QR 5+8=13 PR 13 > x
QR & PR 8+x PQ 8+x > 5
PQ & PR 5+x QR 5+x > 8

The last column of the first row tells us that the length of PR is less than 13. Let's now solve the other two inequalities for x. 8+x &> 5 &⇒ x &> -3 [0.4em] 5+x &> 8 &⇒ x &> 3 Since the value of x represents the length of PR, it must be positive. This means that the inequality x> -3 does not give us any information. However, the last inequality tells us that the length of PR must be greater than 3. The length ofPRis greater than3. The length ofPRis less than13. ⇓ 3 < x < 13 Let's now recall the given four possible lengths for PR. 3 5 10 12 The value 3 is the only one that do not lie between 3 and 13. This means that the length of PR cannot be 3. The following diagram shows the triangles that can be formed with the other three given lengths.


Triangles

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