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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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Student Learning Objectives:
  • Identify triangles by their angle measures and side lengths
  • Identify whether a form is a triangle by given side lengths
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 AB, BC and AC that form three angles. Click and drag point C to change the shape of the figure.

A triangle with a free vertex

What types of angle are formed at vertex C?
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 4, 4.5, and 8.8 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 ABC such that its two sides are segments.

Angle ABC

If the free endpoints of each side A and C 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, △ ABC and △ BCA both denote the triangle whose vertices are A, B, and C.

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 90^(∘).

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 90^(∘).

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 60^(∘) 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 AB=CD, then AB≅CD.
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 ABC, three inequalities hold true.

AB + BC > AC BC + AC > AB AC + AB > BC

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 18 centimeters long and the chopsticks are each 24 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 15, 9, and 5 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 18
Chopstick 1 24
Chopstick 2 24

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 3 Comparison
Straw and Chopstick 1 18+24=42 Chopstick 2 42 > 24 ✓
Straw and Chopstick 2 18+24=42 Chopstick 1 42 > 24 ✓
Chopstick 1 and Chopstick 2 24+24=48 Straw 48 > 18 ✓

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 15
Pencil 9
Crayon 5

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 3 Comparison
Pen and Pencil 15+9=24 Crayon 24 > 5 ✓
Pen and Crayon 15+5=20 Pencil 20 > 9 ✓
Pencil and Crayon 9+5=14 Pen 14 > 15 *

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 1 & Match 2 8.8+4.5=13.3 4.0 13.3 > 4.0 ✓
Match 1 & Match 3 8.8+4.0=12 4.5 12 > 4.5 ✓
Match 2 & Match 3 4.5+4.0=8.5 8.8 8.5 > 8.8 *

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
Exercise 3.1

Diego and Paulina will go to the baseball stadium after school. They will follow different paths. Paulina will follow the path that passes by the hospital. The streets that connect the school, the hospital, and the stadium form a triangle.

Triangle

If Diego and Paulina walk at the same pace and do not stop along the way, who will get to the stadium first?

Diego and Paulina walk at the same pace and do not stop along the way, so whoever has chosen the shortest path will arrive first.

The streets that connect the school, the hospital, and the baseball stadium form a triangle. This means that the lengths of the sides of this triangle satisfy the Triangle Inequality Theorem and so, three inequalities can be written. SH + HB &> SB [0.4em] HB + SB &> SH [0.4em] SB + SH &> HB The first inequality tells us that the path from the school to the stadium that passes by the hospital is longer than the path that directly connects the school and the stadium. We can conclude that Diego will arrive first.

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Hint & Answer
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Solution

Triangles

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