3.3: Triangle Basics
Learning Objectives
- Define triangles.
- Classify triangles as acute, right, obtuse, or equiangular.
- Classify triangles as scalene, isosceles, or equilateral.
Defining Triangles
The first shape to examine is the triangle. Though you have probably heard of triangles before, it is helpful to review the formal definition. A triangle is any closed figure made by three line segments intersecting at their endpoints. All of the following shapes are triangles:
A triangle is a closed shape made by _____________ line segments that ___________________ at their endpoints.
Every triangle has three vertices (points at which the segments meet), three sides (the segments themselves), and three interior angles (formed at each vertex).
A triangle has three _______________________ angles formed at each vertex.
A triangle has three ___________________, which are the line segments that make up the shape.
A triangle has three _________________________, which are the points where the sides meet.
The singular version of the word vertices is vertex.
Vertices is the plural version.
This means you can have two vertices but only one vertex.
The plural form of the word vertex is ______________________________.
The singular form of the word vertices is ___________________________.
You may have learned in the past that the sum of the interior angles in a triangle is always \begin{align*}180^\circ\end{align*}. Later we will prove this property, but for now you can use this fact to find missing angles.
All three interior angles in a triangle add up to _______________.
Reading Check:
Label all parts of the following triangle using the vocabulary words:
vertex, interior angle, or side
Classifications by Angles
Earlier, you learned how to classify angles as acute, obtuse or right. Now that you know how to identify triangles, we can classify them as well. One way to classify a triangle is by the measure of its angles.
In any triangle, two of the angles will always be acute. The third angle, however, can be acute, obtuse, or right.
In all triangles, at least two of the interior angles are __________________________.
Reading Check:
Why do you think two angles in a triangle will always be acute?
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This is how triangles are classified. If a triangle has one right angle, it is called a right triangle. Below are some pictures of right triangles:
If a triangle has one right angle (or one \begin{align*}90^\circ\end{align*} angle), it is called a ___________________ triangle.
If a triangle has one obtuse angle, it is called an obtuse triangle. Some pictures of obtuse triangles are shown:
A triangle with one obtuse angle (or one angle greater than \begin{align*}90^\circ\end{align*}), is called an ________________ triangle.
If all of the interior angles are acute, it is called an acute triangle, such as these:
A triangle with all acute angles (or angles smaller than \begin{align*}90^\circ\end{align*}), is called an __________________ triangle.
A special type of acute triangle occurs when all angles are congruent. This triangle is called an equiangular triangle.
All angles are congruent in an __________________________________ triangle.
Reading Check:
1. True or false: All triangles have two acute angles.
2. True or false: A right triangle has two right angles.
3. True or false: An equiangular triangle is also, by definition, an obtuse triangle.
4. Draw a picture of each of the following types of triangles, labeling all the parts:
a. equiangular:
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b. acute:
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c. obtuse:
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Classifications by Side Lengths
There are more types of triangle classes that are not based on angle measure. Instead, these classifications have to do with the sides of the triangle and their relationships to each other.
When a triangle has all sides of different lengths, it is called a scalene triangle. The triangles below are all scalene:
When all sides of a triangle are different lengths, it is called a _____________________ triangle.
When at least two sides of a triangle are congruent, the triangle is said to be an isosceles triangle.
If a triangle has at least two congruent sides, it is called an _____________________ triangle.
Finally, when a triangle has sides that are all congruent, it is called an equilateral triangle.
[Note that by the definitions, an equilateral triangle is also an isosceles triangle.]
A triangle with all three sides congruent is called an ____________________________ triangle.
An equilateral triangle is the same as an ______________________________ triangle.
Equiangular and equilateral are words with similar parts. Let’s analyze the words:
The prefix “equi-” means “equal”
“angular” means “angled”
so “equiangular” means “having equal angles”
“lateral” means “side to side”
so “equilateral” means “having equal sides”
Reading Check:
Give an example of the side lengths of each of the following triangles:
1. equilateral ________, ________, and ________
2. isosceles ________, ________, and ________
3. scalene ________, ________, and ________
Congruent Sides and Congruent Angles
For any triangle, the number of congruent sides will always equal the number of congruent angles.
The number of congruent sides is the same as the number of congruent ____________________ in all triangles.
For example, a right isosceles triangle will have one right angle, which means the other two angles must add up to \begin{align*}90^\circ\end{align*} (since all three angles add up to \begin{align*}180^\circ\end{align*} and \begin{align*}180^\circ - 90^\circ = 90^\circ\end{align*}). If the triangle is isosceles, these two other angles must be equal to each other. Therefore, a right isosceles triangle has two \begin{align*}45^\circ\end{align*} angles (since \begin{align*}90^\circ \div 2 = 45^\circ\end{align*}) and two congruent sides.
An isosceles triangle has two \begin{align*}45^\circ\end{align*} angles and two ____________________ sides.
Also, any equilateral triangle is also equiangular: because all three interior angles will sum to \begin{align*}180^\circ\end{align*}, each one of them will measure \begin{align*}60^\circ\end{align*} (since \begin{align*}180^\circ \div 3 = 60^\circ\end{align*}).
Every equiangular triangle is also ____________________.
It is important to have these concepts solidified in your mind as you explore other topics of geometry and mathematics.
Reading Check:
1. Why is an equilateral triangle always an isosceles triangle?
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2. Why is an equilateral triangle also equiangular?
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3. True or false: Every isosceles right triangle has two \begin{align*}45^\circ\end{align*} angles and two congruent sides.
4. True or false: Every equilateral triangle has three \begin{align*}60^\circ\end{align*} angles.
Graphic Organizer for Lesson 1
Type of Triangle | Draw a picture | Lists some characteristics of this triangle in your own words |
---|---|---|
Right | ||
Acute | ||
Obtuse | ||
Equiangular | ||
Scalene | ||
Isosceles | ||
Equilateral |
Image Attributions
Concept Nodes:
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