What are angles?
An angle is formed in geometry when two rays are joined at one of their endpoints. These rays are referred to as the sides of an angle, or the arms of an angle.
Arms of an angle:
The arms of an angle are the two rays that meet at a common point to form the angle. Analyze the diagram above, which shows that OP and OQ are the arms of the angle POQ.
The vertex of an angle:
In an angle, two rays share a common endpoint called the vertex. Look at the figure above, and the vertex here is O, showing us the point where the two arms meet.
What is a triangle?
A triangle is a three-sided polygon with three interior angles. It is one of the most basic shapes in geometry, consisting of three vertices joined together, and is denoted by the symbol “∆” while solving problems based on this polygon. The sides and angles of a triangle are used to classify it into different types.
Let’s look at some real-world examples of a triangle:
The most common examples of triangles in our daily lives are sandwiches or pizza slices, pyramids, and the traffic signs that we see everyday.
Components of a triangle:
A triangle is made up of several parts. It has three angles, three sides, and three vertices. Examine the triangle ABC below, which depicts a triangle’s sides (side 1, side 2, and side 3), vertices (A, B, and C), and interior angles (∠A, ∠B, and ∠C).
Types of triangles based on angles
Triangles are classified into the following types based on their angles:
Acute triangle:
A triangle whose three interior angles are acute, is known as an acute triangle. In other words, all the interior angles of an acute-angled triangle are less than 90 degrees. The acute triangle is represented in the diagram below, where we can see that all the three interior angles are less than 90 degrees.
Obtuse triangle:
Obtuse triangles are the ones in which one of the three interior angles is greater than 90 degrees. In other words, an obtuse-angled triangle is formed when one of the angles in a triangle is an obtuse angle. An obtuse triangle is represented in the diagram below, where we can see that one of the interior angles is greater than 90 degrees.
Right triangle:
A right triangle is made up of three angles, one of which is 90 degrees. A right triangle is shown in the diagram below.
Equiangular triangle:
An equiangular triangle is one where all the three interior angles measure 60 degrees. Also, for such a triangle, all sides are of the same length.
Solved Classification of Triangles Examples
Example 1:
Classify the triangle by the angles shown.
Solution:
The triangle has one right angle.
So, it is a right triangle.
Example 2:
Classify the triangle by its angles and its sides.
Solution:
All the angles of the triangle are acute and
all its sides are of the same length.
So, it is an equiangular triangle.
Example 3:
Classify the triangle based on its angles. Determine the measure of each angle to confirm your answer.
Solution:
There is no right or obtuse angle in this triangle.
So, it is an acute triangle.
To verify the answer, let’s measure each angle using a protractor.
The measure of \( \angle ~ABC\) is 60 degrees.
The measure of \( \angle ~BCA\) is 60 degrees.
The measure of \( \angle ~CAB\) is 60 degrees.
Since each angle measures \( 60^\circ\), which is less than \( 90^\circ\), it can be concluded that the triangle is an acute triangle.
Example 4:
Observe the face of the sandwich given in the picture and identify the type of triangle it forms based on the angles that are observed in the sandwich. Verify your answer.
Solution:
We can see that the sandwich face is in the form of a triangle that has a right angle.
So, it is a right triangle.
To verify the answer, let’s measure each angle using a protractor.
The measure of \( \angle ~ABC\) is 90 degrees.
The measure of \( \angle ~BCA\) is 45 degrees.
The measure of \( \angle ~CAB\) is 45 degrees.
Since the measure of \( \angle ~ABC\) is \( 90^\circ\), we can conclude that the triangle is a right triangle. Hence, the answer is verified.
Example 5:
Classify the triangles based on their angles.
Image 1: There are three acute angles in this triangle. As a result, the triangle is acute.
Image 2: There is one obtuse angle in the triangle. As a result, the triangle is obtuse.
Image 3: The triangle has one right angle.
So, it is a right triangle.
In a triangle, there are three sides, three vertices, and three angles.
An equiangular triangle is a triangle with three equal interior angles. Each of the interior angles of an equiangular triangle measures 60 degrees.
A triangle is classified as follows, based on its angles:
1) Acute Triangle
2) Right Triangle
3) Obtuse Triangle
4) Equiangular triangle
Triangles are classified as below:
Acute triangle:
Each of the triangle’s angles is less than 90 degrees in an acute triangle.
Obtuse triangle:
One of the triangle’s angles is greater than 90 degrees, forming an obtuse triangle.
Right triangle:
A right triangle has one 90-degree angle and two acute angles.