What is a Triangle?
A triangle is a polygon made up of three sides. The point where two sides of the triangle meet are called the vertex and form an interior angle. All triangles have three sides, three vertices, and three interior angles.
Consider a triangle with three vertices as A, B and C are denoted as △ABC, as shown below.
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Basic Properties of Triangles
- A triangle has three interior angles, and the sum of all the interior angles in a triangle is 180°. This is known as the angle-sum property of a triangle.
- A triangle has three sides. The total length of any two sides of a triangle is greater than the measure of the third side. This is also known as the triangle inequality property.
- The longest side of the triangle is opposite its biggest angle, and the side opposite the smallest angle is the shortest side of the triangle.
What are the Different Types of Triangles?
A triangle is a closed 2-dimensional shape with three sides. There are different names for the types of triangles. Based on the length of the sides, there are three types of triangles: scalene, isosceles and equilateral triangles. A triangle’s name also depends on the measure of its interior angle: acute (if all angles are less than 90°), right (if one angle is 90°), and obtuse (if one angle is more than 90°)
The figure given below shows the various types of triangles and their properties.
Classification of Triangles Based on Side Lengths
There are three types of triangles based on the measure of their sides.
Scalene Triangles
All sides of a scalene triangle are of different lengths. Consequently, the three interior angles are different from each other.
Isosceles Triangles
An isosceles triangle has two sides of equal length. The angles opposite to these equal sides are equal.
Equilateral Triangles
The three sides of an equilateral triangle are of equal length. Consequently, each interior angle of equilateral triangles is equal to 60°.
Classification of Triangles Based on Interior Angles
In geometry, there are various types of angles based on measurement. The interior angles of a triangle can be of three kinds: namely acute, right, or obtuse angles, as shown below.
There are three types of triangles based on the measure of interior angles.
Acute Triangles
All interior angles in an acute triangle are acute angles, i.e., they measure less than 90°.
Right Triangles
The triangle with one interior angle of 90° or a right angle is called a right triangle. Hypotenuse, the side opposite the right angle, is the longest side in a right triangle.
Obtuse Triangles
The triangle with one interior angle greater than 90° or an obtuse angle is called an obtuse triangle.
Solved Examples
Example 1: Classify the triangle as equilateral, isosceles, or scalene.
Solution:
Examine the lengths of the sides to see if any sides are equal. All sides of the triangle are different from each other.
Therefore, the triangle is scalene.
Example 2: Classify the triangle as acute, right, and obtuse.
Solution:
Examine the measure of each interior angle of the given triangle and compare it to 90°. All the angles of the triangle are less than 90°.
Therefore, the triangle is acute.
3. In an isosceles triangle, ABC, ∠A = 100° and ∠B = ∠C. Find the measure of ∠C.
Solution:
Following the angle sum property of a triangle, we get
∠A + ∠B + ∠C = 180°
So, ∠B + ∠C = 180° – 100° = 80°
Since ∠B = ∠C, measure of ∠C = 80° ÷ 2 = 40°
Practice Problems
Types of Triangles - Definition With Examples
What type of triangle is in the image given below?
Two sides of the triangle are equal. Therefore, it is an isosceles triangle.
If the two angles of a triangle are 45° and 60°, what is the third angle? Classify the triangle.
According to the angle sum property, the third angle must be 180° - 45° - 60° or 75°. All the
interior angles are less than 90°. Therefore, the triangle is acute.
Which of the following cannot be the sides of a scalene triangle?
All sides of a scalene triangle must have different lengths.
In a right triangle, if angle A measures 30°, find B.
According to the angle sum property, ∠B = 180° -∠C -∠A = 180° - 90° - 30° = 60°
Conclusion
In this article, we learned about the types of triangles. They can be classified based on their sides and angles. To learn more math topics in a fun and interactive way, visit SplashLearn, the game-based learning platform.
Frequently Asked Questions
What is a triangle?
A triangle is a closed, 2-dimensional shape with three sides, three angles, and three vertices. A triangle is also a polygon.
Are interior angles of an equilateral triangle always 60°?
Yes. An equilateral triangle has three equal sides, and all three interior angles are equal. Applying the angle sum property, we can say that each angle of an equilateral triangle must be 180°÷ 3 or 60°
Can an isosceles triangle be obtuse?
An isosceles triangle has two equal sides, and angles opposite to these sides are equal. In an isosceles triangle, the equal angles are acute. The third angle can be acute, obtuse, or right. Therefore, an isosceles triangle can be an obtuse triangle.
How many right angles can a triangle have?
A triangle cannot have more than one right angle. Every triangle has to follow the angle sum property, which states that the sum of the three interior angles is 180°. We cannot construct a triangle with more than one right angle. If one angle is 90°, the other two must be acute.