Types of Triangles by Angles with Definitions and Properties
Understanding the Classification of Triangles Based on Angles is crucial for mastering geometry. Triangles can be found everywhere—from art to architecture—and knowing how to classify them helps students solve problems in exams like CBSE Class 9/10, JEE, and other competitive tests. This lesson explains the types of triangles by their angle measures, step-by-step, in an easy and practical way.
What is a Triangle? Basics & Angle Types
A triangle is a closed figure with three straight sides and three angles. The sum of its interior angles is always 180°. Understanding angle types helps in triangle classification:
- Acute Angle: Less than 90°
- Right Angle: Exactly 90°
- Obtuse Angle: More than 90° but less than 180°
Each triangle’s shape is defined by its angles and side lengths. To know more about triangle shape rules, visit Triangle and its Properties.
Classification of Triangles Based on Angles
Triangles are classified by their angles into three main types:
- Acute-Angled Triangle: All three angles are less than 90°
- Right-Angled Triangle: One angle is exactly 90°, the other two are acute
- Obtuse-Angled Triangle: One angle is more than 90° (obtuse), others are acute
| Triangle Type | Angle Measures | Mini-Diagram |
|---|---|---|
| Acute-Angled | All angles < 90° | |
| Right-Angled | One angle = 90° | |
| Obtuse-Angled | One angle > 90° |
Properties of Triangle Types by Angle
| Type | Defining Property | Example Angles |
|---|---|---|
| Acute-Angled | All angles < 90° | 60°, 70°, 50° |
| Right-Angled | One angle = 90° | 90°, 40°, 50° |
| Obtuse-Angled | One angle > 90° | 120°, 30°, 30° |
Worked Examples
Let’s see how to identify triangles by angle type, step-by-step:
Example 1
Classify the triangle with angles 80°, 60°, 40°.
- Check if all angles are less than 90° ✔️
- Answer: Acute-angled triangle
Example 2
Given a triangle with angles 100°, 40°, and 40°. What type is it?
- One angle > 90° (100°) ✔️
- Answer: Obtuse-angled triangle
Example 3
A triangle has angles 90°, 35°, and 55°. Classify it.
- One angle = 90° ✔️
- Answer: Right-angled triangle
Practice Problems
- A triangle’s angles are 88°, 89°, 3°. Classify this triangle.
- True or False: A triangle can have more than one right angle.
- Find all possible types for a triangle with angles 95°, 50°, and 35°.
- Draw and label an acute-angled triangle and mark all its angles.
- If a triangle has two equal angles of 45°, what type of triangle is it by angles?
For more practice, try the Classification of Triangles Worksheet (PDF).
Common Mistakes to Avoid
- Assuming a triangle can have more than one right or obtuse angle (impossible as angle sum exceeds 180°)
- Getting confused between angle-based and side-based classification
- Forgetting that all three angles in an equilateral triangle are ALWAYS acute (60°)
- Mixing up right-angled triangles with isosceles triangles: An isosceles triangle can also be right-angled, but not always.
Need more triangle tips? Explore Properties of Triangle on Vedantu.
Real-World Applications
Classifying triangles by angles is useful in carpentry, engineering, and construction. For example, right-angled triangles are crucial when designing ramps or stairs (see Area of a Triangle), while obtuse-angled triangles often appear in roof and bridge designs. Acute triangles are common in decorative patterns and trusses.
Page Summary
In this lesson, we learned how to classify triangles based on angles: acute-angled, right-angled, and obtuse-angled. We discussed their properties, provided clear diagrams, worked examples, and real-life uses. Knowing these types will help you answer geometry questions quickly for exams and understand shapes in the world around you. For deeper knowledge, explore more topics on Vedantu.
- Isosceles Triangle
- Scalene Triangle
- Acute Angle Triangle
- Obtuse Angled Triangle
- Congruence of Triangles
- Triangle and its Properties
FAQs on Classification of Triangles Based on Angles
1. How do you classify triangles based on their angles?
Triangles are classified into three types based on their angles: acute-angled, right-angled, and obtuse-angled triangles. The classification depends on the measure of each interior angle.
2. What are the three types of triangles by angles?
The three types of triangles based on their angles are: acute-angled triangles (all angles less than 90°), right-angled triangles (one angle equal to 90°), and obtuse-angled triangles (one angle greater than 90°).
3. What are the names of the triangles classified by angles?
Triangles classified by angles are named based on the largest angle present: acute-angled, right-angled, or obtuse-angled triangles. Understanding this classification is crucial for Geometry and exam preparation.
4. What is the difference between angle-based and side-based classification of triangles?
Angle-based classification categorizes triangles by the measures of their interior angles (acute, right, obtuse). Side-based classification categorizes triangles by the lengths of their sides (equilateral, isosceles, scalene). Both methods are important for a complete understanding of triangles.
5. Can a triangle have more than one right or obtuse angle?
No. The sum of angles in any triangle is always 180°. If a triangle had two or more right angles (90° each), or two or more obtuse angles (greater than 90° each), the sum would exceed 180°, which is impossible. Therefore, a triangle can have at most one right angle or one obtuse angle.
6. What is an example of a real-world obtuse triangle?
Many real-world objects form obtuse triangles. For example, a slanted roof of a house often forms an obtuse triangle. Other examples include certain types of ramps or wedges.
7. Can a triangle be both isosceles and obtuse?
Yes, a triangle can be both isosceles (two sides of equal length) and obtuse (one angle greater than 90°). Imagine an isosceles triangle where the two equal sides are shorter than the base; the angle opposite the base would be obtuse.
8. How are triangle angles linked to triangle area calculation?
Knowing the angles and lengths of at least two sides of a triangle allows you to calculate its area using trigonometric formulas, specifically involving the sine function. For example, the area can be calculated using Area = (1/2)ab sin C, where a and b are the lengths of two sides and C is the angle between them.
9. Are all equilateral triangles acute-angled?
Yes. An equilateral triangle has all three sides equal in length, and each interior angle measures 60°. Since 60° is less than 90°, all equilateral triangles are also acute-angled triangles.
10. Does the classification by angles affect triangle congruence?
Yes, the classification of triangles by angles influences congruence rules. While not a direct criterion for congruence, the type of triangle (acute, right, obtuse) can help determine which congruence theorems (SSS, SAS, ASA, RHS) are applicable to a particular problem. For example, the RHS (Right-angle, Hypotenuse, Side) theorem only applies to right-angled triangles.
11. What is the sum of the angles in any triangle?
The sum of the interior angles in any triangle, regardless of its classification by angles or sides, always equals 180°. This is a fundamental property of triangles.
