Angle Sum Property of Triangle
Every triangle, no matter its shape or size, has a special property: the sum of its three interior angles is always 180°. This is called the Angle Sum Property of a Triangle.
This property is one of the most useful facts in geometry. It helps us find unknown angles in triangles and verify whether a set of angles can form a triangle.
Whether the triangle is tiny (drawn on a notebook) or enormous (a plot of land), whether it is equilateral, isosceles, or scalene, the three angles always add up to exactly 180°. This is a fundamental rule that never changes.
What is Angle Sum Property of Triangle - Class 5 Maths (Geometry)?
The Angle Sum Property of a Triangle states that the sum of all three interior angles of a triangle is always 180°.
If the three angles of a triangle are ∠A, ∠B, and ∠C, then:
∠A + ∠B + ∠C = 180°
This property holds true for all types of triangles — acute, right, obtuse, equilateral, isosceles, and scalene.
Types and Properties
Verifying the Angle Sum Property:
Method 1 — Measurement:
- Draw any triangle on paper.
- Measure each angle with a protractor.
- Add the three measurements. The sum will be 180° (allowing for small measurement errors).
Method 2 — Paper tearing:
- Draw a triangle and cut it out.
- Tear off all three corners.
- Place the three corners together with their vertices at one point.
- The three angles will form a straight line (180°).
Using the property to find unknown angles:
- If two angles are known, the third angle = 180° − (sum of the two known angles).
- In a right triangle, one angle is 90°. So the other two angles add up to 90°.
- In an equilateral triangle, all angles are equal: 180° ÷ 3 = 60° each.
Solved Examples
Example 1: Example 1: Finding the Third Angle
Problem: Two angles of a triangle are 50° and 70°. Find the third angle.
Solution:
Step 1: Sum of all three angles = 180°
Step 2: Third angle = 180° − (50° + 70°)
Step 3: Third angle = 180° − 120° = 60°
Answer: The third angle is 60°.
Example 2: Example 2: Right Triangle
Problem: In a right triangle, one of the other angles is 35°. Find the remaining angle.
Solution:
Step 1: One angle = 90° (right angle).
Step 2: Second angle = 35°.
Step 3: Third angle = 180° − (90° + 35°) = 180° − 125° = 55°.
Answer: The remaining angle is 55°.
Example 3: Example 3: Equilateral Triangle
Problem: All three angles of an equilateral triangle are equal. Find each angle.
Solution:
Step 1: Let each angle = x°.
Step 2: x + x + x = 180°
Step 3: 3x = 180°
Step 4: x = 180° ÷ 3 = 60°
Answer: Each angle of an equilateral triangle is 60°.
Example 4: Example 4: Isosceles Triangle
Problem: An isosceles triangle has a top angle of 40°. The two base angles are equal. Find each base angle.
Solution:
Step 1: Let each base angle = x°.
Step 2: 40° + x + x = 180°
Step 3: 40° + 2x = 180°
Step 4: 2x = 140°
Step 5: x = 70°
Answer: Each base angle is 70°.
Example 5: Example 5: Can These Angles Form a Triangle?
Problem: Can angles 60°, 80°, and 50° form a triangle?
Solution:
Step 1: Add the three angles: 60° + 80° + 50° = 190°.
Step 2: For a triangle, the sum must be exactly 180°.
Step 3: 190° ≠ 180°.
Answer: No, these angles cannot form a triangle.
Example 6: Example 6: Checking Validity
Problem: Can angles 45°, 55°, and 80° form a triangle?
Solution:
Step 1: Sum = 45° + 55° + 80° = 180°.
Step 2: The sum is exactly 180°.
Answer: Yes, these angles can form a triangle.
Example 7: Example 7: Finding Angles Using Variables
Problem: The angles of a triangle are x°, 2x°, and 3x°. Find all three angles.
Solution:
Step 1: x + 2x + 3x = 180°
Step 2: 6x = 180°
Step 3: x = 30°
Step 4: The three angles are: 30°, 60°, and 90°.
Answer: The angles are 30°, 60°, and 90°. This is a right triangle.
Example 8: Example 8: Word Problem
Problem: Ria draws a triangle. The first angle is twice the second angle. The third angle is 20° more than the second angle. Find all three angles.
Solution:
Step 1: Let the second angle = x°.
Step 2: First angle = 2x°. Third angle = (x + 20)°.
Step 3: 2x + x + (x + 20) = 180
Step 4: 4x + 20 = 180
Step 5: 4x = 160
Step 6: x = 40
Step 7: Angles: 80°, 40°, 60°.
Answer: The three angles are 80°, 40°, and 60°.
Example 9: Example 9: Obtuse Triangle
Problem: A triangle has angles 25°, 30°, and x°. Find x and classify the triangle.
Solution:
Step 1: x = 180° − (25° + 30°) = 180° − 55° = 125°.
Step 2: Since 125° > 90°, the triangle has one obtuse angle.
Answer: x = 125°. The triangle is an obtuse triangle.
Example 10: Example 10: Can a Triangle Have Two Right Angles?
Problem: Dev says a triangle can have two right angles. Is he correct?
Solution:
Step 1: If two angles are 90° each, their sum = 90° + 90° = 180°.
Step 2: The third angle = 180° − 180° = 0°.
Step 3: An angle of 0° is not possible in a triangle.
Answer: No, Dev is incorrect. A triangle cannot have two right angles.
Real-World Applications
Where is the angle sum property used?
- Construction: Builders use this property to verify that triangular roof trusses have correct angles.
- Navigation: Sailors and surveyors use triangulation — measuring two angles and using the property to find the third — to determine distances.
- Art and design: Designers ensure triangular patterns have angles that add up correctly for symmetry.
- Problem solving: When only two angles are given in a triangle, the third can always be found using this property.
- Checking work: After measuring all three angles with a protractor, you can verify accuracy by checking if the sum is 180°.
Key Points to Remember
- The sum of the three interior angles of any triangle is always 180°.
- If two angles are known, the third = 180° minus the sum of the two known angles.
- An equilateral triangle has three equal angles of 60° each.
- A right triangle has one 90° angle; the other two add up to 90°.
- A triangle cannot have more than one right angle or more than one obtuse angle.
- A triangle cannot have all three angles as obtuse (each > 90°) because the sum would exceed 180°.
- To verify: tear off three corners of a paper triangle and place them together — they form a straight line.
- Three given angles form a valid triangle only if their sum is exactly 180° and each angle is greater than 0°.
Practice Problems
- The angles of a triangle are 65°, 75°, and x°. Find x.
- Can 90°, 60°, and 40° be the angles of a triangle? Why or why not?
- In a right triangle, one acute angle is 48°. Find the other acute angle.
- An isosceles triangle has a vertex angle of 100°. Find each base angle.
- The three angles of a triangle are 2x°, 3x°, and 4x°. Find all three angles.
- Neha says a triangle can have all three obtuse angles. Is she right? Explain.
- A triangle has angles in the ratio 1 : 2 : 3. Find each angle and classify the triangle.
- Two angles of a triangle are 55° and 55°. What type of triangle is this?
Frequently Asked Questions
Q1. What is the angle sum property of a triangle?
It states that the sum of the three interior angles of any triangle is always 180°. This is true for every type of triangle — acute, right, obtuse, equilateral, isosceles, or scalene.
Q2. How do you find the third angle of a triangle?
Subtract the sum of the two known angles from 180°. For example, if two angles are 50° and 60°, the third angle = 180° − 50° − 60° = 70°.
Q3. Can a triangle have two obtuse angles?
No. Two obtuse angles would each be more than 90°, so their sum alone would exceed 180°. There would be no room for a third positive angle.
Q4. Can a triangle have all acute angles?
Yes. A triangle with all three angles less than 90° is called an acute triangle. For example, 60°, 70°, and 50°.
Q5. What are the angles of an equilateral triangle?
All three angles are 60°. Since they are equal and must add to 180°, each angle = 180° ÷ 3 = 60°.
Q6. Why can't a triangle have an angle of 0 degrees or 180 degrees?
An angle of 0° would mean two sides overlap completely (no triangle formed). An angle of 180° would make the three vertices lie in a straight line (again, no triangle).
Q7. How can I verify the angle sum property by paper folding?
Cut out a triangle, tear off all three corners, and place them together at a single point. The three angles will fit together to form a straight line (180°).
Q8. Does this property work for all triangles?
Yes. It applies to every triangle — large or small, regular or irregular, right-angled or not. The sum of interior angles is always 180°.
Q9. What if the sum of three given angles is not 180 degrees?
Then those three angles cannot be the angles of a triangle. For example, 70°, 80°, and 40° add up to 190°, so they cannot form a triangle.
Q10. Is this topic in the NCERT Class 5 syllabus?
Yes. The angle sum property is introduced in the Geometry chapter of the NCERT/CBSE Class 5 Maths curriculum and is built upon in higher classes.
Related Topics
- Angles (Grade 5)
- Types of Triangles (Grade 5)
- Lines, Line Segments and Rays
- Parallel and Perpendicular Lines
- Quadrilaterals (Grade 5)
- Circles (Grade 5)
- Symmetry (Grade 5)
- Reflection and Rotation
- Nets of 3D Shapes (Grade 5)
- Views of 3D Shapes (Grade 5)
- Complementary and Supplementary Angles
- Constructing Triangles
