Triangle Angle Sum

Discover why the three angles in any triangle always add up to 180 degrees.

Intermediate25 minLesson
Printable worksheet

Definition

The Triangle Angle Sum Theorem states that the sum of the three interior angles in any triangle is always 180°180°.
If a triangle has angles AA, BB, and CC:
A+B+C=180°A + B + C = 180°
This works for every triangle - no matter its shape or size:
  • Equilateral triangles: 60°+60°+60°=180°60° + 60° + 60° = 180°
  • Right triangles: 90°+45°+45°=180°90° + 45° + 45° = 180°
  • Scalene triangles: 30°+50°+100°=180°30° + 50° + 100° = 180°

Try it now

What is the sum of all interior angles in any triangle?

Worked Examples

A triangle has angles of 65°65° and 45°45°. What is the measure of the third angle?
1

Write the angle sum equation

A+B+C=180°A + B + C = 180° → Set up the equation
2

Substitute known angles

65°+45°+C=180°65° + 45° + C = 180° → Plug in the values
3

Add the known angles

110°+C=180°110° + C = 180° → 110°110°
4

Solve for the unknown

C=180°−110°C = 180° - 110° → C=70°C = 70°

Common Mistakes

Using 360°360° instead of 180°180°

Why it's wrong: 360°360° is for the sum of angles around a point or in a quadrilateral. Triangle angles always sum to 180°180°.

Correct: Remember: Triangle = 3 sides = 180°180° (half of 360°360°)

Forgetting to account for the right angle in right triangles

Why it's wrong: Students sometimes forget that the right angle is 90°90° and try to find all three angles from scratch.

Correct: In a right triangle, you already know one angle is 90°90°, so the other two must sum to 90°90°.

Getting an angle larger than 180°180° or negative

Why it's wrong: This indicates a calculation error - each angle in a triangle must be between 0°0° and 180°180°.

Correct: If your answer is negative or greater than 180°180°, check your arithmetic and equation setup.

Interactive Visual

Triangle Explorer

By sides:Scalene
By angles:Acute

Area:

0.3 sq units

Explore the relationship between angles in a triangle.

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the sum of all interior angles in any triangle?

Why It Matters

The Triangle Angle Sum Theorem is one of the most useful facts in geometry:
  • Architecture: Engineers use this theorem to design stable roof structures and bridges
  • Navigation: Pilots and sailors calculate angles to plot courses
  • Art and Design: Artists use triangular compositions for balanced layouts
  • Problem Solving: If you know two angles, you can always find the third!
This theorem is the foundation for more advanced geometry, including working with polygons and trigonometry.

Real World Applications

Roof Construction

Builders use the Triangle Angle Sum Theorem when designing roof trusses to ensure structural stability.

Example:

A roof truss has a peak angle of 40°40° and equal base angles. Each base angle must be 180°−40°2=70°\frac{180° - 40°}{2} = 70°.

1Try It Yourself

An architect designs a triangular roof section. The peak angle is 50°50° and the two base angles are equal.

What is each base angle?

Step 1: Write the mathematical expression

If the base angles are equal, find: (180°−50°)÷2(180° - 50°) \div 2

Navigation and Surveying

Surveyors measure angles to map land. Knowing the Triangle Angle Sum helps verify measurements.

Example:

A surveyor measures two angles of a triangular plot as 48°48° and 67°67°. The third angle must be 180°−48°−67°=65°180° - 48° - 67° = 65°.

2Try It Yourself

A navigator plots a triangular course. At two of its corners, the angles inside the triangle are 72°72° and 58°58°.

What is the third angle of the course?

Step 1: Write the mathematical expression

Calculate: 180°−72°−58°180° - 72° - 58°

Key Takeaways

  • 1The sum of the three interior angles in any triangle is always 180°180°
  • 2Formula: A+B+C=180°A + B + C = 180°
  • 3To find a missing angle: subtract the known angles from 180°180°
  • 4In a right triangle, the two acute angles sum to 90°90°
  • 5This theorem works for all triangles: equilateral, isosceles, scalene, acute, right, and obtuse

Frequently Asked Questions

Yes! The Triangle Angle Sum Theorem applies to every triangle - equilateral, isosceles, scalene, acute, right, and obtuse. No matter the shape, the angles always sum to 180°180°.
Yes! The Triangle Angle Sum Theorem applies to every triangle - equilateral, isosceles, scalene, acute, right, and obtuse. No matter the shape, the angles always sum to 180°180°.
Walk around the edges of a triangle. At each corner you turn by the exterior angle, and after three turns you face the way you started: one full turn, 360°360°. At each corner the interior and exterior angles form a straight angle, 180°180°, so the three corners give 3×180°=540°3 \times 180° = 540°. Take away the 360°360° of turning and 540°−360°=180°540° - 360° = 180° is left for the three interior angles. You can also tear off the three corners of a paper triangle and put them side by side: together they form a straight line.
No! If two angles were 90°90° each, their sum would already be 180°180°, leaving 0°0° for the third angle. Since every angle must be greater than 0°0°, a triangle can have at most one right angle.

Glossary

Interior angle
An angle inside a polygon, formed by two adjacent sides
Triangle Angle Sum Theorem
The mathematical rule stating that the three interior angles of any triangle add up to 180°180°
Acute angle
An angle measuring less than 90°90°
Right angle
An angle measuring exactly 90°90°
Obtuse angle
An angle measuring more than 90°90° but less than 180°180°

Formula Card

Triangle Angle Sum

A+B+C=180°A + B + C = 180°

The sum of all three interior angles in any triangle equals 180 degrees

Finding Missing Angle

C=180°−A−BC = 180° - A - B

Subtract the known angles from 180° to find the missing angle

Today's challengeOne problem, three tries

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