Triangle Angle Sum Theorem

The sum of angles in a triangle

Written and reviewed by the Study Mondo Education TeamLast updated February 16, 2026

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Triangle Angle Sum Theorem

The Fundamental Theorem

The sum of the interior angles of any triangle is $180°$ .

$\angle A + \angle B + \angle C = 180°$

Triangle Classification by Angles

Acute Triangle: All three angles are acute (< 90°)

Right Triangle: One angle is exactly 90°

Obtuse Triangle: One angle is obtuse (> 90°)

Equiangular Triangle: All three angles equal 60°

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two remote interior angles.

Example: If an exterior angle measures $130°$ , and one remote interior angle is $70°$ , the other remote interior angle is: $130° - 70° = 60°$

Corollary

The measure of each angle of an equilateral triangle is $60°$ .

Applications

This theorem is used to:

Find missing angle measures
Prove triangle congruence
Solve geometric proofs

📚 Practice Problems

1Problem 1easy

❓ Question:

In a triangle, two angles measure 45° and 70°. Find the measure of the third angle.

💡 Show Solution

Step 1: Recall Triangle Angle Sum Theorem: The sum of the three interior angles of any triangle is 180°

Step 2: Set up the equation: 45° + 70° + x = 180°

Step 3: Simplify: 115° + x = 180°

Step 4: Solve for x: x = 180° - 115° x = 65°

Step 5: Verify: 45° + 70° + 65° = 180° ✓

Answer: The third angle measures 65°

2Problem 2easy

❓ Question:

Two angles of a triangle measure $45°$ and $65°$ . Find the third angle.

💡 Show Solution

Use the Triangle Angle Sum Theorem:

$45° + 65° + x = 180°$

$110° + x = 180°$

$x = 70°$

Answer: $70°$

3Problem 3easy

❓ Question:

Two angles of a triangle measure $45°$ and $65°$ . Find the third angle.

💡 Show Solution

Use the Triangle Angle Sum Theorem:

$45° + 65° + x = 180°$

$110° + x = 180°$

$x = 70°$

Answer: $70°$

4Problem 4easy

❓ Question:

In triangle ABC, angle A = 3x, angle B = 2x, and angle C = x. Find the value of x and the measure of each angle.

💡 Show Solution

Step 1: Use Triangle Angle Sum: The sum of angles = 180° A + B + C = 180°

Step 2: Substitute the expressions: 3x + 2x + x = 180°

Step 3: Combine like terms: 6x = 180°

Step 4: Solve for x: x = 180°/6 x = 30°

Step 5: Find each angle: Angle A = 3x = 3(30°) = 90° Angle B = 2x = 2(30°) = 60° Angle C = x = 30°

Step 6: Verify: 90° + 60° + 30° = 180° ✓

Answer: x = 30°, angles are 90°, 60°, and 30°

5Problem 5medium

❓ Question:

An exterior angle of a triangle measures 125°. One of the non-adjacent interior angles is 55°. Find the other non-adjacent interior angle.

💡 Show Solution

Step 1: Recall Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles

Step 2: Set up the equation: Exterior angle = Sum of two remote interior angles 125° = 55° + x

Step 3: Solve for x: x = 125° - 55° x = 70°

Step 4: Verify using Triangle Angle Sum: Adjacent interior angle = 180° - 125° = 55° All three interior angles: 55° + 55° + 70° = 180° ✓

Answer: The other non-adjacent interior angle is 70°

6Problem 6medium

❓ Question:

In a triangle, the angles are in the ratio $2:3:4$ . Find all three angle measures.

💡 Show Solution

Let the angles be $2x$ , $3x$ , and $4x$ .

$2x + 3x + 4x = 180$

$9x = 180$

$x = 20$

The three angles are:

$2(20) = 40°$
$3(20) = 60°$
$4(20) = 80°$

Check: $40 + 60 + 80 = 180$ ✓

Answer: $40°$ , $60°$ , $80°$

7Problem 7medium

❓ Question:

In a triangle, the angles are in the ratio $2:3:4$ . Find all three angle measures.

💡 Show Solution

Let the angles be $2x$ , $3x$ , and $4x$ .

$2x + 3x + 4x = 180$

$9x = 180$

$x = 20$

The three angles are:

$2(20) = 40°$
$3(20) = 60°$
$4(20) = 80°$

Check: $40 + 60 + 80 = 180$ ✓

Answer: $40°$ , $60°$ , $80°$

8Problem 8hard

❓ Question:

An exterior angle of a triangle measures $125°$ . One of the remote interior angles measures $55°$ . Find the other two angles of the triangle.

💡 Show Solution

Step 1: Use Exterior Angle Theorem

The exterior angle equals the sum of remote interior angles: $125° = 55° + x$ $x = 70°$

So one remote interior angle is $70°$ .

Step 2: Find the third angle (adjacent to exterior)

The exterior angle and its adjacent interior angle are supplementary: $125° + y = 180°$ $y = 55°$

The three angles are: $55°$ , $70°$ , $55°$

Check: $55 + 70 + 55 = 180$ ✓

Answer: The three angles are $55°$ , $70°$ , and $55°$

9Problem 9hard

❓ Question:

An exterior angle of a triangle measures $125°$ . One of the remote interior angles measures $55°$ . Find the other two angles of the triangle.

💡 Show Solution

Step 1: Use Exterior Angle Theorem

The exterior angle equals the sum of remote interior angles: $125° = 55° + x$ $x = 70°$

So one remote interior angle is $70°$ .

Step 2: Find the third angle (adjacent to exterior)

The exterior angle and its adjacent interior angle are supplementary: $125° + y = 180°$ $y = 55°$

The three angles are: $55°$ , $70°$ , $55°$

Check: $55 + 70 + 55 = 180$ ✓

Answer: The three angles are $55°$ , $70°$ , and $55°$

10Problem 10medium

❓ Question:

In triangle XYZ, angle X = (2a + 10)°, angle Y = (3a - 5)°, and angle Z = (a + 25)°. Find the value of a and all three angle measures.

💡 Show Solution

Step 1: Use Triangle Angle Sum Theorem: X + Y + Z = 180°

Step 2: Substitute the expressions: (2a + 10) + (3a - 5) + (a + 25) = 180

Step 3: Combine like terms: 2a + 3a + a + 10 - 5 + 25 = 180 6a + 30 = 180

Step 4: Solve for a: 6a = 180 - 30 6a = 150 a = 25

Step 5: Find each angle: Angle X = 2a + 10 = 2(25) + 10 = 50 + 10 = 60° Angle Y = 3a - 5 = 3(25) - 5 = 75 - 5 = 70° Angle Z = a + 25 = 25 + 25 = 50°

Step 6: Verify: 60° + 70° + 50° = 180° ✓

Answer: a = 25, angles are 60°, 70°, and 50°

11Problem 11hard

❓ Question:

In an isosceles triangle, the vertex angle is twice the measure of each base angle. Find all three angles.

💡 Show Solution

Step 1: Define variables: Let x = measure of each base angle In an isosceles triangle, the two base angles are equal Vertex angle = 2x (given as twice a base angle)

Step 2: Apply Triangle Angle Sum: Base angle + Base angle + Vertex angle = 180° x + x + 2x = 180°

Step 3: Solve: 4x = 180° x = 45°

Step 4: Find all angles: Each base angle = x = 45° Vertex angle = 2x = 2(45°) = 90°

Step 5: Verify: 45° + 45° + 90° = 180° ✓ This is a 45-45-90 triangle (an isosceles right triangle)

Step 6: Check the relationship: Vertex angle = 90° = 2(45°) ✓ The vertex angle is indeed twice each base angle

Answer: The base angles are 45° each, and the vertex angle is 90°

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❓ Frequently Asked Questions

What is Triangle Angle Sum Theorem?▾

The sum of angles in a triangle

How can I study Triangle Angle Sum Theorem effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 11 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

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What course covers Triangle Angle Sum Theorem?▾

Triangle Angle Sum Theorem is part of the Geometry course on Study Mondo, specifically in the Triangles section. You can explore the full course for more related topics and practice resources.

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Yes, this page includes 11 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Triangle Angle Sum Theorem

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Triangle Angle Sum Theorem

The Fundamental Theorem

Triangle Classification by Angles

Exterior Angle Theorem

Corollary

Applications

📚 Practice Problems

1Problem 1easy

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2Problem 2easy

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3Problem 3easy

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4Problem 4easy

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5Problem 5medium

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6Problem 6medium

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7Problem 7medium

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8Problem 8hard

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9Problem 9hard

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10Problem 10medium

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11Problem 11hard

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