Triangle Angle Sum Theorem

The sum of angles in a triangle

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:

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°

4Problem 4medium

❓ 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: $40°$ , $60°$ , $80°$

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 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°

7Problem 7hard

❓ 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 $55°$

8Problem 8hard

❓ 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°

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics