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:

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: $70°$

2Problem 2medium

❓ 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: $40°$ , $60°$ , $80°$

3Problem 3hard

❓ 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 $55°$

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