Angles in a Triangle — Properties, Types, and Sum Rule

Grade: 7–8 | Topic: Geometry

What You Will Learn

By the end of this page, you will know that the three interior angles of any triangle sum to 180°, be able to find any missing angle given the other two, classify triangles by their angles, and use the exterior angle theorem.

Theory

The Triangle Angle Sum Property

The three interior angles of any triangle always add up to exactly 180°.

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

This means: if you know any two angles, you can always find the third by subtracting from 180°.

Why 180°? If you draw a straight line through a triangle's vertex parallel to the opposite side, the three angles at that vertex form a straight line — which is exactly 180°.

Types of Triangles by Angles

Type	Angle property
Acute triangle	All three angles $< 90°$
Right triangle	Exactly one angle $= 90°$
Obtuse triangle	Exactly one angle $> 90°$
Equilateral triangle	All angles $= 60°$

The Exterior Angle Theorem

An exterior angle is formed by extending one side of a triangle beyond the vertex. It equals the sum of the two non-adjacent interior angles (also called remote interior angles).

$\text{Exterior angle} = \angle A + \angle B$

where $\angle A$ and $\angle B$ are the two interior angles not next to the exterior angle.

Special Triangle Properties

Isosceles triangle: two equal sides → the two base angles (angles opposite the equal sides) are also equal.

Equilateral triangle: all three sides equal → all three angles are $60°$.

Right triangle: one angle is $90°$ → the other two angles are complementary (they add up to $90°$).

Worked Examples

Example 1: Finding a Missing Angle

Problem: A triangle has angles of $47°$ and $68°$. Find the third angle.

Step 1: Use the angle sum property. $\angle C = 180° - 47° - 68°$

Step 2: Calculate. $\angle C = 180° - 115° = 65°$

Answer: The third angle is $\mathbf{65°}$.

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Example 2: Isosceles Triangle

Problem: An isosceles triangle has a vertex angle of $40°$. Find the two base angles.

Step 1: The two base angles are equal. Let each base angle $= x$.

Step 2: Apply the angle sum. $40° + x + x = 180°$ $2x = 140°$ $x = 70°$

Answer: Each base angle is $\mathbf{70°}$.

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Example 3: Using the Exterior Angle Theorem

Problem: Two interior angles of a triangle are $52°$ and $74°$. Find the exterior angle at the third vertex.

Step 1: The exterior angle equals the sum of the two non-adjacent interior angles. $\text{Exterior angle} = 52° + 74° = 126°$

Verify: The third interior angle $= 180° - 52° - 74° = 54°$. The exterior angle $= 180° - 54° = 126°$. Consistent.

Answer: The exterior angle is $\mathbf{126°}$.

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Example 4: Algebraic Angle Problem

Problem: The angles of a triangle are $x$, $2x$, and $3x$. Find each angle and classify the triangle.

Step 1: Set up the equation. $x + 2x + 3x = 180°$ $6x = 180°$ $x = 30°$

Step 2: Find each angle. $x = 30°, \quad 2x = 60°, \quad 3x = 90°$

Step 3: Classify. One angle is exactly $90°$ — this is a right triangle.

Answer: Angles are $30°$, $60°$, $90°$ — a right triangle.

Common Mistakes

Mistake 1: Setting Angles Equal to 90° Instead of 180°

❌ $\angle A + \angle B + \angle C = 90°$

✅ The angles of a triangle sum to $180°$, not $90°$. (Two complementary angles sum to $90°$, but that is only for a pair of angles, not a triangle.)

Mistake 2: Using the Exterior Angle as an Interior Angle

❌ An exterior angle of $110°$ is used as one of the triangle's interior angles.

✅ The exterior angle is outside the triangle. The interior angle at that vertex is $180° - 110° = 70°$.

Mistake 3: Assuming All Triangles Can Have Two Obtuse Angles

❌ A triangle with angles $100°$, $95°$, and an unknown third angle.

✅ $100° + 95° = 195° > 180°$, which is impossible. A triangle can have at most one obtuse angle.

Practice Problems

Try these on your own before checking the answers:

1. A triangle has angles $33°$ and $91°$. Find the third angle and classify the triangle.
2. An equilateral triangle has one angle labelled $(2x + 10)°$. Find $x$.
3. Two angles of a triangle are $x$ and $3x$. The third angle is $60°$. Find $x$.
4. An exterior angle of a triangle measures $135°$. One of the non-adjacent interior angles is $80°$. Find the other non-adjacent interior angle.
5. Can a triangle have angles of $90°$, $90°$, and $0°$? Explain.

Click to see answers

1. $180° - 33° - 91° = 56°$. Triangle has an angle $> 90°$ (the $91°$ angle), so it is an obtuse triangle.
2. Each angle of an equilateral triangle $= 60°$. So $2x + 10 = 60 \implies 2x = 50 \implies x = 25$.
3. $x + 3x + 60° = 180° \implies 4x = 120° \implies x = 30°$. Angles: $30°$, $90°$, $60°$.
4. Exterior angle $=$ sum of non-adjacent angles: $135° = 80° + \text{other} \implies \text{other} = 55°$.
5. No. $90° + 90° + 0° = 180°$ mathematically, but a triangle cannot have a $0°$ angle (a side would collapse to a point). Triangles must have three positive interior angles.

Summary

• The three interior angles of any triangle sum to $180°$.
• To find a missing angle: subtract the known angles from $180°$.
• Exterior angle theorem: exterior angle = sum of the two non-adjacent interior angles.
• Triangle types: acute (all $< 90°$), right (one $= 90°$), obtuse (one $> 90°$), equilateral (all $= 60°$).

Related Topics

• Area and Perimeter — Formulas, Examples, and Practice
• Area of a Triangle — Formula and Examples
• Pythagorean Theorem — Formula, Proof, and Examples

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