Overview

Most geometry problems start with angle relationships. The triangle angle sum and exterior angle rule are the first tools to use, and they combine naturally with linear pairs and straight lines.

Key Ideas

• In any triangle, $\angle A + \angle B + \angle C = 180^\circ$.
• An exterior angle equals the sum of the two remote interior angles.
• A straight line is $180^\circ$, and a full turn is $360^\circ$.

Core Skills

Angle Chasing

Set up a consistent direction around the triangle, write the angle sum, and solve the resulting equation. Use linear pairs at extensions to relate exterior and interior angles.

Exterior Angle Rule

If one side is extended, the exterior angle equals the sum of the two remote interior angles. This often turns a one-step chase into a direct equation.

Linear Pairs

If two angles form a straight line, their measures add to $180^\circ$. This is the bridge between a triangle and any surrounding diagram.

Worked Example

If a triangle has angles $45^\circ$ and $70^\circ$, find the third angle.

The third angle is $180^\circ - 45^\circ - 70^\circ = 65^\circ$.

More Examples

Example 1: Exterior Angle

In triangle $ABC$, side $BC$ is extended to $D$. If $\angle A = 40^\circ$ and $\angle B = 55^\circ$, find exterior angle $\angle ACD$.

$\angle ACD = \angle A + \angle B = 95^\circ$.

Example 2: Linear Pair

If $\angle A = 2x$ and $\angle B = 3x$ in triangle $ABC$, and the exterior angle at $C$ is $150^\circ$, find $x$.

Exterior angle equals $\angle A + \angle B$, so $2x + 3x = 150^\circ$, giving $x = 30^\circ$.

Strategy Checklist

• Mark which angles are interior, exterior, or linear pairs.
• Use triangle angle sum early to reduce variables.
• Prefer the exterior angle rule when an extension is drawn.
• Keep a running total to avoid sign mistakes.

Common Pitfalls

• Forgetting to subtract both known angles.
• Mixing interior and exterior angles.
• Adding all three angles to an exterior angle instead of just the two remote.

Practice Problems