Finding Exterior Angles from Interior Angles

A triangle has interior angles of $50°$, $60°$, and $70°$. Find all three exterior angles.

Use the supplementary relationship: Exterior angle $= 180° - $ Interior angle = Formula identified

Find exterior angle at $50°$ vertex: $180° - 50° = 130°$ = $130°$

Find exterior angle at $60°$ vertex: $180° - 60° = 120°$ = $120°$

Find exterior angle at $70°$ vertex: $180° - 70° = 110°$ = $110°$

Verify the sum: $130° + 120° + 110° = 360°$ = Confirmed!

Answer: The exterior angles are $130°$, $120°$, and $110°$, which sum to $360°$.

Exterior Angle of a Regular Hexagon

Find the measure of each exterior angle of a regular hexagon.

Identify the number of sides: A hexagon has $n = 6$ sides = $n = 6$

Apply the formula for regular polygons: Exterior angle $= \frac{360°}{n}$ = Formula ready

Substitute and calculate: $\frac{360°}{6} = 60°$ = $60°$

Verify with interior angle: Interior $= 180° - 60° = 120°$. Check: $\frac{(6-2) \times 180°}{6} = 120°$ ✓ = Verified!

Answer: Each exterior angle of a regular hexagon measures $60°$.

Finding the Number of Sides

A regular polygon has exterior angles of $30°$ each. How many sides does it have?

Recall the exterior angle formula: Each exterior angle $= \frac{360°}{n}$ = Formula identified

Set up the equation: $30° = \frac{360°}{n}$ = Equation ready

Solve for $n$: $n = \frac{360°}{30°} = 12$ = $n = 12$

Identify the polygon: A polygon with 12 sides is a dodecagon = Dodecagon

Answer: The polygon has $12$ sides. It is a regular dodecagon.

Mixed Interior and Exterior Problem

In a quadrilateral, three exterior angles measure $85°$, $95°$, and $70°$. Find the fourth exterior angle and all interior angles.

Find the sum of known exterior angles: $85° + 95° + 70° = 250°$ = $250°$

Find the fourth exterior angle: $360° - 250° = 110°$ = $110°$

Convert to interior angles: $180° - 85° = 95°$, $180° - 95° = 85°$, $180° - 70° = 110°$, $180° - 110° = 70°$ = $95°, 85°, 110°, 70°$

Verify interior angle sum: $95° + 85° + 110° + 70° = 360°$ ✓ = Confirmed!

Answer: The fourth exterior angle is $110°$. The interior angles are $95°$, $85°$, $110°$, and $70°$.

Mistake: Confusing interior and exterior angles

Why: Students sometimes calculate interior angles when asked for exterior, or vice versa.

Correct: Remember: exterior angles are formed by extending a side OUTSIDE the polygon. Interior + Exterior = $180°$ at each vertex.

Mistake: Thinking the exterior angle sum depends on the number of sides

Why: Since interior angle sums change with the number of sides, students assume exterior sums do too.

Correct: The sum of exterior angles is ALWAYS $360°$ for any convex polygon, regardless of how many sides it has.

Mistake: Forgetting to extend only one side at each vertex

Why: At each vertex, you can extend either side, but you should only count one exterior angle per vertex.

Correct: Choose one side to extend at each vertex. Each vertex contributes exactly one exterior angle to the sum.

Robot Navigation

When programming a robot to trace a polygon path, it needs to turn at each vertex. The turn angle equals the exterior angle!

A robot tracing a regular pentagon turns $\frac{360°}{5} = 72°$ at each corner. After 5 turns, it has rotated $360°$ total and faces its original direction.

Satellite Dish Installation

Satellite installers use angle measurements to properly aim dishes. Understanding exterior angles helps calculate mounting positions.

If a satellite dish needs to point at an angle that creates a $150°$ interior angle with the roof, the exterior angle (from the ground reference) is $180° - 150° = 30°$.

Q: Why do exterior angles always sum to 360 degrees?

A: Imagine walking around the polygon, turning at each vertex. Each turn is an exterior angle. When you return to your starting point facing the same direction, you have turned a full circle: $360°$!

Q: Does this work for concave (non-convex) polygons?

A: The $360°$ rule applies to convex polygons. For concave polygons, some exterior angles are measured differently (they can be negative in some conventions), but the concept still applies with proper definitions.

Q: What is the smallest exterior angle possible?

A: For a regular polygon, the exterior angle approaches $0°$ as the number of sides approaches infinity (like a circle). The largest is $120°$ for an equilateral triangle.