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Study Guide: How to Solve: Exterior Angles
Source: https://www.fatskills.com/k-12-assessment-tests/chapter/how-to-solve-exterior-angles

How to Solve: Exterior Angles

By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.

⏱️ ~6 min read

How to Solve: Exterior Angles

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you can find an exterior angle in 10 seconds, you’ll save minutes on your exam—and minutes mean marks. Let’s make sure you never lose a point on this again."

What You Need To Know First

Before tackling exterior angles, you must already understand: 1. Angles in a triangle – The sum of interior angles in any triangle is 180°. 2. Linear pairs – Two angles on a straight line add up to 180°. 3. Polygon basics – A polygon is a closed shape with straight sides (e.g., triangle, quadrilateral, pentagon).

If any of these are shaky, pause and review them first.

Key Vocabulary

Term Plain-English Definition Quick Example
Exterior Angle An angle formed by one side of a polygon and the extension of an adjacent side. If you extend one side of a triangle, the angle outside the triangle is exterior.
Interior Angle An angle inside the polygon at a vertex. The three angles inside a triangle are interior angles.
Remote Interior Angles The two interior angles of a triangle that are not adjacent to the exterior angle. In a triangle, if the exterior angle is at vertex A, the remote interior angles are at B and C.
Polygon A closed 2D shape with straight sides (e.g., triangle, quadrilateral, pentagon). A stop sign is an octagon (8-sided polygon).
Regular Polygon A polygon where all sides and all angles are equal. A square is a regular quadrilateral.
Sum of Exterior Angles The total of all exterior angles in any polygon (one at each vertex) is always 360°. A pentagon has 5 exterior angles that add up to 360°.

Formulas To Know

1. Exterior Angle of a Triangle

Formula: Exterior Angle = Sum of the Two Remote Interior Angles

Variables: - Let the exterior angle be at vertex A. - The two remote interior angles are at vertices B and C.

Example: If ∠B = 50° and ∠C = 70°, then the exterior angle at A = 50° + 70° = 120°.

MEMORISE THIS – This is not usually given on exam sheets.

2. Sum of Exterior Angles of Any Polygon

Formula: Sum of Exterior Angles = 360° (for any polygon, no matter how many sides)

Variables: - Works for triangles, quadrilaterals, pentagons, etc.

Example: A hexagon has 6 exterior angles. Their sum is 360°.

GIVEN ON EXAM SHEET (but memorise it anyway—it saves time).

3. Measure of One Exterior Angle in a Regular Polygon

Formula: Each Exterior Angle = 360° ÷ Number of Sides

Variables: - Number of Sides = How many sides the polygon has (e.g., 5 for a pentagon).

Example: A regular octagon (8 sides) has exterior angles of 360° ÷ 8 = 45° each.

MEMORISE THIS – Examiners love testing this.

Step-by-Step Method

How to Find an Exterior Angle of a Triangle

When to use: Given two interior angles, or one interior angle and the exterior angle.

Steps: 1. Identify the exterior angle – It’s the angle outside the triangle, formed by extending one side. 2. Find the two remote interior angles – These are the two angles not next to the exterior angle. 3. Add the two remote interior angles – Their sum equals the exterior angle. 4. Check with linear pair (optional) – The exterior angle + its adjacent interior angle = 180°.

Worked Example (Using Steps): Question: In triangle ABC, ∠A = 60° and ∠B = 50°. Find the exterior angle at C.

  1. Exterior angle at C is formed by extending side BC.
  2. Remote interior angles are ∠A (60°) and ∠B (50°).
  3. Add them: 60° + 50° = 110° (exterior angle at C).
  4. Check: Exterior angle (110°) + interior angle at C (70°) = 180° ✔️

How to Find the Sum of Exterior Angles of Any Polygon

When to use: Any polygon (triangle, quadrilateral, etc.).

Steps: 1. Count the number of sides – e.g., pentagon = 5 sides. 2. Remember the rule – Sum of exterior angles = 360° (always). 3. Write the answer – No calculation needed!

Worked Example: Question: What is the sum of the exterior angles of a 12-sided polygon (dodecagon)? 1. Number of sides = 12. 2. Sum of exterior angles = 360° (always). 3. Answer: 360°.

How to Find One Exterior Angle in a Regular Polygon

When to use: Given a regular polygon (all sides and angles equal).

Steps: 1. Count the number of sides – e.g., hexagon = 6 sides. 2. Use the formula: Each exterior angle = 360° ÷ number of sides. 3. Calculate and simplify – Give the answer in degrees.

Worked Example: Question: Find the measure of one exterior angle of a regular decagon (10 sides). 1. Number of sides = 10. 2. Formula: 360° ÷ 10 = 36°. 3. Answer: 36°.

Worked Examples

Example 1 – Basic (Triangle Exterior Angle)

Question: In triangle PQR, ∠P = 40° and ∠Q = 60°. Find the exterior angle at R.

Solution: 1. Exterior angle at R is formed by extending QR. 2. Remote interior angles are ∠P (40°) and ∠Q (60°). 3. Add them: 40° + 60° = 100°. 4. Check: Exterior angle (100°) + interior angle at R (80°) = 180° ✔️

What we did and why: - We used the rule that an exterior angle equals the sum of the two remote interior angles. - This works because the three interior angles of a triangle add to 180°, and the exterior angle + its adjacent interior angle also add to 180°.

Example 2 – Medium (Regular Polygon)

Question: A regular polygon has an exterior angle of 24°. How many sides does it have?

Solution: 1. Formula: Each exterior angle = 360° ÷ number of sides. 2. Rearrange: Number of sides = 360° ÷ exterior angle. 3. Plug in: 360° ÷ 24° = 15. 4. Answer: 15 sides.

What we did and why: - We used the formula for exterior angles in a regular polygon. - Since all exterior angles are equal, dividing 360° by one exterior angle gives the number of sides.

Example 3 – Exam Style (Disguised Problem)

Question: In the figure below, AB is extended to D. If ∠CAB = 55° and ∠ABC = 45°, find ∠CBD.

(Note: ∠CBD is the exterior angle at B.)

Solution: 1. Identify the exterior angle – ∠CBD is outside the triangle, formed by extending AB. 2. Remote interior angles – ∠CAB (55°) and ∠ACB. 3. Find ∠ACB first – In triangle ABC, 55° + 45° + ∠ACB = 180° → ∠ACB = 80°. 4. Add remote angles: 55° + 80° = 135°. 5. Answer: ∠CBD = 135°.

What we did and why: - The question didn’t say "exterior angle," but we recognised ∠CBD as one. - We first found the missing interior angle (∠ACB) using the triangle angle sum. - Then we applied the exterior angle rule.

Common Mistakes

Mistake Why it Happens Correct Approach
Adding the wrong angles Students add the adjacent interior angle instead of the remote ones. Remote angles are the two not next to the exterior angle.
Forgetting the sum is 360° Students think the sum of exterior angles changes with the number of sides. 360° is always the sum, no matter the polygon.
Misidentifying the exterior angle Students pick an interior angle instead of the one outside. Exterior angle is outside the shape, formed by extending a side.
Assuming all polygons have equal exterior angles Students forget this only applies to regular polygons. Only regular polygons have equal exterior angles.
Not checking with linear pair Students forget that exterior angle + adjacent interior angle = 180°. Always verify: exterior angle + adjacent interior angle = 180°.

Exam Traps

Trap How to Spot it How to Avoid it
Disguised exterior angle The question doesn’t say "exterior angle" but shows a diagram with an extended side. Look for extended sides—any angle outside the shape is an exterior angle.
Irregular polygon The question asks for an exterior angle but doesn’t say the polygon is regular. Don’t assume equal angles—only regular polygons have equal exterior angles.
Missing interior angle The question gives two angles but asks for an exterior angle, hiding the third interior angle. Find all interior angles first using the triangle sum (180°).

1-Minute Recap

"Okay, let’s lock this in—30 seconds to master exterior angles, right before your exam.

  1. For triangles: An exterior angle equals the sum of the two remote interior angles. If you see an extended side, add the two angles not next to it.
  2. For any polygon: The sum of all exterior angles is always 360°. No exceptions.
  3. For regular polygons: One exterior angle = 360° ÷ number of sides. Easy.
  4. Watch for traps: Extended sides mean exterior angles—don’t ignore them! And if the polygon isn’t regular, don’t assume all exterior angles are equal.

Now go crush that exam. You’ve got this!


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