Reason with triangle and polygon interior and exterior angles
Use triangle and quadrilateral sums · Find polygon sums by triangulation · Use regular convex polygon exterior angles
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Line-angle relationships
- Regular polygon vocabulary
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use triangle and quadrilateral sums
- Find polygon sums by triangulation
- Use regular convex polygon exterior angles
Build the polygon sum
Triangle interior angles sum to 180°; a quadrilateral can be split into two triangles, so its sum is 360°. A simple convex n-gon can be divided from one vertex into n−2 triangles, giving (n−2)×180°.
An isosceles triangle has equal angles opposite its equal sides; an equilateral triangle has three 60° angles. A triangle exterior angle equals the sum of the two remote interior angles because both derive from the 180° straight-line and triangle sums.
Regular convex polygons
A full walk around a convex polygon turns through 360°. A regular convex n-gon therefore has exterior angle 360°/n and interior angle 180°−360°/n. Regularity is needed to divide equally; an irregular polygon's individual angles may differ.
Do not apply ordinary simple-polygon formulas to self-intersecting star figures without specifying a different convention. Label which interior or adjacent exterior angle is being calculated.
Sources for this lesson
Pause and explain
What is the interior-angle sum of a convex pentagon?
Worked example
Find each interior and exterior angle of a regular octagon.
Show the worked solution
- Exterior = 360°/8 = 45°.
- Interior = 180°−45° = 135°; eight interior angles total 1080°.
Answer Interior 135°; exterior 45°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
An isosceles triangle has vertex angle 40°. Find its two equal base angles.
Give me a hint
Subtract the vertex angle from the triangle sum.
Compare my reasoning
- The base angles total 140°.
- Equal sharing gives 70° each.
70° and 70°.
Look for these in your work
- Use equality opposite equal sides
- Check the sum is 180°
A regular convex polygon has exterior angle 30°. Find its number of sides and each interior angle.
Give me a hint
The exterior angles total 360°.
Compare my reasoning
- n = 360/30 = 12.
- Interior = 180−30 = 150°.
12 sides; interior angle 150°.
Look for these in your work
- Use the regularity condition
- Distinguish interior from exterior
A triangle's two remote interior angles are 48° and 62°. Find the exterior angle at the third vertex, then its adjacent interior angle.
Give me a hint
Use the exterior theorem and supplement.
Compare my reasoning
- Exterior = 48+62 = 110°.
- Third interior = 180−110 = 70°; all three interiors total 180°.
Exterior 110°; adjacent interior 70°.
Look for these in your work
- Identify the remote angles
- Give a reasoned check
Common mistakes
- Equal-angle calculations require regularity.
- An exterior angle is supplementary to its adjacent interior angle.
What can you explain now?
What is the interior-angle sum of a convex pentagon?
Compare with the explanation
540°.
It divides into 5−2 = 3 triangles.
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Make it stick
What is the interior-angle sum of a convex pentagon?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.