Angles, Parallel Lines and Polygons
Angle facts are most powerful when every step has a reason. This lesson builds from straight lines and vertically opposite angles to parallel lines, polygon sums and short geometric proofs.
Foundation & Higher. Algebraic angle questions may use skills from Solving Linear Equations.
Core angle facts
| Fact | Total or relationship |
|---|---|
| Angles on a straight line | 180° |
| Angles around a point | 360° |
| Vertically opposite angles | Equal |
| Angles in a triangle | 180° |
| Exterior angle of a triangle | Equals the sum of the two opposite interior angles |
Give the mathematical reason, not just a visual nickname.
Parallel lines and transversals
When a transversal crosses parallel lines, corresponding and alternate angles are equal, while co-interior angles sum to 180°. The lines must actually be parallel for these facts to apply.
Worked example — two reasons
Parallel lines are cut by a transversal. One angle is 68°. Its corresponding angle is also 68°. The co-interior partner of a 68° angle is 180°−68°=112°. State which relationship you use at each step; equality and a sum of 180° are different facts.
Triangle and exterior-angle reasoning
Triangle angles sum to 180°. One way to justify this is to draw a line through a vertex parallel to the opposite side; alternate angles transfer the other two angles onto a straight line.
The exterior angle of a triangle therefore equals the sum of the two opposite interior angles.
Polygon interior angles
An n-sided polygon can be split from one vertex into n−2 triangles, so its interior-angle sum is (n−2)×180°.
For a regular polygon, divide that total by n to find one interior angle. The exterior angles of any convex polygon make one full turn, so they total 360°.
Worked example — find the number of sides
A regular polygon has exterior angle 24°. Since exterior angles total 360°, number of sides = 360 ÷ 24 = 15.
If you are given the interior angle instead, first find the exterior angle using interior + exterior = 180°.
Algebraic angles and proof
Use geometric facts to form an equation, solve it, then return to the geometry. A proof should state each fact and reason in sequence.
Example: if an isosceles triangle has base angles (x+10)° and (x+10)° and vertex angle 40°, then 2(x+10)+40=180, so x=60.
Think Like a Mathematician
A good geometry solution is a chain of justified statements. Writing ‘because they look equal’ is never enough; writing ‘alternate angles are equal because the lines are parallel’ is.
Common Mistakes
Using parallel-line facts without parallel lines.
Calling an angle ‘Z angle’ instead of giving the formal reason: alternate angles.
Using 360÷n for an irregular polygon’s individual exterior angles.
Forgetting that an exterior angle and its adjacent interior angle sum to 180°.
Practice — angles and polygons
1. Two angles on a straight line are 68° and x°. Find x.
x=180−68=112°.
2. Vertically opposite angles are labelled (3x+5)° and 80°. Find x.
3x+5=80, so 3x=75 and x=25.
3. A triangle has angles 47°, 63° and x°. Find x.
x=180−47−63=70°.
4. An exterior angle of a triangle is 125°. One opposite interior angle is 54°. Find the other.
125=54+x, so x=71°.
5. Parallel lines are cut by a transversal. One corresponding angle is 118°. Find the matching corresponding angle.
Corresponding angles are equal, so 118°.
6. Two parallel lines are cut by a transversal. The co-interior angles are (3x)° and (x+20)°. Find x.
3x+x+20=180 → 4x=160 → x=40.
7. Find the interior-angle sum of an octagon.
(8−2)×180=1080°.
8. Find one interior angle of a regular decagon.
Interior sum=(10−2)×180=1440°. Divide by 10: 144°.
9. A regular polygon has exterior angle 30°. How many sides?
360÷30=12 sides.
10. A regular polygon has interior angle 165°. How many sides?
Exterior=180−165=15°. 360÷15=24 sides.
11. Explain why alternate angles are not automatically equal in every Z-shaped diagram.
They are equal only when the two relevant lines are parallel.
12. An isosceles triangle has equal base angles (4x−5)° and vertex angle 50°. Find x.
2(4x−5)+50=180 → 8x+40=180 → 8x=140 → x=17.5.
Frequently Asked Questions
Do all exterior angles have to be equal?
No. The exterior angles of a convex polygon total 360°. They are all equal when the polygon is equiangular; a regular polygon has equal sides as well as equal angles. For example, a non-square rectangle has four equal exterior angles but is not regular.
Can I use triangle angle sum without proving it every time?
Yes. Once established, it is a standard fact you may use.
Are algebraic angle questions Higher-only?
No. Difficulty does not determine tier by itself; many algebraic angle problems are accessible at Foundation.