Curriculum fit: GCSE Maths Geometry and measures (Foundation and Higher). US: Grade 7-8 and Geometry (Congruence).
On this page
What Is It?
Many geometry problems ask you to find a missing angle and to give a reason. The key skill is knowing which angle rule to use and naming it clearly.
When a line crosses a pair of parallel lines it creates several angles that are equal or add to 180°. Polygons also follow set rules for the sum of their angles.
Key Formulas and Ideas
- Angles on a straight line add to 180°. Angles around a point add to 360°.
- Angles in a triangle add to 180°. Angles in a quadrilateral add to 360°.
- Alternate angles (Z shape) and corresponding angles (F shape) are equal; co-interior angles (C shape) add to 180°.
- Sum of interior angles of an n-sided polygon = (n − 2) × 180°. Each exterior angle of a regular polygon = 360° ÷ n.
Worked Examples
Parallel Lines
A transversal crosses two parallel lines. One angle is 65°. Find the alternate angle and the co-interior angle.
- Alternate angles are equal, so the alternate angle is 65°.
- Co-interior angles add to 180°: 180° − 65° = 115°.
✓ Answer 65° and 115°
Interior Angles
Find the sum of the interior angles of a pentagon.
- n = 5.
- (5 − 2) × 180° = 3 × 180°.
✓ Answer 540°
Regular Polygon
Find each interior angle of a regular octagon.
- Exterior angle = 360° ÷ 8 = 45°.
- Interior angle = 180° − 45° = 135°.
✓ Answer 135°
Common Mistakes to Avoid
- Giving a reason such as 'Z angles' with no mention of parallel lines.
- Using the interior angle sum formula with n instead of n − 2.
- Mixing up alternate and co-interior angles.
- Forgetting that the exterior and interior angles at a vertex add to 180°.
Practice Questions
Frequently Asked Questions
What reasons will examiners accept?
Use full terms like 'alternate angles are equal', 'angles on a straight line add to 180°' and 'angles in a triangle add to 180°'.
How many sides does a polygon have if each exterior angle is 30°?
360° ÷ 30° = 12 sides.
Keep Going With These Guides
Congruent Triangles: SSS, SAS, ASA and RHS Explained
Learn the congruence conditions for triangles, why AAA does not prove congruence and how to use them in simple proofs, with worked examples.
GCSEHigh SchoolArea of 2D Shapes: Rectangle, Triangle, Trapezium and More
Calculate the perimeter and area of rectangles, triangles, parallelograms, trapeziums (trapezoids) and compound shapes, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolCircle Theorems Explained: Rules, Examples and Reasons
Learn the main circle theorems and use them to find angles with a reason for every step. GCSE Higher and high school Geometry, with worked examples.
GCSEA-LevelHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.