Geometry · Grade 8
Angle Relationships and Parallel Lines
Angle rules let you calculate unknown angles without measuring them. Learn to identify the relationship before choosing a rule.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Identify the mathematical relationship or rule.
- Apply it accurately in worked and new problems.
- Check calculations and explain your reasoning.
1. Angles on a straight line
Angles on a straight line add to 180°. If one angle is 127°, the adjacent angle is 53°.
2. Angles around a point
Angles around a point add to 360°. Add the known angles and subtract their total from 360° to find the missing angle.
3. Vertically opposite angles
When two straight lines cross, vertically opposite angles are equal. Adjacent angles on a straight line are supplementary.
4. Parallel lines and a transversal
When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior angles add to 180°.
5. Triangle and polygon angle sums
The interior angles of a triangle add to 180°. For an n-sided polygon, the interior-angle sum is (n − 2) × 180°. A regular polygon has equal interior angles.
Reasoning with angle relationships
When a transversal crosses two parallel lines, corresponding angles are equal and alternate interior angles are equal. Co-interior angles on the same side of the transversal add to 180°. These relationships let you find missing angles even when the diagram is not drawn to scale.
Worked reasoning: Suppose two corresponding angles are (3x + 10)° and (2x + 35)°. Set them equal: 3x + 10 = 2x + 35, so x = 25. Each angle is 85°. Check by substituting 25 into both expressions.
- Do not assume two angles are equal just because they look equal in a sketch.
- Use 180° for a straight line and 360° for a full turn.
- Write the angle relationship before solving for an unknown.
Self-check: If co-interior angles are 4x° and (2x + 30)°, then 4x + 2x + 30 = 180, so x = 25.
Worked examples
- A straight-line pair: 180° − 68° = 112°.
- Angles around a point: 90° + 120° + 80° + x = 360°, so x = 70°.
- A triangle has angles 45° and 65°; the third is 70°.
- A quadrilateral's interior angles add to (4 − 2) × 180° = 360°.
Independent practice
Attempt each question before reviewing the answer.
- Find the supplement of 38°. Answer: 142°.
- Find the angle around a point if the other angles total 275°. Answer: 85°.
- Vertically opposite to 73° is what angle? Answer: 73°.
- A triangle has angles 35° and 85°. Find the third. Answer: 60°.
- Find the interior-angle sum of a pentagon. Answer: 540°.
- Co-interior angle with 112° on parallel lines? Answer: 68°.
- A regular hexagon's interior-angle sum is? Answer: 720°; each angle is 120°.
- Corresponding angle to 64° on parallel lines? Answer: 64°.
Knowledge check
Choose one answer for each question, then check for explanations.
Printable worksheet and answer key
Use the independent-practice questions above as your worksheet. Show each step and check your answers after completing the set.
Study reminder
Write the rule used beside each solution to make your reasoning clear.