Angles, triangles and parallel lines
Calculate unknown angles and give the geometric reason for every step.
Section Paper 3 · Core · Calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Classify acute and obtuse angles.
- Recognise parallel lines.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What do angles on a straight line total?
What you will learn
- Use point, straight-line, triangle and quadrilateral angle sums.
- Recognise vertically opposite, corresponding, alternate and co-interior angles.
- Write reasons with correct three-letter angle notation.
Start with a known total or equality
Angles around a point total 360°, and adjacent angles on a straight line total 180°. Opposite angles made by two intersecting straight lines are vertically opposite and equal; adjacent ones are generally supplementary rather than equal. In angle ABC, B is the vertex, so keep the letter order tied to the actual diagram.
The angles in a triangle total 180°; in a quadrilateral they total 360°. An isosceles triangle's base angles are equal, and an equilateral triangle's three angles are each 60°. If an isosceles triangle has apex angle 44°, its base angles are (180 − 44)/2 = 68°. State both the sum rule and the equal-base-angle reason.
Parallel lines provide new equalities
When a transversal crosses parallel lines, corresponding angles in matching positions are equal, alternate interior angles on opposite sides of the transversal are equal, and co-interior angles inside the parallels on the same side total 180°. The diagram shows a 67° angle and its alternate partner; the adjacent co-interior angle is 113°.
The parallel condition is essential. A drawing that merely looks parallel does not justify these rules without stated information or arrow markings. Identify the actual angle pair before calculating, then write a geometric reason such as 'alternate angles between parallel lines', not just 'Z angles' or 'because the picture shows it'. A multi-step solution can combine a parallel equality with a triangle sum.
Pause and explain
A triangle has angles 38° and 57°. What is the third angle?
Worked example
Two parallel lines are crossed by a transversal. One interior angle is 67°. Find its alternate interior angle and its co-interior partner.
Show the worked solution
- The alternate interior angle is 67° because alternate angles between parallel lines are equal.
- The co-interior partner is 180 − 67 = 113° because those angles are supplementary.
- The two values serve different angle positions; do not interchange their reasons.
Answer Alternate 67°; co-interior 113°.
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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 apex angle 44°. Find both base angles and give reasons.
Give me a hint
Subtract the apex from 180° before sharing equally.
Compare my reasoning
- The remaining sum is 180 − 44 = 136° by the triangle angle sum.
- The two base angles are equal because the triangle is isosceles.
- Each is 136 ÷ 2 = 68°.
68° and 68°.
Look for these in your work
- Used the triangle total.
- Explained why the remaining angles are equal.
A quadrilateral has angles 72°, 96° and 110°. Find the fourth. Then find the angle adjacent to it on a straight line.
Give me a hint
Use different totals for the two tasks.
Compare my reasoning
- The known angles total 72 + 96 + 110 = 278°.
- The fourth angle is 360 − 278 = 82° by the quadrilateral sum.
- Its straight-line neighbour is 180 − 82 = 98°.
Fourth angle 82°; straight-line neighbour 98°.
Look for these in your work
- Used the correct total at each step.
- Gave both geometric reasons.
A transversal makes an interior 58° angle with one of two parallel rails. Find the co-interior angle at the other rail and its vertically opposite angle.
Give me a hint
First use the parallel relationship, then the intersection equality.
Compare my reasoning
- Co-interior angles total 180°, so the angle at the other rail is 122°.
- Its vertically opposite angle is equal to it.
- That opposite angle is also 122°; name each relationship separately.
122° and 122°.
Look for these in your work
- Checked that the rails are stated parallel.
- Distinguished supplementation from vertical equality.
Common mistakes
- Applying parallel angle rules without a parallel condition.
- Calling adjacent intersecting angles vertically opposite.
- Using a number without its geometric reason.
What can you explain now?
Angles around a point are 85°, 140° and x°. Find x.
Compare with the explanation
135°
The full-turn total is 360°, so x = 360 − 85 − 140 = 135°.
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Make it stick
Tomorrow, draw two parallel lines and a transversal. Mark one corresponding, one alternate and one co-interior pair with reasons.
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.