Angles, triangles and parallel lines
Calculate unknown angles and give the geometric reason for every step.
Core and Extended.
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°.
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°.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources