Measuring angles and constructing triangles
Measure accurately, then construct a triangle from three side lengths with ruler and compasses.
Core and Extended.
Before you begin
- Read millimetres and degrees.
- Compare and add lengths.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A drawn angle looks acute. Which protractor reading is consistent?
What you will learn
- Use a ruler and protractor with the correct starting scale.
- Construct an SSS triangle with visible compass arcs.
- Check whether three lengths can form a non-degenerate triangle.
Measure from the correct zero
Align the ruler's zero mark with one endpoint, rather than its physical edge, and read the other endpoint. To measure an angle, put the protractor's centre at the vertex and its zero line along one arm. Read the scale that starts at zero on that arm. Estimate acute or obtuse first to catch a wrong-scale reading.
To draw a 65° angle, draw a straight base ray, position the protractor, mark 65° and join the vertex to the mark with a ruler. Draw and label the requested line segment length accurately. A protractor is suitable for drawing a given angle, but the three-side construction below specifically uses a ruler and compasses only.
Three lengths locate the third vertex
For AB = 7 cm, AC = 5 cm and BC = 6 cm, draw AB first. Open the compasses to 5 cm and draw an arc centred at A. Without confusing the centres, draw a 6 cm arc centred at B. Their intersection is C; join AC and BC with a ruler. Leave the construction arcs visible as evidence of the method.
The arcs may meet above and below AB. Either intersection gives a congruent triangle unless a location is specified. Each pair of side lengths must sum to more than the third: lengths 2, 3 and 5 would lie on one line rather than form a triangle. This syllabus does not require compass constructions of angle bisectors or perpendicular bisectors; do not substitute those tasks for the specified three-side construction.
Pause and explain
To construct AC = 5 cm, which arc is needed?
Worked example
Construct triangle ABC with AB = 7 cm, AC = 5 cm and BC = 6 cm.
Show the worked solution
- Draw and label the 7 cm base AB accurately.
- Draw a 5 cm arc from A and a 6 cm arc from B; label an intersection C.
- Join AC and BC using a ruler, retain both arcs and check all three lengths.
Answer A labelled 5–6–7 cm triangle with visible construction arcs.
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.
On paper construct sides 4 cm, 5 cm and 6 cm. Use the 6 cm side as the base.
Give me a hint
Each remaining side is a compass radius from one base endpoint.
Compare my reasoning
- Draw a 6 cm base with a ruler.
- Draw arcs of radii 4 cm and 5 cm from its opposite endpoints.
- Join one intersection to both endpoints and check lengths; keep the arcs.
A 4–5–6 cm triangle with both construction arcs visible.
Look for these in your work
- Used compasses for the two distances.
- Retained arcs and accurate straight edges.
Draw a 4.8 cm line segment, then an angle of 125° at one end using a protractor.
Give me a hint
Choose the protractor scale beginning at your base ray's zero.
Compare my reasoning
- Measure the segment from the ruler's zero mark.
- Place the protractor centre exactly at the chosen endpoint and mark 125°.
- Draw the second ray with a ruler; check the angle is obtuse, not 55°.
A measured 4.8 cm segment with an obtuse 125° angle.
Look for these in your work
- Aligned the vertex and baseline.
- Read the correct scale; checked with tools on paper.
Can a triangular frame have sides 3 cm, 4 cm and 8 cm? Explain what happens to construction arcs.
Give me a hint
Compare the two shorter sides with the longest.
Compare my reasoning
- 3 + 4 = 7 cm, which is less than 8 cm.
- Arcs of radii 3 and 4 from endpoints 8 cm apart cannot meet.
- No non-degenerate triangle has these side lengths.
No; the two shorter sides cannot span the 8 cm separation.
Look for these in your work
- Checked the strict triangle inequality.
- Connected the numerical result to the construction.
Common mistakes
- Using the ruler's physical edge instead of its zero mark.
- Reading the wrong protractor scale.
- Erasing the arcs or guessing the third vertex.
What can you explain now?
Why should the compass arcs remain visible?
Compare with the explanation
They show how the third vertex was located.
The two fixed-distance arcs demonstrate construction from the supplied side lengths rather than an approximate sketch.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Construct a different three-side triangle tomorrow. Explain what each compass arc fixes before drawing it.
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