Construct an angle bisector
Construct angle bisectors with equal-radius arcs and a straight edge, select a non-vertex crossing and distinguish the required small or reflex ray.
Exam scope and assessment notes
Shared Foundation/Higher Construction covers F4.5A–D: measurement and drawing, triangles and other 2D shapes, scale drawings, and both straight-edge-and-compasses bisectors. These six original lessons are linked to those syllabus rows; the links do not establish complete textbook coverage or specialist approval. Use the named instruments and retain your marks, arcs and joins. A numerical answer or displayed reference alone does not show instrument skill. Physical millimetre work needs a real ruler and a verified object or calibrated print; recheck calibration after printing, resizing or zooming. Measure only diagrams explicitly supplied as accurate for measurement. Virtual drawing units do not certify physical millimetres. Written answers and model comparisons are for self-checking, with no marks, mastery or formal assessment approval. The source is linear 4MA1 Issue 2, checked against November 2026/June 2027 entry context; regional options depend on centre eligibility. Modular qualifications remain separate. Publisher chapter links are navigation aids: the relevant final chapter body and Teaching Hub content remain unverified. Other syllabus sections remain separate.
Before you begin
- Use an unchanged compass opening and a straight edge to join constructed points.
- Recognise the vertex, boundary rays and a marked small or reflex arc.
- Use matching side lengths and a common side to explain equal angles, without claiming a full formal-proof programme.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is established by the single first arc centred at O meeting the two rays at A and B?
What you will learn
- Create equal first-arc distances from the vertex and equal second-arc distances from their endpoints.
- Use the non-vertex crossing to draw and justify the bisector instead of measuring half an angle.
- Select the ray inside the marked angle, including a reflex arc, and identify unusable or degenerate crossings.
Place your own instruments and retain the joins, rays and arcs. The givens are shown first; your construction stays at the positions you choose.
Worked construction
Construct the bisector of the supplied 60° angle using straight edge and compasses. Use a first arc of radius 4 cm centred at O, then the same 4 cm opening for the two endpoint arcs. Retain all arcs, label the first crossings A and B, and identify the correct second-arc crossing P away from O. Do not use a precompleted bisector or a protractor-halving shortcut.
Drawing unit: cm. Printed graduations calibrate the virtual drawing; they do not certify physical paper or monitor size.
All submitted joins and arcs stay visible. Select a recorded ruler or protractor to inspect its graduations; other instruments retain their actual footprints. Long lines and rays are cropped to this viewing frame.
Coordinates describe the virtual drawing in cm; they are placement controls, not endpoint readings. Positive angles turn anticlockwise; negative sweeps turn clockwise. Clicking chooses a visible position. It does not move an instrument onto a reference answer.
Read or use my recorded ruler
These readings come from your actual ruler placement. A misplaced ruler keeps its offset; this tool does not report the ideal endpoint reading.
Find intersections of my retained traces
Intersections use the actual arc sweeps and line extents. A full circle or supporting line cannot supply a crossing outside a traced arc, ray or segment.
This is a fixed mathematical comparison. A complete written response records an attempt; a teacher still needs to review the reasoning. Virtual graduations do not establish physical ruler, protractor or compass skill, and no marks or mastery are awarded. Verify physical calibration and recheck it after any print or zoom change.
Build two pairs of equal distances
Begin at the angle’s vertex O. Draw one arc centred at O that meets both given rays; label the crossings A and B. Because that arc has one unchanged radius, OA=OB. From A and B draw arcs with the same chosen second opening, large enough to give a usable crossing away from O. Keep both stages of arcs visible.
Do not assume that any two marks on the rays are equally far from O, and do not change the opening between the endpoint arcs. In the selected cases the second opening equals the first, so O itself is one of the second-arc crossings; do not select it as the new point. Choose the other crossing P, distinct from O, before joining anything.
Join and explain the angle division
Draw OP with a straight edge. The triangles OAP and OBP have OA=OB from the first arc, AP=BP from the second arcs, and OP in common. Their matching side lengths give equal corresponding angles at O, so the constructed ray bisects the requested small opening. Preserve this chain of reasons rather than writing only a halved angle number.
A protractor can check the two angle parts after construction, but measuring the original angle, dividing its number by two and drawing that new protractor angle is a different method. Here the straight-edge-and-compasses evidence is required. The diagram can be tilted: none of the compass steps depends on a horizontal arm or a compass-north direction.
Choose the requested ray and check feasibility
The full bisector line has two opposite rays from O. For a clearly marked reflex arc between the same boundary rays, the smaller-angle bisector ray lies in the wrong opening. Extend the constructed line through O and choose its opposite ray inside the marked long arc. Explain the chosen arc as well as the line; a correct line with the wrong ray does not bisect the requested angle.
The arc stages need nonzero radii, distinct boundary rays and a crossing away from O. If endpoint arcs fail to meet, or touch only at an unusable point, adjust the common opening rather than inventing a bisector. For a straight angle the same-size second opening would touch only at the vertex, so a larger second opening can provide an off-vertex crossing. Coincident rays at zero or a full turn need an explicit convention rather than a fabricated unique interior; these fixed tasks use distinct rays and marked arcs.
Sources for this lesson
Pause and explain
A constructed bisector divides a given 146° small opening equally. What two angle parts should a later check find?
Worked example
Construct the bisector of the supplied 60° angle using straight edge and compasses. Use a first arc of radius 4 cm centred at O, then the same 4 cm opening for the two endpoint arcs. Retain all arcs, label the first crossings A and B, and identify the correct second-arc crossing P away from O. Do not use a precompleted bisector or a protractor-halving shortcut.
Show the worked solution
- Draw the 4 cm first arc from O across both boundary rays, locating A and B. This gives OA=OB=4 cm without measuring a midpoint angle.
- Draw 4 cm endpoint arcs centred at A and B. In this equal-opening setup they meet at O and at a second point P; choose P, because O again would not define a joining direction.
- Use the straight edge to draw OP through that non-vertex crossing. The equal first radii, equal endpoint radii and shared OP make the two corresponding triangles match.
- The two parts of the given 60° angle are therefore 30° and 30°. Those values check the division after construction; the retained arcs and OP supply the requested method.
Answer Construct OP using the non-vertex intersection of the equal endpoint arcs. It divides the supplied 60° angle into 30° and 30°, with both arc stages retained.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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.
Construct the internal bisector of the supplied 100° angle on the turned diagram. Use a first arc of radius 3 cm from O and equal 3 cm endpoint arcs. Keep the arcs, select the non-vertex crossing and state the two resulting angle parts. Explain why a horizontal ray is unnecessary.
Give me a hint
The compass centres and unchanged openings follow the rays, whatever their page orientation.
Compare my reasoning
- From O draw the 3 cm first arc meeting the two rays at A and B. Keep the equality OA=OB visible through that common radius.
- Draw 3 cm arcs from A and B. Choose their crossing P away from O and draw OP with a straight edge.
- Use OA=OB, AP=BP and common OP to explain the equal angle parts. The given 100° opening is divided into 50° and 50°.
- The whole diagram is turned, but all distance relationships and the selected interior ray are unchanged. No private orientation number or north bearing is needed.
Construct OP through the non-vertex crossing. It divides the marked 100° opening into 50° and 50°; a turned baseline does not change the method.
Look for these in your work
- Retain the equal first-arc and endpoint-arc constructions.
- Choose the non-vertex crossing and join it to O with a straight edge.
- Explain the two 50° parts and why page orientation is irrelevant.
Independently construct the internal bisector of the supplied fresh 124° obtuse angle with a tilted baseline. Use 4 cm for each of the two arc stages and retain all arcs. Select the non-vertex crossing, draw the bisecting ray and explain the equal-side reasoning. Private page-orientation values are not initial givens.
Give me a hint
Construct equal distances from O first, then equal distances from the two new arc endpoints; do not start by plotting a calculated half-angle.
Compare my reasoning
- Draw the 4 cm first arc centred at O, meeting the tilted rays at A and B. This establishes OA=OB.
- With the same 4 cm opening at A and B, draw endpoint arcs and choose their crossing P away from O. Draw OP inside the marked obtuse opening.
- The triangles OAP and OBP have the two equal radius pairs and the common side OP, so the angle parts at O are equal.
- The numerical check is 124÷2=62°, giving 62° and 62°. Submit the actual retained construction, not merely the number 62° or a protractor-made half-angle.
Construct the interior ray OP using the non-vertex equal-arc crossing. It divides the supplied 124° angle into 62° and 62°, justified by matching side lengths.
Look for these in your work
- Carry out both equal-radius arc stages independently on the tilted diagram.
- Retain the arcs and select the interior non-vertex ray.
- Give the equal-side reason and the two 62° checks without replacing the construction by calculation.
Use the two supplied rays and their clearly marked clockwise 240° reflex arc. Construct the bisector of that requested long opening using 3 cm arcs for both stages. Retain the arcs and full joining line, select the correct ray from O, and explain why using the smaller-angle bisector ray would answer a different question. The rays’ private reference-coordinate directions are not given for calculation.
Give me a hint
Construct the bisector line first, then identify which of its opposite rays lies halfway inside the marked long arc.
Compare my reasoning
- Draw the 3 cm first arc centred at O to locate equal-distance points A and B on the boundary rays. Draw equal 3 cm endpoint arcs and select the crossing P away from O.
- Join O to P and extend that straight line through O. The OP ray lies inside the smaller companion opening, so it is not the requested reflex-angle bisector ray.
- Choose the opposite ray of the same constructed line, which lies inside the marked clockwise 240° arc. It divides that long opening into 240÷2=120° and 120°.
- Keep the two rays distinguishable and retain all arc evidence. The correct full line alone is insufficient unless the selected ray and requested reflex arc are identified; private orientation numbers and a numerical half-angle are not substitutes for the construction.
Select the ray opposite the smaller-angle OP ray on the constructed bisector line. It bisects the marked clockwise 240° reflex arc into 120° and 120°, with all arc stages retained.
Look for these in your work
- Construct the equal-radius bisector line from a non-vertex crossing.
- Select and label the opposite ray inside the marked reflex arc, not the smaller opening.
- Explain the two 120° parts and retain construction evidence without using private coordinate directions.
Common mistakes
- Choosing the vertex again as the new arc crossing, or changing the opening between corresponding arcs.
- Measuring half an angle with a protractor instead of performing the straight-edge-and-compasses construction.
- Selecting the smaller-angle bisector ray when the marked opening is reflex, or inventing a unique ray for degenerate data.
What can you explain now?
One crossing of the two endpoint arcs is the original vertex O. Can joining O to that same point supply the required bisector direction?
Compare with the explanation
No. Choose a crossing P distinct from O and draw OP.
Joining the vertex to itself gives no line or ray direction. A usable non-vertex crossing, together with the equal-radius evidence, is needed for the straight-edge-and-compasses construction.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Describe both equal-radius arc stages and the non-vertex joining step. Give the equal-side reason for the angle division, then explain why a reflex problem needs the correct ray and not merely the correct full line.
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.