Identify and draw acute, right, obtuse and reflex angles
Name a selected angle from its rays, marked arc and exact measure, including rotated diagrams and reflex sweeps.
Exam scope and assessment notes
Foundation and Higher share all four section4.1 requirements. The Higher table says See Foundation Tier; there are no additional H4.1 rows. These six original lessons supply selected angle types, local point/straight/intersection rules, marked parallel-transversal relations, triangle sums and exterior angles, isosceles/equilateral properties and right/mixed triangle angles. F:4.1B requires both the local-line and parallel-line lessons; F:4.1D requires both special-triangle and right-triangle lessons with substantive teaching, a worked example and every Guided, Independent and Transfer stage. Six lessons and24 selected seed cases are StudyForA’s design, not a syllabus-prescribed count. Draw or copy each explicitly labelled configuration, retain the selected reflex or nonreflex exterior arc and mark only given or justified collinearity, parallel arrows, equality ticks and right squares. Diagrams need not be to scale; do not measure them unless instructed. State geometric reasons before equations, substitute back and reject impossible or insufficient data. The triangle-sum explanation uses an auxiliary parallel through a vertex with transported alternate angles and a straight-angle partition; the exterior relationship requires an actual straight side extension. Equilateral is also isosceles under the inclusive at-least-two convention, and right-isosceles overlap remains valid. A fixed reference picture and a correct number do not establish that the learner drew or justified the configuration. Written work is not automatically marked. Attempt acknowledgements and revealed references do not award marks or establish mastery. These structural links are not exhaustive subskill approval; specialist approval and formal assessment validation remain unestablished. Section4.5 instrument constructions and4.4 measurement/bearings remain separate, as does the full H4.7 reasoning scope across lines, polygons and circles. No blanket formal-proof waiver is imported from unrelated circle-theorem notes. The wider Maths A course remains incomplete. Publisher book navigation is separate evidence: Book1 Shape and Space1 has literal F4.1B/C/D chapter tags, not F4.1A; Hub18 literally tags B/D, not A/C. Map pages60–99 and proof chapter start60 agree here, with the next proof chapter at100; do not invent a pagination conflict or claim final-edition approval. Only selected public uncorrected proof pages60–65 were previously inspected. Full final-book and Teaching Hub body/pagination alignment remains unverified; chapter links confer no full-book or modular teaching credit.
Before you begin
- Read positive degree labels and add or subtract whole-number angle measures.
- Recognise a common endpoint and labelled rays.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Rays OA and OB have a selected arc labelled82°. Which classification fits that arc?
What you will learn
- Identify the vertex, two rays and selected angle arc in a labelled diagram.
- Distinguish acute, right, obtuse and reflex angles using exact degree boundaries.
- Retain the selected sweep when a figure is rotated or the same rays enclose two different arcs.
Use four separate vertices O₁,O₂,O₃,O₄, with each starting ray OᵢAᵢ and second ray OᵢBᵢ clearly labelled. Mark the chosen arc, write its exact degree label and use a right-angle square only for the90° case. In the reflex case draw the long arc outside the smaller wedge. Make a second, rigidly rotated copy of one entire ray-and-arc diagram. Draw and classify selected angles 37°,90°,128° and247°. For the247° example, mark its long arc and find the other sweep between the same rays.
Copy and annotate the labelled figures on paper. Retain every given, chosen arc, straight extension and geometric reason. Exact givens decide deductions; these screen diagrams must not be measured unless a task explicitly asks. Unknown values and classifications stay hidden until you attempt your reasoning.
Selected sweep 1
- selected: given 37°
Selected sweep 2
- selected: given 90°
Given right-angle square at O2.
Selected sweep 3
- selected: given 128°
Selected sweep 4
- selected: given 247°
- other sweep: unknown
Name the vertex and choose the arc
An angle is formed by two rays with a common endpoint, the vertex. In ∠AOB the vertex is O, so name the rays OA and OB before deciding which part of the turn is meant. Mark an arc in the selected region and place its degree label beside that arc. A ray pair alone can enclose a smaller angle and a larger reflex sweep.
Use the degree symbol ° on angle measures. A right-angle square is a mark of90°, not a decoration to add whenever a drawing looks nearly square. In deduction problems use the stated labels and marks rather than measuring an unscaled sketch. Drawing a labelled example helps identify the angle, but its appearance does not override exact data.
Use exact boundaries for the four types
An acute angle satisfies0°<θ<90°. A right angle is exactly90°. An obtuse angle satisfies90°<θ<180°, and a reflex angle satisfies180°<θ<360°. The inequalities are strict at the boundaries:89° is acute and181° is reflex, however close the sketches look to familiar boundary shapes.
The contrast values0°,180° and360° are not members of those four named categories. They describe a zero turn, a straight angle and a complete turn respectively. In particular180° is neither obtuse nor reflex, and360° is not a reflex angle. Compare the selected exact value with the boundaries before giving its name.
Preserve a reflex sweep and a rotated figure
For two distinct nonopposite rays, the smaller and larger sweeps complete one360° turn. If the selected long arc is θ, the other sweep is360°−θ. This is a complete-turn relationship, not the90° relationship called complementary angles. Do not replace a requested reflex measure by its smaller counterpart and keep the same answer label.
Rotating or reflecting the whole ray-and-arc diagram preserves the angle measure and type. Keep the marked arc attached to the same region during that change. To illustrate a given angle, copy or draw an appropriately labelled ray pair, using an instrument only when the activity calls for it; these sketches do not by themselves constitute the separate full programme of formal instrument constructions.
Sources for this lesson
Pause and explain
The long selected arc between OP and OQ is271°. What is its type and the other sweep?
Worked example
Use four separate vertices O₁,O₂,O₃,O₄, with each starting ray OᵢAᵢ and second ray OᵢBᵢ clearly labelled. Mark the chosen arc, write its exact degree label and use a right-angle square only for the90° case. In the reflex case draw the long arc outside the smaller wedge. Make a second, rigidly rotated copy of one entire ray-and-arc diagram. Draw and classify selected angles 37°,90°,128° and247°. For the247° example, mark its long arc and find the other sweep between the same rays.
Show the worked solution
- Annotate the four chosen arcs as 37°, 90°, 128°, 247°. The first angle is between0° and90°, the second equals90°, the third is between90° and180°, and the fourth is between180° and360°. Match those comparisons to the selected arcs, not to a horizontal or vertical appearance.
- The classifications in order are acute, right, obtuse and reflex. For the last ray pair, one complete turn is360°, so the other sweep is360°−247°=113°. Keep the label247° on the long arc and113° on the other arc; they answer different selected-angle questions.
- Rotate the whole rays-and-arc sketch without moving the arc label to another sector. Rotation changes its direction on the page, not its measure or type. Exact labels decide near-boundary cases; an unscaled drawing is not permission to round an angle into a different category.
Answer 37°, 90°, 128°, 247°: acute, right, obtuse, reflex respectively. Other sweep for the last pair: 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.
Use four separate vertices O₁,O₂,O₃,O₄, with each starting ray OᵢAᵢ and second ray OᵢBᵢ clearly labelled. Mark the chosen arc, write its exact degree label and use a right-angle square only for the90° case. In the reflex case draw the long arc outside the smaller wedge. Make a second, rigidly rotated copy of one entire ray-and-arc diagram. For the two measures sharing one ray pair, draw both arcs in opposite sweep directions rather than silently changing which arc is selected. Draw and classify selected sweeps74°,90°,153° and286°. Explain how74° and286° can label different arcs between the same two rays.
Give me a hint
Locate the selected arc before comparing its degree value with90°,180° and360°. Keep the smaller and larger sweeps separately labelled.
Compare my reasoning
- Annotate the four chosen arcs as 74°, 90°, 153°, 286°. The first angle is between0° and90°, the second equals90°, the third is between90° and180°, and the fourth is between180° and360°. Match those comparisons to the selected arcs, not to a horizontal or vertical appearance.
- The classifications in order are acute, right, obtuse and reflex. For the last ray pair, one complete turn is360°, so the other sweep is360°−286°=74°. Keep the label286° on the long arc and74° on the other arc; they answer different selected-angle questions.
- For the same two rays, the74° smaller arc and286° larger arc add to360°. Mark both arcs explicitly. A rigid rotation of that labelled pair preserves both classifications; changing the selected arc does not. The other two cases remain right at90° and obtuse at153°.
74°, 90°, 153°, 286°: acute, right, obtuse, reflex respectively. Other sweep for the last pair: 74°.
Look for these in your work
- Draw the named rays and selected arcs, with degree labels and the right-angle mark where justified.
- Classify each exact measure using the correct strict boundaries.
- Explain the other sweep and preserve arc identity when rotating the figure.
Use four separate vertices O₁,O₂,O₃,O₄, with each starting ray OᵢAᵢ and second ray OᵢBᵢ clearly labelled. Mark the chosen arc, write its exact degree label and use a right-angle square only for the90° case. In the reflex case draw the long arc outside the smaller wedge. Make a second, rigidly rotated copy of one entire ray-and-arc diagram. Draw and classify the exact labels89°,90°,179° and181°. Explain why an unscaled sketch must not turn89° into a right angle or181° into an obtuse angle.
Give me a hint
Locate the selected arc before comparing its degree value with90°,180° and360°. Keep the smaller and larger sweeps separately labelled.
Compare my reasoning
- Annotate the four chosen arcs as 89°, 90°, 179°, 181°. The first angle is between0° and90°, the second equals90°, the third is between90° and180°, and the fourth is between180° and360°. Match those comparisons to the selected arcs, not to a horizontal or vertical appearance.
- The classifications in order are acute, right, obtuse and reflex. For the last ray pair, one complete turn is360°, so the other sweep is360°−181°=179°. Keep the label181° on the long arc and179° on the other arc; they answer different selected-angle questions.
- The exact inequalities matter:89<90,90 is the boundary itself,179<180 and181>180. Thus89° remains acute and181° remains reflex even if the unscaled sketch looks almost right or straight. The181° long arc has a179° smaller counterpart, not the same selected angle.
89°, 90°, 179°, 181°: acute, right, obtuse, reflex respectively. Other sweep for the last pair: 179°.
Look for these in your work
- Draw the named rays and selected arcs, with degree labels and the right-angle mark where justified.
- Classify each exact measure using the correct strict boundaries.
- Explain the other sweep and preserve arc identity when rotating the figure.
Use four separate vertices O₁,O₂,O₃,O₄, with each starting ray OᵢAᵢ and second ray OᵢBᵢ clearly labelled. Mark the chosen arc, write its exact degree label and use a right-angle square only for the90° case. In the reflex case draw the long arc outside the smaller wedge. Make a second, rigidly rotated copy of one entire ray-and-arc diagram. For the two measures sharing one ray pair, draw both arcs in opposite sweep directions rather than silently changing which arc is selected. A fictional hinge display shows56°,90°,124° and a deliberately selected long sweep236°. Draw and classify all four states, distinguishing the124° and236° arcs between the same two hinge rays.
Give me a hint
Locate the selected arc before comparing its degree value with90°,180° and360°. Keep the smaller and larger sweeps separately labelled.
Compare my reasoning
- Annotate the four chosen arcs as 56°, 90°, 124°, 236°. The first angle is between0° and90°, the second equals90°, the third is between90° and180°, and the fourth is between180° and360°. Match those comparisons to the selected arcs, not to a horizontal or vertical appearance.
- The classifications in order are acute, right, obtuse and reflex. For the last ray pair, one complete turn is360°, so the other sweep is360°−236°=124°. Keep the label236° on the long arc and124° on the other arc; they answer different selected-angle questions.
- The124° opening and236° long sweep share the same ray pair because124+236=360, but only the first is obtuse and only the second is reflex. Keep separate arc labels on the hinge diagram; a rotation of the complete display changes neither result.
56°, 90°, 124°, 236°: acute, right, obtuse, reflex respectively. Other sweep for the last pair: 124°.
Look for these in your work
- Draw the named rays and selected arcs, with degree labels and the right-angle mark where justified.
- Classify each exact measure using the correct strict boundaries.
- Explain the other sweep and preserve arc identity when rotating the figure.
Common mistakes
- Naming the angle from page orientation rather than its exact selected arc.
- Replacing a reflex sweep by the smaller angle between the same rays.
- Including180° or360° in an open-range type, or treating an unscaled near-boundary sketch as an exact measurement.
What can you explain now?
Draw opposite rays OA and OB on one straight line, with the selected arc labelled180°. Is that angle obtuse or reflex?
Compare with the explanation
Neither: it is a straight angle of180°.
Obtuse angles are strictly below180° and reflex angles are strictly above180°. Mark the straight selected arc rather than choosing a different sweep.
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Make it stick
Define all four angle types using exact degree boundaries. Show two different marked sweeps between the same rays and explain what a rigid rotation preserves.
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.