At a glance. One lesson of about 60 minutes. It assumes isosceles triangles, exterior angles and angles on parallel lines. Students prove the centre-angle theorem in three cases, then derive the semicircle, same-segment, cyclic quadrilateral and alternate segment results.
Calculator: no calculator. No calculator is used: the lesson is proof and angle chasing, and every number is a whole number of degrees.
What students will learn
Know
- The angle at the centre is twice the angle at the circumference on the same arc.
- Angles on the same arc are equal; the angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral add to 180°.
Understand
- The theorem needs three cases because the centre can be inside, on a side of, or outside the angle.
- The angle on the minor arc is the supplement of the angle on the major arc.
Do
- Prove the centre-angle theorem in all three cases.
- Use the theorems to chase angles in a circle figure.
- Derive the alternate segment theorem from the semicircle.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the circle theorems correctly, including reflex angles (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe how the angle behaves on each arc and for each position of the centre.
- Criterion C. Communicating: proofs with every reason written and every case covered (worked examples 1 and 3).
- Criterion D. Applying mathematics in real-life contexts: the pond wall and surveyors.
The one thing to do with the figure
Make students predict, then reveal. Before touching the first figure, ask for two numbers, what P reads and what Q reads. Take a show of hands for equal or different.
In the second figure, ask students to name which case of the proof each position uses, before reading the caption.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the two surveyors. Collect predictions and reasons. |
| 12 min | First figure; key idea with the three cases; settle the hook; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or C, return to which arc P stands on. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 65° each; 63°; 360°. |
|---|---|
| Worked examples | The proof of case 3; 224° and 68°; the alternate segment proof. |
| Faded example | 113°. |
| Something to think about | A diameter: every point sees it at 90°. |
| 1. From the minor arc back to the centre | B. A is the major-arc angle; C is the reflex centre angle; D halves instead of doubling. |
| 2. Base angle of the centre triangle | 28°. (180 − 124) ÷ 2. |
| 3. Centre angle from a triangle in the circle | 142°. 2 × 71. |
| 4. A reflex centre angle from a cyclic quadrilateral | 270°. ABC = 45, ADC = 135, then 2 × 135. |
| 5. Centre angle from a tangent-chord angle | 128°. 2 × 64. |
| 6. A cyclic trapezium | 102°. 180 − 78. |
| 7. Two tangents from a point | 66°. 132 ÷ 2. |
| 8. A tangent and a cyclic quadrilateral | 48°. ADB = 52, BAD = 80, then 180 − 132. |
| 9. A tangent-chord angle seen from the minor arc | C. A is the alternate-segment angle; B is the centre angle; D is the angle between the chord and the radius. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gave the major-arc angle, the reflex centre angle, or halved instead of doubling. | Ask which arc P stands on, and whether the centre angle is twice or half the angle at the circumference. |
| 62 or 56 in question 2 | Gave the angle at the circumference, or the whole remaining angle. | Mark OA and OB equal on a sketch and name the triangle isosceles. |
| 76 or 71 in question 3 | Doubled the wrong angle, or gave the circumference angle. | Shade the chord AB and ask which inscribed angle stands on it. |
| 90 or 135 in question 4 | Doubled the wrong angle, or gave the angle at D. | Write ABC + ADC = 180 first, then ask which angle stands on the long arc. |
| 64 or 116 in question 5 | Gave the tangent-chord angle, or its supplement. | Write: alternate segment gives 64° at the circumference; the centre angle is twice. |
| 78 or 51 in question 6 | Gave the given angle, or halved. | Mark D and B as opposite vertices. |
| 132 or 114 in question 7 | Gave the centre angle, or the minor-arc angle. | Say which arc C is on before halving. |
| 52 or 28 in question 8 | Stopped at the alternate-segment angle, or used angle BCD inside triangle ABD. | Shade triangle ABD and name its three angles before using a theorem. |
| A, B or D in question 9 | Gave the alternate-segment angle, the centre angle, or the angle between the chord and the radius. | Shade the alternate segment, then ask: is P in it, or on the other side of TA? |
Questions worth asking
- Why does the proof need three cases? Listen for “the centre can be inside the angle, on a side, or outside, and the pictures differ”.
- Why does P on the minor arc read 180° minus half the centre angle? Listen for “the quadrilateral APBQ is cyclic and opposite angles add to 180°”.
- How does the alternate segment theorem follow from the semicircle? Listen for “the diameter through the point of contact makes a right angle with the tangent and a right angle in the semicircle”.
- Why does a reflex angle at the centre not break the theorem? Listen for “the proof used only that radii are equal; it never needed the angle to be less than 180°”.
Differentiation
A way in for anyone stuck
- Give the three diagrams of the cases, already drawn, with the isosceles triangles shaded.
- Provide a table: angle at centre, arc, angle at circumference; fill in for both arcs.
- Restrict to questions 1 to 4 at first.
Stretch
- Prove the cyclic quadrilateral theorem using the centre-angle theorem and the full turn at O, carefully including the case where one centre angle is reflex.
- Prove that the angle between two chords meeting inside a circle equals half the sum of the arcs they cut off.
Students learning in English as an additional language
“Arc” is a curved part of the circle. “Major” is the larger arc and “minor” the smaller. “Inscribed” means drawn inside with its corners on the circle. Use a drawn circle and colour each arc in turn.
| Word | Plain meaning | Example |
|---|---|---|
| major arc | the longer of the two arcs between two points on a circle | the long way round |
| minor arc | the shorter of the two arcs between two points on a circle | the short way round |
| subtend | to make an angle at a point from the two ends of a line | the chord subtends 140° at the centre |
| inscribed angle | an angle at the circumference between two chords | angle APB in the figure |
| alternate segment | the part of the circle on the other side of the chord from the tangent-chord angle | the side the angle does not open towards |
Sentence starters for explaining aloud or in writing:
- P is on the … arc, so the angle is …
- The centre is … the angle, so I use case …
- The two radii are equal, so the triangle is …
Two things not to say
“The angle at the circumference is always half.” Say which arc.
“Just learn the theorems.” Ask them to derive each from the centre-angle result.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- CentreThe angle at the circumference on an arc is 37°. Find the angle at the centre on the same arc.
74°. - Earlier work (Year 9)An isosceles triangle has base angles of 52°. Find its apex angle.
76°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- SemicircleAB is a diameter, C is on the circle and angle CAB = 39°. Find angle CBA.
51°. - Minor arcThe centre angle on a chord is 100°. Find the angle at a point on the minor arc.
130°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why a cyclic quadrilateral with a diameter as one side has two right angles.
Each of the other two vertices is in a semicircle. - OpenFind a quadrilateral with all four angles different that cannot be cyclic and say how you know.
Any whose opposite angles do not add to 180°, such as 70°, 80°, 95°, 115° in order: 70 + 95 = 165.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Compasses, a protractor and a ruler are enough to draw any of the cases.