At a glance. One lesson of about 60 minutes, with no calculator. It assumes angle facts and isosceles triangles. Students prove the centre-angle theorem, derive the same-segment and semicircle results, and apply them with algebra.
Calculator: no calculator. No calculator is needed: every angle is a whole number of degrees and the point is the reasoning.
What students will learn
Know
- The angle at the centre is twice the angle at the circumference on the same arc.
- Angles at the circumference on the same arc are equal.
- The angle in a semicircle is 90°.
Understand
- The theorem is a chain of two isosceles triangles and two exterior angles, so it is proved, not remembered.
- The semicircle and same-segment results are special cases of the centre-angle theorem.
Do
- Write the proof with a reason for each line.
- Apply the theorem to find angles, with and without algebra.
- Combine the theorem with isosceles triangles and Pythagoras.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the circle theorems and their use with algebra (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe the pattern that APB stays fixed and why it is half of AOB.
- Criterion C. Communicating: a proof where each line has a stated reason, using the standard names of the theorems (worked example 1).
- Criterion D. Applying mathematics in real-life contexts: the pond and the posts.
The one thing to do with the figure
Move P first, then ask. In the first figure, let students move P through all nine positions before they say anything about the angle. Ask them to predict the centre angle's effect, then set it to 180°.
In the second figure, ask students to name which triangles are isosceles and which theorem gives 2x, before reading the text.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the surveyor on the pond. Collect predictions. |
| 10 min | First figure; settle the hook; the corollaries. |
| 12 min | Second figure and the proof, worked example 1 line by line. |
| 7 min | Worked examples 2 and 3; pairs finish the faded example; the warning card. |
| 5 min | Question 1, the check. If many choose A or B, return to which angle is larger. |
| 11 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 | 52°; 55°; 75°. |
|---|---|
| Hook | It stays the same, at half the centre angle. |
| Worked examples | Proof; 58° and 58°; 66°. |
| Faded example | 56°. |
| 1. Angle at the centre and the circumference | C. A doubles; B repeats the centre angle; D takes it from 180. |
| 2. Angle at the circumference | 65°. |
| 3. Angle in a semicircle | 55°. |
| 4. Angle theorem with algebra | 20. Then ACB = 60°. |
| 5. Semicircle with algebra | 45°. x = 15. |
| 6. Same segment, then the centre | 76°. |
| 7. Isosceles then the circumference angle | 70°. |
| 8. A semicircle with Pythagoras | 8 cm. |
| 9. Centre angle from a circumference angle | B. A repeats the angle; C uses 90 − 31; D uses 180 − 31. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Doubled, repeated the angle, or subtracted from 180. | Say: which angle is bigger, the one at the centre or at the circumference? |
| 260 or 50 in question 2 | Doubled, or took 180 away. | Put the two angles on the diagram first, and label the larger as the centre angle. |
| 145 or 35 in question 3 | Ignored the right angle, or treated the triangle as isosceles. | Mark the 90° at P before using the angle sum. |
| 50 or 200 in question 4 | Used equal angles, or doubled the wrong one. | Write centre = 2 × circumference with the expressions in place. |
| 99 or 15 in question 5 | Forgot the right angle, or gave x. | Underline the angle the question wants before answering. |
| 38 or 19 in question 6 | Repeated, or halved. | Ask: is the centre angle bigger or smaller than the circumference angle? |
| 140 or 80 in question 7 | Stopped at the centre angle, or took only one base angle from 180. | Draw triangle OAB first, mark OA = OB, then use the theorem. |
| 4 or 11.7 in question 8 | Subtracted lengths, or added squares. | Mark the right angle at C and find the hypotenuse. |
| A, C or D in question 9 | Repeated, used 90 − 31, or used 180 − 31. | Ask which theorem applies: is AC a diameter? Is there a right angle? |
Questions worth asking
- Why does the angle APB not depend on where P is on the arc? Listen for “it is always half of AOB, and AOB does not move”.
- What special case do you get when AB is a diameter? The centre angle is 180°, so the circumference angle is 90°.
- Which two earlier facts does the proof use? Isosceles triangles and the exterior angle of a triangle.
- What if the centre is outside the angle APB? Does the theorem still hold? Yes, but the proof is different: subtract rather than add.
Differentiation
A way in for anyone stuck
- Give the proof with each line in a separate box and the reasons on separate cards to match.
- Provide a diagram with OP, OA, OB marked as equal and the angles x and y placed.
- Restrict to the first four questions and the check, then add algebra.
Stretch
- Prove the theorem when PA is a diameter, so there is only one isosceles triangle. (AOB is the exterior angle of triangle OPB, so AOB = 2y = 2 × APB.)
- Show that if the angle subtended at the circumference by a chord is 90° for every point on the major arc, the chord is a diameter.
Students learning in English as an additional language
An “angle at the circumference” has its vertex on the circle. “Subtend” means to make an angle with: a chord subtends an angle at a point. “Segment” is the region between a chord and an arc. “Semicircle” is half a circle. Draw each on the board and name the two arcs.
| Word | Plain meaning | Example |
|---|---|---|
| subtend | to make an angle at a point from two other points | a chord subtends an angle at the centre |
| circumference | the edge of a circle, or the distance round it | the rim of a plate |
| segment | the region of a circle between a chord and its arc | the cap cut off by a straight cut |
| semicircle | half of a circle, cut off by a diameter | half a plate |
| major arc | the longer arc between two points on a circle | the long way round |
| corollary | a result that follows quickly from one already proved | the semicircle result |
Sentence starters for explaining aloud or in writing:
- The angle at the centre is … because …
- These two angles are equal because they are in the same …
- Triangle OAB is isosceles because …
Two things not to say
“Just learn the theorems.” Ask students to prove one in their own words.
“It looks equal.” A drawing is evidence, not proof.
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.
- Centre angleThe angle at the centre is 70°. What is the angle at the circumference on the same arc?
35°. - Earlier work (Year 7)Two angles of a triangle are 50° and 70°. Find the third.
60°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo posts A and B on the rim of a round pond subtend 100° at the centre. What angle do they subtend at a point on the far rim?
50°. - Working backAB is a diameter and P is on the circle with AP = 12 cm and PB = 5 cm. Find the radius.
6.5 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningUse the centre-angle theorem to explain why angles in the same segment are equal.
Both are half of the same centre angle. - OpenFind a case with a reflex centre angle and say what angle it subtends on the minor arc.
For example 200° gives 100°.
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? A paper circle, a pin at the centre and two strings give the angles to measure with a protractor. No calculator is used.