At a glance. One lesson of about 60 minutes, with no calculator. It assumes the centre-angle theorem from Lesson 7. Students prove that opposite angles of a cyclic quadrilateral add to 180° and that the tangent-chord angle equals the angle in the alternate segment, and then solve mixed problems.
Calculator: no calculator. No calculator is needed: every angle is a whole number of degrees and the point is the reasoning and the algebra.
What students will learn
Know
- Opposite angles of a cyclic quadrilateral add to 180°.
- An exterior angle equals the interior opposite angle.
- The angle between a tangent and a chord equals the angle in the alternate segment.
Understand
- Both results follow from the centre-angle theorem and from the tangent being perpendicular to the radius.
- A cyclic quadrilateral's angles are halves of arcs, and opposite arcs make a whole circle.
Do
- Write each proof with a reason for each line.
- Use the results to find angles, with and without algebra.
- Combine them with the semicircle and isosceles-triangle results.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: cyclic quadrilateral and alternate segment results (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students explain why the sums of opposite angles stay the same when the arcs change.
- Criterion C. Communicating: proofs with stated reasons (worked examples 1 and 3) and solutions where each angle has a reason beside it.
- Criterion D. Applying mathematics in real-life contexts: the designer's posts round a pond (the hook) and the lamp posts on a plaza (extra questions).
The one thing to do with the figure
Ask students to find the invariant. In the first figure, ask: which numbers change as you move the arcs, and which stay the same? Let students write the sums for both opposite pairs on paper, before you reveal the rule; the figure prints the four angles but not their sums.
The second figure keeps P in the alternate segment. Ask students to sketch a point Q on the minor arc, on the same side as the angle, and to explain why angle TQA is 180° − θ (TPAQ is a cyclic quadrilateral).
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the four posts. Collect predictions. |
| 10 min | First figure; settle the hook; worked example 1 together. |
| 10 min | Second figure; worked example 3 together; the warning card. |
| 7 min | Worked example 2; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the sum 180. |
| 13 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 | 90°; 56°; x = 22. |
|---|---|
| Hook | 80°: opposite angles add to 180°. |
| Worked examples | Proof; 102° and 76°; proof. |
| Faded example | 110°. |
| 1. Opposite angles of a cyclic quadrilateral | B. A repeats the angle; C uses 90; D uses 360. |
| 2. Opposite angle | 85°. |
| 3. Alternate segment | 54°. |
| 4. Cyclic quadrilateral with algebra | 35. |
| 5. Exterior angle of a cyclic quadrilateral | 68°. |
| 6. Alternate segment in a triangle | 67°. |
| 7. A diameter in a cyclic quadrilateral | 120°. |
| 8. Alternate segment with algebra | 43°. x = 13. |
| 9. Opposite angle, with the angle at D in two parts | A. B repeats the angle at D; C and D use only part of it. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Repeated the angle, used 90, or used 360. | Ask: which angle is opposite? What do opposite angles add to? |
| 95 or 265 in question 2 | Said opposite angles are equal, or subtracted from 360. | Draw the quadrilateral, mark A and C, and write A + C = ? |
| 126 or 27 in question 3 | Used supplementary, or halved. | Mark both angles on the diagram: tangent-chord angle and the one in the alternate segment. |
| 15 or 36 in question 4 | Equated them, or dropped both constants. | Write the sum equals 180 in full, then collect terms. |
| 112 or 34 in question 5 | Gave the interior angle BCD, or halved. | Mark the straight line BCE and the opposite angle. |
| 115 or 132 in question 6 | Left out one angle of the triangle. | Mark all three angles of triangle ABC before solving. |
| 60 or 150 in question 7 | Stopped at ABC, or used the wrong vertex. | Find the right angle first, then ABC, then the opposite angle. |
| 13 or 39 in question 8 | Gave x, or left out the +4. | Underline what the question asks for. |
| B, C or D in question 9 | Treated opposite angles as equal, or used only part of the angle at D. | Shade the whole angle ADC first, then name the angle opposite it. |
Questions worth asking
- Why must the opposite angles of a cyclic quadrilateral add to 180°? Listen for “their centre angles make a full turn”.
- Why must a cyclic parallelogram be a rectangle? Opposite angles are equal and add to 180°, so each is 90°.
- Which segment is the alternate segment? The one on the other side of the chord from the angle you started with.
- How does the alternate segment proof use the tangent being perpendicular to the radius? It gives the base angle 90° minus the tangent-chord angle.
Differentiation
A way in for anyone stuck
- Give a cyclic quadrilateral template with the opposite pairs shaded in two colours.
- Provide the proofs with the reasons on cards to put in order.
- Restrict to numerical questions first, and bring in the algebra in the last two.
Stretch
- Prove that if opposite angles of a quadrilateral add to 180°, its corners lie on a circle.
- Two circles meet at P and Q, and a line through P meets the circles at A and B, and another through Q meets them at C and D. Show that AC is parallel to BD.
Students learning in English as an additional language
“Cyclic” means on a circle. “Alternate” means on the other side. “Opposite” means facing: A faces C. “Exterior” means outside. Point to each on a diagram as you say it, and keep the proofs on the board.
| Word | Plain meaning | Example |
|---|---|---|
| cyclic quadrilateral | a four-sided shape whose corners all lie on one circle | four posts round a round pond |
| opposite angles | angles at corners that do not share a side | A and C in ABCD |
| exterior angle | an angle formed outside a shape by extending one side | the angle at C after extending BC |
| alternate segment | the segment of a circle on the other side of a chord | the segment across the chord TA from the marked angle at T |
| tangent-chord angle | the angle between a tangent and a chord at the point of contact | the angle at T |
| converse | the result with its cause and effect swapped | if opposite angles add to 180, the corners are on a circle |
Sentence starters for explaining aloud or in writing:
- Angle … and angle … are opposite, so …
- The angle in the alternate segment is … because …
- I know the tangent is perpendicular to the …, so …
Two things not to say
“They are equal because they look equal.” Ask for the theorem and the reason.
“Opposite angles are equal.” That is parallelograms; here they add to 180°.
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.
- Cyclic quadrilateralA cyclic quadrilateral has an angle of 65°. What is the opposite angle?
115°. - Earlier work (Year 7)Three angles of a quadrilateral are 80°, 90° and 100°. Find the fourth.
90°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextFour lamp posts stand on the rim of a circular plaza and form a quadrilateral ABCD. The angle at A is 124°. What is the angle at C?
56°. - Working backA tangent and a chord make (2x + 5)° and the angle in the alternate segment is (3x − 10)°. Find the angle.
35°, with x = 15.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the exterior angle of a cyclic quadrilateral equals the interior opposite angle.
Interior + opposite = 180° and interior + exterior = 180°. - OpenA quadrilateral has angles 70°, 100°, 110° and 80° in order. Can it be cyclic? Give a reason.
Yes: 70 + 110 = 180 and 100 + 80 = 180.
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 circle drawn with a compass and four points joined by a ruler give a quadrilateral to measure. No calculator is used.