The shape of the lesson
By the end of the lesson, learners can
- Points, lines and vertically opposite: In the angle facts tab, drag the lines and watch the angle readouts.
- Parallel lines: In the parallel lines tab, drag the transversal to change its angle.
- Triangles and quadrilaterals: In the triangles and quadrilaterals tab, drag the vertices to any position.
- Mark every angle you find on the diagram as you go.
- Look for straight lines, points, crossings, triangles and parallel arrows.
- Write equations when angles are given in terms of \(x\).
- State the reason in full words: 'alternate angles are equal'.
- Check: every triangle 180°, every straight line 180°.
One picture every idea returns to
Angles at a point add to \(360^\circ\); on a straight line to \(180^\circ\); vertically opposite angles are equal; a triangle's angles add to \(180^\circ\) and a quadrilateral's to \(360^\circ\). With parallel lines, corresponding and alternate angles are equal and co-interior angles add to \(180^\circ\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Every angle has a reason.. Seven facts unlock almost every angle question: angles at a point, on a straight line, vertically opposite, in a triangle, in a quadrilateral, and the three parallel-line rules. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Points, lines and vertically opposite | In the angle facts tab, drag the lines and watch the angle readouts. Project tab 2 of the lab (Angles lab, angle facts) and let a student drive it while the class predicts. Model the worked example On a straight line on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Parallel lines | In the parallel lines tab, drag the transversal to change its angle. Project tab 3 of the lab (Angles lab, parallel lines) and let a student drive it while the class predicts. Model the worked example Co-interior angles on the board, one line at a time. Your turn: In Fig. 1, explain why the angle below-left of P is \(65^\circ\), using a parallel-line reason. | Pairs |
| 21 to 30 min | Teach and check: Triangles and quadrilaterals | In the triangles and quadrilaterals tab, drag the vertices to any position. Project tab 4 of the lab (Angles lab, triangles and quadrilaterals) and let a student drive it while the class predicts. Model the worked example A quadrilateral on the board, one line at a time. Your turn: Triangle ABC is isosceles with AB = AC and angle BAC = \(36^\circ\). Find angle ABC. | Round Table |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 17 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Triangles and quadrilaterals, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Write the reason in words, using the proper term: 'angles on a straight line add to 180°'. | A correct number with no reason, or with 'because it is', often loses the reason mark. |
| Co-interior angles are NOT equal: they add to \(180^\circ\). | Setting \(2x + 15 = 3x + 15\) gives \(x = 0\), a sure sign the wrong rule was used. |
| In an isosceles triangle, the equal angles are the ones OPPOSITE the equal sides, not the angle between them. | Check which sides are marked equal before you halve anything. |
Angles: lines, parallel lines and triangles: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Four angles meet at a point. They are \(105^\circ\), \(87^\circ\), \(2x\) and \(x\). Find \(x\).
The angles \(4x\), \(5x\) and \(27^\circ\) lie on a straight line. Find \(x\).
Two parallel lines are crossed by a transversal. One of the angles formed is \(58^\circ\).
Find (a) the angle corresponding to it, (b) the angle co-interior with it, (c) the angle vertically opposite the co-interior angle. Give a reason for each.
The angles of a quadrilateral are \(x\), \(x + 30^\circ\), \(2x\) and \(2x - 30^\circ\). Find each angle.
Triangle ABC has interior angles \(a\), \(b\) and \(c\) at A, B and C. Side BC is extended beyond C to D.
Prove that the exterior angle ACD is equal to \(a + b\).
PQ and RS are parallel lines, with Q to the right of P and S to the right of R. Point T lies between the lines, to the right of P and R. Angle QPT = \(38^\circ\) and angle SRT = \(47^\circ\).
Find angle PTR, giving reasons.
Answer key and marking notes
- Angles at a point add to \(360^\circ\): \(105 + 87 + 3x = 360\).
- \(3x = 168\), so \(x = 56\).
- Angles on a straight line add to \(180^\circ\): \(9x + 27 = 180\).
- \(9x = 153\), so \(x = 17\).
- (a) \(58^\circ\): corresponding angles are equal.
- (b) \(180^\circ - 58^\circ = 122^\circ\): co-interior angles add to \(180^\circ\).
- (c) \(122^\circ\): vertically opposite angles are equal.
- Angle sum of a quadrilateral is \(360^\circ\): \(6x = 360\), so \(x = 60\).
- The angles are \(60^\circ\), \(90^\circ\), \(120^\circ\) and \(90^\circ\). Check: total \(360^\circ\).
- Angles in a triangle add to \(180^\circ\): \(a + b + c = 180^\circ\), so \(a + b = 180^\circ - c\).
- BCD is a straight line, so angle ACD \(= 180^\circ - c\) (angles on a straight line).
- Therefore angle ACD \(= a + b\).
- Draw a line through T parallel to PQ and RS, extending LEFT from T to a point U (towards the side of P and R).
- Angle PTU = angle QPT = \(38^\circ\) (alternate angles).
- Angle RTU = angle SRT = \(47^\circ\) (alternate angles).
- Angle PTR \(= 38^\circ + 47^\circ = 85^\circ\).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Two straight lines cross. Two vertically opposite angles are \(3x + 10^\circ\) and \(5x - 30^\circ\). Find \(x\).
- A\(20\)
- B\(25\)
- C\(47.5\)
- D\(10\)
The diagram shows two parallel lines crossed by a transversal. Find the angle marked \(x\).
- A\(116^\circ\)
- B\(64^\circ\)
- C\(26^\circ\)
- D\(244^\circ\)
One angle of an isosceles triangle is \(110^\circ\). Find the size of each of the other two angles.
- A\(35^\circ\)
- B\(110^\circ\)
- C\(70^\circ\)
- D\(55^\circ\)
A side of a triangle is extended. The exterior angle formed is \(128^\circ\). One of the two interior angles opposite it is \(51^\circ\). Find the other one.
- A\(77^\circ\)
- B\(52^\circ\)
- C\(1^\circ\)
- D\(179^\circ\)
In triangle ABC, AB = AC. The side BC is extended beyond C to D, and angle ACD \(= 112^\circ\). Find angle BAC.
- A\(44^\circ\)
- B\(68^\circ\)
- C\(56^\circ\)
- D\(34^\circ\)
The angles of a quadrilateral are \(x + 20^\circ\), \(2x\), \(3x - 10^\circ\) and \(110^\circ\). Find the smallest angle.
- A\(60^\circ\)
- B\(40^\circ\)
- C\(30^\circ\)
- D\(90^\circ\)
Quiz key
\(3x + 10 = 5x - 30\), so \(40 = 2x\) and \(x = 20\) (both angles are \(70^\circ\)). The tempting 25 treats them as angles on a straight line, \(8x - 20 = 180\). 47.5 sets their sum to \(360^\circ\), and 10 comes from a sign slip, \(2x = 20\).
The two angles lie between the parallel lines on the same side of the transversal, so they are co-interior angles and add to \(180^\circ\): \(x = 180^\circ - 64^\circ = 116^\circ\). The tempting \(64^\circ\) treats them as alternate angles, which are on OPPOSITE sides of the transversal. \(26^\circ\) subtracts from \(90^\circ\), and \(244^\circ\) is the reflex angle at that point.
Two angles of \(110^\circ\) would add to more than \(180^\circ\), so \(110^\circ\) must be the angle between the equal sides. The other two are equal: \((180^\circ - 110^\circ) \div 2 = 35^\circ\). The tempting \(110^\circ\) makes it one of the base angles, which is impossible. \(70^\circ\) forgets to halve, and \(55^\circ\) halves the \(110^\circ\).
\(128^\circ = 51^\circ + y\), so \(y = 77^\circ\). Check: the third angle is \(180^\circ - 128^\circ = 52^\circ\), and \(51^\circ + 77^\circ + 52^\circ = 180^\circ\). The tempting \(52^\circ\) is the interior angle NEXT to the exterior angle, not the one opposite. \(1^\circ\) subtracts both \(128^\circ\) and \(51^\circ\) from \(180^\circ\), and \(179^\circ\) adds them.
Angle ACB \(= 180^\circ - 112^\circ = 68^\circ\) (angles on a straight line). AB = AC, so angle ABC = angle ACB \(= 68^\circ\) (base angles of an isosceles triangle). Angle BAC \(= 180^\circ - 68^\circ - 68^\circ = 44^\circ\). The tempting \(68^\circ\) stops at angle ACB. \(56^\circ\) halves the exterior angle, and \(34^\circ\) shares the \(68^\circ\) between A and B as if they were the equal angles.
\(x + 20 + 2x + 3x - 10 + 110 = 360\), so \(6x + 120 = 360\) and \(x = 40\). The angles are \(60^\circ, 80^\circ, 110^\circ, 110^\circ\), so the smallest is \(60^\circ\). The tempting \(40^\circ\) is \(x\), not an angle. \(30^\circ\) uses an angle sum of \(180^\circ\) (\(x = 10\)), and \(90^\circ\) uses \(540^\circ\) (\(x = 70\)).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.