At a glance. One lesson of about 60 minutes. Students find corresponding, alternate and co-interior angles with reasons, solve equations from them, and use equal corresponding angles as the test for parallel lines.
Calculator: no calculator. Nothing here needs one. The arithmetic is subtraction from 180 and one- and two-step equations.
What students will learn
Know
- Corresponding and alternate angles are equal and co-interior angles add up to 180° when lines are parallel.
- Arrows mark parallel lines.
- Equal corresponding angles show that two lines are parallel.
Understand
- The pairs are consequences of one idea: a crossing copied along the transversal, plus vertically opposite angles and a straight line.
- The facts are conditional: without parallel lines they do not hold.
- A reason makes a step in a chain valid.
Do
- Name pairs of angles from a diagram.
- Find missing angles in two or more steps and give reasons.
- Solve equations that come from angle pairs.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: naming pairs and finding angles (questions 1 to 5).
- Criterion B. Investigating patterns: the tilt figure. Students record the angles for three tilts and say which pairs are always equal, then state a general rule.
- Criterion C. Communicating: writing a reason for each step in an angle chase and using the shapes as hints, not as reasons (worked example, questions 6 and 8).
- Criterion D. Applying mathematics in real-life contexts: stripes and slanted bands in weaving, road crossings and design, where parallel is assumed or checked.
The one thing to do with the figure
Ask for the position before the number. “Where is the matching angle at the bottom?” Students point first and then check with the button.
In the second figure, set the tilt to 0 and ask why the angles are equal. Then move to 1 and ask what has gone: the equal angles. Lines that are almost parallel are not parallel, and equal corresponding angles is the test.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure: students predict the matching angle, then press each button. |
| 10 min | Key idea with the reasons for F, Z and C; the second figure on the test for parallel. |
| 10 min | Worked example; students finish the faded one in pairs, with a reason for each line. |
| 5 min | Question 1, the check. If many choose C, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 110°; 50; x = 30. |
|---|---|
| Worked example | 52°, 52° and 128°. |
| Faded example | 117°, 117° and 63°. |
| 1. Which angle is equal | A. B and C are the two ways of getting 110°; D forgets that some of the angles are the 110° ones. |
| 2. Corresponding angles | 112. Corresponding angles are equal. |
| 3. Co-interior angles | 105. 180 − 75 = 105. |
| 4. Two-step angle chase | 132. b = 180 − 48 = 132, and c = b. |
| 5. Alternate angles with an unknown | 30. 2x = 60, x = 30. |
| 6. The reason that earns the mark | B. A gives the shape instead of the name, C gives the look of the diagram, D names the wrong pair. |
| 7. Co-interior with unknowns | 25. 6x + 30 = 180, 6x = 150, x = 25. |
| 8. Which pair is it? | 110. x = 30, and 3 × 30 + 20 = 110. |
| 9. Always equal | C. A, B and D all need parallel lines; C needs only the crossing. If many choose A, return to the second figure. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Mixes co-interior with corresponding, or picks a neighbour on the line. | Have them mark the F, Z and C on one diagram in three colours. |
| 68 in question 2 | Takes the neighbour on a line for the corresponding angle. | Ask where the angle sits relative to the transversal and line. |
| 75 or 15 in question 3 | Treats co-interior as equal, or uses 90. | Draw the pair as a C and write the total 180 beside it. |
| 48 or 42 in question 4 | Stops after one step, or uses 90 for a line. | Ask them to name each step's reason before calculating it. |
| 35 or 40 in question 5 | Mishandles the +10 when solving. | Check by putting x back into 2x + 10. |
| A, C or D in question 6 | Gives the letter shape, or the look of the diagram, as the reason. | Ask which words would convince someone who has not seen the diagram: the pair name and “parallel lines”. |
| 30 or 35 in question 7 | Drops the constants or adds them to the wrong side. | Collect like terms in two columns first. |
| 30 or 95 in question 8 | Gives x instead of the angle, or treats alternate as co-interior. | Ask what the question asks for, and which pair name was used. |
Questions worth asking
- Why are corresponding angles equal when the lines are parallel? The second line is a copy of the first moved along the transversal. Listen for “slide” or “same slant”.
- Which of the three pair names do you actually need? Any two will do, because the third follows from them and a straight line. Which is quickest depends on the diagram.
- Could the equal angles be a coincidence in a drawing? In a drawing, yes; in a proof, no. Arrows or given angles are the evidence.
- What is special about a 90° transversal? All eight angles are 90°; no rule breaks.
Differentiation
A way in for anyone stuck
- Provide a printed diagram with the F, Z and C pairs in three colours and the reason sentence beside each.
- Give tracing paper: students trace an angle at the top and slide it down to see it fit.
- Reduce to one diagram with one given angle and ask for one other angle at a time.
Stretch
- Question 8, then: set up the same angles as co-interior and find x. (8x − 20 = 180, so x = 25 and the angles are 95° and 85°.)
- Give a diagram with a transversal and two lines that are not parallel and ask by how many degrees they differ from parallel.
Students learning in English as an additional language
Say each pair name with its shape drawn: corresponding (F), alternate (Z), co-interior (C). “Transversal” is a new word; link it to “transport”, to carry across.
| Word | Plain meaning | Example |
|---|---|---|
| transversal | a line that crosses two or more other lines | The diagonal band across two stripes |
| corresponding angles | angles in the same position at the two crossings | The F shape; equal for parallel lines |
| alternate angles | angles inside the lines on opposite sides of the transversal | The Z shape; equal for parallel lines |
| co-interior angles | angles inside the lines on the same side of the transversal | The C shape; add to 180° for parallel lines |
| parallel | in the same direction and never meeting | Two rails of a track |
| converse | the same statement the other way round | If angles are equal, the lines are parallel |
Sentence starters for explaining aloud or in writing:
- … and … are corresponding angles, so they are … because …
- The lines are parallel because …
- These angles add up to 180° because they are …
Two things not to say
“It's a Z, so it's equal.” The shape is a hint to find the pair; the reason is that the lines are parallel and the angles are alternate.
“These lines are parallel, I can see.” Seeing is not a reason. Arrows or given equal angles are.
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.
- PairsParallel lines, one angle 64°. Write the corresponding, alternate and co-interior angles at the other crossing.
64°, 64° and 116°. - EquationsAlternate angles are (x + 15)° and 80°. Find x.
x = 65.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ChainTwo parallel lines are crossed by a transversal; one acute angle is 38°. What are all eight angles?
Four angles of 38° and four of 142°. - TestCorresponding angles are 73° and 71°. Are the lines parallel? What do you expect to see further away?
No; they would meet on one side, where the lines lean together.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningFrom Unit 4, lesson 2: two lines cross and one angle is (5x − 20)° while the opposite angle is (3x + 10)°. Find x.
x = 15. - GeneralizingA transversal makes an angle t with two parallel lines. Write the other seven angles in terms of t.
Four are t and four are 180 − t.
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. Under each one, “Show me the steps” gives the same states in words.
No device? Use two rulers and a pencil on squared paper.