Angles at parallel lines

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

• Slides Download Copy address for Prepare Present from the browser, or export to PowerPoint or PDF.
• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can find corresponding, alternate and co-interior angles where a line crosses two parallel lines.
• I can tell these pairs apart from vertically opposite angles.
• I can explain that the names say where the angles are, and parallel lines make the pairs equal or add to 180°.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board and the Investigate board yourself first. Students need angles on a straight line and vertically opposite angles (Year 6).

The one idea: each pair takes one angle from each crossing, and the name says where they are (same place; between and opposite sides; between and same side). The common mistake is naming a pair at one crossing (vertically opposite) as alternate; it is planted after the names, in Mia's reasoning.

The boards in Investigate, Modify and Make do not name the pairs: students pick, and the check says whether the pair is right. F, Z and C shapes are given as memory aids only; the Modify board leans the other way so students do not rely on them.

Running the lesson

1. Predict: take a show of hands. Students who chose 'one grows, the other shrinks' may be thinking of a and b (a straight line).
2. Run: ask students to say why the bottom crossing looks like the top one (the lines go the same way).
3. Investigate: draw the eight angles on the board and colour each pair as you name it. Ask: does each pair use both crossings?
4. Modify: before the tilt task, ask what changes if the lines are not parallel. Let the board show it.
5. Make: collect the pairs students found and ask why only the pairs between the lines always work.

When students get stuck

Answers to the checks

1. turning the crossing line: $a$ is above the top line, on the left of the crossing line. $e$ is in the same place at the bottom crossing. If you turn the crossing line, what happens to $a$ and $e$?
   They stay the same size as each other. The parallel lines cross the transversal the same way, so the bottom crossing is a copy of the top one, slid along. a and e are corresponding angles.
2. turn the crossing line: $a$ and $e$ are picked. Turn the crossing line ($t$) to $120^\circ$ and watch both. Was your prediction right?
   
3. Pick a pair: This board does not name the pairs. Pick two alternate angles.
   
   c and f, or d and e: between the lines, on opposite sides of the crossing line.
4. Find the mistake: Mia names a pair of angles. Click the line where her reasoning goes wrong.
   line 3
   Line 3: b and c are at the same crossing, opposite each other, so they are vertically opposite. Alternate angles are one at each crossing, between the parallel lines. (Vertically opposite angles are equal too, so line 4's b = c is true, for a different reason.)
5. Co-interior: Which pair is co-interior?
   $c$ and $e$
   c and e are both between the lines and both on the left of the crossing line: co-interior.
6. Corresponding: The crossing line leans the other way now. Pick two corresponding angles.
   
   a and e, b and f, c and g, or d and h.
7. Co-interior: Pick two co-interior angles.
   
   c and e, or d and f.
8. Vertically opposite: Pick two vertically opposite angles.
   
   a and d, b and c, e and h, or f and g: at one crossing, opposite each other.
9. Not parallel: The lower line is tilted by $w$, so the lines are not parallel. Are $c$ and $f$ still alternate angles?
   Yes: they are in alternate places, but they need not be equal
   The names say where the angles are. Only when the lines are parallel are alternate (and corresponding) angles equal, and co-interior angles add to 180°.
10. A right angle: Edge case: turn the crossing line until all eight angles are the same size.
    
    At t = 90° the crossing line is perpendicular to the parallel lines: all eight angles are right angles.
11. Name it: $d$ and $e$ are
    alternate
    Between the lines, on opposite sides: alternate.
12. Name it again: $b$ and $f$ are
    corresponding
    Both above their line, on the right: the same place at each crossing, so corresponding.
13. One crossing: $a$ and $d$ are
    vertically opposite
    At one crossing, opposite each other: vertically opposite.
14. Review: a straight line: Review: $a$ and $b$ are on a straight line. $a = 115^\circ$. What is $b$?
    65
    Angles on a straight line add to 180°: 180 − 115 = 65°.
15. a pair that adds to 180°: Pick two angles between the parallel lines, at different crossings, that add to $180^\circ$.
    
    c and e, or d and f: co-interior angles. (At t = 90° the alternate pairs c and f, d and e add to 180° too, but co-interior pairs always do.)

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Corresponding

Open task: any program that does what the step asks.

one change at a time › Co-interior

Open task: any program that does what the step asks.

one change at a time › Vertically opposite

Open task: any program that does what the step asks.

one change at a time › Not parallel

Open task: any program that does what the step asks.

one change at a time › A right angle

Open task: any program that does what the step asks.

a pair that adds to 180°

Open task: any program that does what the step asks.

Stretch and support

Stretch: Draw a diagram with three parallel lines and one transversal. How many corresponding pairs are there? (Each of the 4 positions gives a pair for each two of the three lines (top and middle, middle and bottom, top and bottom): 4 × 3 = 12 pairs.)

Support: Use the board with names on (the Run board): pick pairs and read the names before trying the boards without them.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Angles at parallel lines".

The idea: Parallel lines and transversals: I can Students identify parallel lines and transversals in diagrams and name the eight angles formed.

What understanding looks like:
- Correctly identifies and labels the eight angles formed.
- Correctly identifies at least one example of each: corresponding, alternate, co-interior pair.
- Does not confuse vertically opposite angles with the parallel line angle relationships.

Mistakes students often make here:
- A student names b and c as alternate (Mia's mistake).
- A student mixes up alternate and co-interior.
- A student cannot find pairs when the diagram is turned.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Angles at parallel lines".

Exit ticket

A transversal crosses two parallel lines. Which angle pairs are equal, and which pair adds to 180°?

Corresponding angles and alternate angles are equal (so are vertically opposite angles); co-interior angles add to 180°.

What each step is for

• turning the crossing line: Committing to what happens to a and e (the same place at each crossing) as the line turns tests whether they see the two crossings as copies of each other.
• turn the crossing line: Turning the transversal with a and e picked shows that the same place at each crossing gives the same size, whatever the angle.
• the three pairs: Corresponding (same place), alternate (between, opposite sides) and co-interior (between, same side) each take one angle from each crossing. Mia's mistake names a pair at one crossing, which is vertically opposite.
• one change at a time: One change at a time: the crossing line leaning the other way (corresponding, co-interior, then vertically opposite to keep it apart), lines that are not parallel, then the edge case of a right angle.
• name the pair: Naming pairs from a fixed diagram, mixed, with vertically opposite as the planted mix-up; the review keeps angles on a straight line fresh.
• a pair that adds to 180°: Finding a pair between the lines that adds to 180° reverses the lesson: from what the angles do back to where they are.

Where it leads

Next: finding angles with parallel lines (node 2), which adds the rules: corresponding and alternate equal, co-interior add to 180°.

Where it leads: angle sums of triangles and polygons (the next unit), and geometric reasoning with reasons in Years 8 to 10.