Finding angles with parallel lines

Classroom resources

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Students will be able to

• I can state the rules: with parallel lines, corresponding and alternate angles are equal and co-interior angles add to 180°.
• I can find an unknown angle using one of these rules, and say which rule.
• I can find angles in two steps, giving a reason for each step.

Timing

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

Before class

Do the Run board yourself first. Students need the pair names (Angles at parallel lines, the lesson before), angles on a straight line and vertically opposite angles (Year 6).

The one idea: with parallel lines, one angle gives all eight, each by a named rule. The common mistake is making co-interior angles equal, because the other two pairs are; it is planted in the Predict and again in Leo's working.

Answers are numbers; reasons are asked for in the worked example, the explanations and one question. Ask for them aloud too: an angle without a reason cannot be checked.

Running the lesson

1. Predict: take a show of hands. Students who chose 65° are using 'equal' for every pair. Do not correct it yet.
2. Run: after setting c = 65°, ask students to turn the line and say what stays true (c + e = 180°).
3. Investigate: model the worked example live, writing the reason beside each angle. Then ask: could you have found e another way?
4. Modify: in the two-step task, collect different routes to g and the reasons each uses.
5. Make: ask why c and e can only be equal at 90°.

When students get stuck

Answers to the checks

1. a co-interior pair: Two parallel lines are crossed by a transversal. $c = 65^\circ$, and $e$ is co-interior with $c$. What is $e$?
   115°. Co-interior angles are not equal: they add to 180°, so e = 180° − 65° = 115°.
2. test it: Turn the crossing line until $c = 65^\circ$, then pick $c$ and $e$. Was your prediction right?
   
3. Equal pairs: Find $f$.
   72
   b and f are corresponding angles, so f = 72°.
4. Find the mistake: Leo finds $f$ when $d = 110^\circ$. Click the line where his working goes wrong.
   line 4
   Line 4: co-interior angles add to 180°, so f = 180° − 110° = 70°. Only corresponding and alternate angles are equal.
5. Co-interior: Find $f$ correctly.
   70
   Co-interior angles add to 180°: f = 180° − 110° = 70°.
6. Alternate: Alternate: find $f$.
   50
   Alternate angles are equal: f = 50°.
7. Co-interior from below: Co-interior, starting at the bottom: find $c$.
   55
   Co-interior angles add to 180°: c = 180° − 125° = 55°.
8. Two steps: Two steps: find $g$.
   40
   e = 140° (corresponding to a), then g = 180° − 140° = 40° (a straight line). Or: b = 40° (a straight line), f = 40° (corresponding), g = 40° (vertically opposite).
9. Check a chain: Turn the crossing line until $h = 130^\circ$. Work out $b$ first, then pick $h$ and $b$ to check.
   
   f = 50° (a straight line with h), so b = 50° (corresponding).
10. Not parallel: Edge case: can you use "alternate angles are equal" to find $f$ here?
    No: the lines are not parallel
    The rules hold only for parallel lines. Look for the arrow marks (>>) before you use them.
11. Corresponding: Find $c$.
    80
    Corresponding angles are equal: c = 80°.
12. Co-interior: Find $d$.
    118
    Co-interior angles add to 180°: d = 180° − 62° = 118°.
13. The reason: $e = 120^\circ$ and $c = 60^\circ$. Which reason links them?
    Co-interior angles add to 180°
    c and e are between the lines on the same side, and 60° + 120° = 180°.
14. Review: names: Review: which pair is alternate?
    $d$ and $e$
    d and e: between the lines, on opposite sides.
15. equal co-interior angles: Turn the crossing line so that the co-interior angles $c$ and $e$ are equal.
    
    They add to 180°, so equal means 90° each: the crossing line is perpendicular to the parallel lines.

Solutions

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

one change at a time › Alternate

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

one change at a time › Co-interior from below

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

one change at a time › Two steps

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

one change at a time › Check a chain

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.

equal co-interior angles

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

Stretch and support

Stretch: In a diagram, a = 3x and f = 2x + 30 with parallel lines (a and f are not a named pair: use a straight line first). Find x. (b = 180 − 3x = f, so x = 30.)

Support: Use the Run board to check each answer: set the given angle, pick the pair and read the size.

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 "Finding angles with parallel lines".

The idea: Properties of corresponding, alternate and co-interior angles: I can Students state and apply the angle properties: corresponding angles are equal, alternate angles are equal, co-interior angles are supplementary.

What understanding looks like:
- States each angle property correctly.
- Calculates missing angles using the correct property.
- Does not apply the equal-angle property to co-interior pairs.

Mistakes students often make here:
- A student makes co-interior angles equal (Leo's mistake).
- A student finds the right size but cannot name the reason.
- A student uses the rules on lines that are not parallel.

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.
- If I seem to be missing the idea that comes before this one (Parallel lines and transversals), check it quickly, then come back.
- 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 "Finding angles with parallel lines".

Exit ticket

A transversal crosses two parallel lines. One angle is 75°. Find the angle corresponding to it, and the angle next to that one on a straight line, with a reason for each.

75° (corresponding angles are equal), then 180° − 75° = 105° (angles on a straight line).

What each step is for

• a co-interior pair: Committing to e when c = 65° tests whether they apply 'equal' to every pair, the most common slip with co-interior angles.
• test it: Setting c to 65° and picking c and e shows the co-interior pair adding to 180°, not equal; turning the line shows it holds for every angle.
• from one angle to all eight: Each angle follows from a known one by a named rule: corresponding or alternate (equal), co-interior or a straight line (add to 180°), vertically opposite (equal). Leo's mistake makes co-interior angles equal.
• one change at a time: One change at a time: alternate, co-interior from the lower crossing, two steps, the board to check a chain, then the edge case where the lines are not parallel.
• angle and reason: Finding angles with each rule and naming the reason, mixed, with the co-interior slip planted again; the review keeps the pair names fresh.
• equal co-interior angles: Making a co-interior pair equal reverses the rule: two angles that add to 180° are equal only when both are 90°.

Where it leads

Next: angle sums of triangles and polygons (the next unit). Then Reasoning with angles (AC9M7M04 node 3) and Unknown angles in shapes (AC9M7M05 node 3) chain these facts with reasons.

Where it leads: geometric proof in Years 8 to 10 (congruent triangles, properties of quadrilaterals).