Reasoning with angles

Classroom resources

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Students will be able to

• I can find an unknown angle in two or more steps, using parallel-line, straight-line and vertically opposite angles.
• I can give a reason for each step, naming the rule.
• I can use angles to show whether two lines are parallel.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

Before class

Do the first reasoning task yourself first, then compare with its model. Students need both parallel-lines lessons and Year 6 angle facts (straight line, vertically opposite, at a point).

The one idea: know, choose the rule, say why. Every angle found needs a named reason. The common mistake is linking two angles that only look related (Kai's b and e); it is planted after the routine.

The reasoning tasks do not mark the free working: students write it their own way (notes, abbreviations such as "vert. opp." and "co-int." are fine). The answer is marked, and at one key step students choose the reason from a list, so the misconception gets feedback. Then they compare their working with a model and tick a short self-check. Look at the working in class: it is where reasoning shows.

Running the lesson

1. Predict: take a show of hands. Students who chose 70° are assuming b and h are linked directly.
2. Run: ask students to say the two rules that make h = 180° − b true for every crossing angle.
3. Investigate: model know, choose, say why on the board, writing the reason in short form. Accept any correct route.
4. Modify: collect two different routes to f and ask which is shorter.
5. Make: ask students to say their two steps before they turn the line.

When students get stuck

Answers to the checks

1. two steps: Two parallel lines are crossed by a transversal. $b = 70^\circ$. What is $h$?
   110°. b and h are not a named pair, so it takes two steps: f = 70° (corresponding to b), then h = 180° − 70° = 110° (f and h are on a straight line).
2. check the chain: Turn the crossing line until $b = 70^\circ$, then pick $b$ and $h$. Was your prediction right?
   
3. Know, choose, say why: Find $h$. Write each step with its reason.
   $f = 70^\circ$ (corresponding to $b$: corresponding angles are equal); $h = 180^\circ - 70^\circ = 110^\circ$ ($f$ and $h$ are on a straight line)
   f = 70° (corresponding to b), h = 110° (a straight line with f). Another route: d = 110° (a straight line with b), h = 110° (corresponding to d).
4. Find the mistake: Kai finds $e$ when $b = 70^\circ$. Click the line where his reasoning goes wrong.
   line 2
   Line 2: b and e are not a named pair (b is above the top line, e above the bottom line, on different sides). Looking related is not a reason. e = 110°: f = 70° (corresponding to b), then e = 180° − 70° (a straight line).
5. Another chain: Find $g$. Write each step with its reason.
   $e = 115^\circ$ (alternate to $d$: alternate angles are equal); $g = 180^\circ - 115^\circ = 65^\circ$ ($e$ and $g$ are on a straight line)
   e = 115° (alternate to d), g = 65° (a straight line with e). Or: f = 65° (co-interior with d), g = 65° (vertically opposite f).
6. A different start: Find $f$, with a reason for each step.
   $e = 130^\circ$ (corresponding to $a$); $f = 180^\circ - 130^\circ = 50^\circ$ ($e$ and $f$ are on a straight line)
   e = 130° (corresponding to a), f = 50° (a straight line with e). Or b = 50° (a straight line with a), f = 50° (corresponding to b).
7. Up from the bottom: Find $b$, going up from the bottom line.
   $c = 180^\circ - 140^\circ = 40^\circ$ (co-interior with $e$: co-interior angles add to $180^\circ$); $b = 40^\circ$ (vertically opposite $c$)
   c = 40° (co-interior with e), b = 40° (vertically opposite c).
8. Your own route: Find $c$. (Use your own steps; two are enough.)
   55
   d = 125° (corresponding to h), then c = 55° (a straight line with d).
9. Not parallel: Are these two lines parallel?
   No: c and f are alternate angles, but they are not equal
   If the lines were parallel, the alternate angles c and f would be equal. They are not, so the lines are not parallel.
10. Showing lines are parallel: Backwards: tilt the lower line until the alternate angles $c$ and $f$ are equal. What does that tell you about the lines?
    
    When alternate angles are equal, the lines are parallel (w = 0). Equal corresponding or alternate angles, or co-interior angles adding to 180°, show that lines are parallel.
11. Down the page: Find $a$, with a reason for each step.
    $c = 75^\circ$ (corresponding to $g$); $a = 180^\circ - 75^\circ = 105^\circ$ ($a$ and $c$ are on a straight line)
    c = 75° (corresponding to g), a = 105° (a straight line with c).
12. Through the middle: Find $h$, with a reason for each step.
    $e = 180^\circ - 80^\circ = 100^\circ$ (co-interior with $c$); $h = 100^\circ$ (vertically opposite $e$)
    e = 100° (co-interior with c), h = 100° (vertically opposite e). Or g = 80° (corresponding to c), h = 100° (a straight line with g).
13. Review: one crossing: Review: Tom says "$b$ and $c$ are alternate angles". Which is right?
    They are vertically opposite: both at the top crossing
    b and c are at the same crossing, opposite each other: vertically opposite (and equal for that reason).
14. set it up: Work out what $b$ must be for $h = 125^\circ$. Then turn the crossing line until $b$ is that size: only $b$ is shown, and this ticks when $h = 125^\circ$.
    
    f = 55° (a straight line with h), so b = 55° (corresponding to f).

Solutions

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

one change at a time › A different start

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

one change at a time › Up from the bottom

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

one change at a time › Your own route

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 › Showing lines are parallel

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

set it up

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

Stretch and support

Stretch: Write a three-step puzzle for a partner: give one angle, ask for an angle three rules away, and write the model reasons.

Support: Allow the Run board for checking each step: set the given angle, pick pairs and read the names.

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 "Reasoning with angles".

The idea: Using angle reasoning to solve problems: I can Students solve multi-step angle problems involving parallel lines, giving geometric reasons for each step.

What understanding looks like:
- Solves a multi-step angle problem correctly.
- States a geometric reason for each step.
- Does not assume angle values without justification.

Mistakes students often make here:
- A student links two angles that only look related (Kai's mistake).
- A student gets the right angle with no reasons.
- A student cannot find a first step.

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 (Properties of corresponding, alternate and co-interior angles), 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 "Reasoning with angles".

Exit ticket

A transversal crosses two parallel lines, making a 120° angle. Find the angle alternate to it and the angle co-interior to it, with reasons.

120° (alternate angles are equal) and 180° − 120° = 60° (co-interior angles add to 180°).

What each step is for

• two steps: Committing to h when only b is known shows whether they look for a chain (b to f to h) or assume a link between b and h.
• check the chain: Setting b to 70° and comparing h shows the two-step answer, and turning the line shows h is always 180° minus b: the chain works for every angle.
• reasons for every step: Know, choose the rule, state the reason: each step links an angle you know to the next by a named rule. Kai's mistake links two angles that only look related.
• one change at a time: One change at a time: a different starting angle, a different route, the reverse direction, then lines that are not parallel and showing lines are parallel from the angles.
• angle chains: Two-step chains with each rule, mixed, with a reason named every time; the review keeps the pair names apart from vertically opposite angles.
• set it up: Working backwards from a target angle two steps away, then checking on the board, reverses the chain: reasoning first, the board only confirms.

Where it leads

Next: angle sums of triangles and polygons, then Unknown angles in shapes, which combines them with these rules.

Where it leads: geometric proof (Years 8 to 10), where a chain of reasons is the proof.