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# Reasoning with angles

Angles and shapes · Parallel lines

<!-- Speaker note: Students find unknown angles at parallel lines in two-step chains, with a reason for each step, writing their working their own way; the platform marks the answer and the reason at the key step, then shows a model to compare with (AC9M7M04, node 3). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** find unknown angles in steps, giving a reason for each: what you know, the rule that links it to the next angle, and the reason in words. Parallel-line rules join straight-line and vertically opposite angles.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A detective does not guess: every step has evidence. Finding an angle is the same. Each angle you find needs a reason, and the reasons are what make the answer trustworthy.

<!-- Speaker note: You know the rules: corresponding and alternate angles are equal, co-interior angles add to $180^\circ$, and from Year 6, vertically opposite angles are equal and angles on a straight line add to $180^\circ$. Most problems need more than one rule. In this lesson you find angles **in steps**, and for each step you say which rule you used. Write your working your own way: short notes are fine. The checker marks your answer and the reason at the step that matters; you compare the rest with a model. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Two parallel lines are crossed by a transversal. $b = 70^\circ$. What is $h$?**

- $20^\circ$
- $70^\circ$
- $110^\circ$
- $250^\circ$

<!-- Speaker note: What happens: 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). Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 70° are assuming b and h are linked directly. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Fangle-reasoning%2Fstep-2&code=widget%3A+parallel%0Aslider+t%3A+30%2C+150%2C+60%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/angle-reasoning/?full=step-2#step-2)

<!-- Speaker note: Then turn the crossing line. Is $h$ always $180^\circ - b$? Which two rules make that true? -->

<!-- Speaker note: Teacher: ask students to say the two rules that make h = 180° − b true for every crossing angle. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 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).

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Know, choose, say why

Here is the routine for every step: what you know, the rule that links it to a new angle, and the reason.

Now find $h$ yourself. Write it your own way; the checker marks your answer and the reason for the first step, then shows you a model to compare.

$$b = 70^\circ$$

1) $f = 70^\circ$ — Choose: $f$ is in the same place as $b$ at the other crossing. Say why: corresponding angles are equal.
2) $h = 180^\circ - 70^\circ$ — Choose: $h$ is next to $f$ on the bottom line. Say why: angles on a straight line add to $180^\circ$.
3) $h = 110^\circ$ — Done: two steps, two reasons.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write b = 70^ and say "Know: the given angle." Line 1: write f = 70^ and say "Choose: f is in the same place as b at the other crossing. Say why: corresponding angles are equal." Line 2: write h = 180^ - 70^ and say "Choose: h is next to f on the bottom line. Say why: angles on a straight line add to 180^." Line 3: write h = 110^ and say "Done: two steps, two reasons." -->

<!-- Speaker note: Teacher: model know, choose, say why on the board, writing the reason in short form. Accept any correct route. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Know, choose, say why

**Find $h$. Write each step with its reason.**

* Answer: $f = 70^\circ$ (corresponding to $b$: corresponding angles are equal); $h = 180^\circ - 70^\circ = 110^\circ$ ($f$ and $h$ are on a straight line)

<!-- Speaker note: 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). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Kai finds $e$ when $b = 70^\circ$. Click the line where his reasoning goes wrong.**

```
 1  b = 70° (given).
 2  b and e look like a pair, so b and e are alternate angles.
 3  Alternate angles are equal, so e = 70°.
```

* Answer: line 2

<!-- Speaker note: 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). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Another chain

**Find $g$. Write each step with its reason.**

* Answer: $e = 115^\circ$ (alternate to $d$: alternate angles are equal); $g = 180^\circ - 115^\circ = 65^\circ$ ($e$ and $g$ are on a straight line)

<!-- Speaker note: e = 115° (alternate to d), g = 65° (a straight line with e). Or: f = 65° (co-interior with d), g = 65° (vertically opposite f). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **A different start:** Each task changes one thing. Your working is yours; the marked parts are the answer and one reason.
- **Up from the bottom:** 
- **Your own route:** 
- **Not parallel:** 
- **Showing lines are parallel:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/angle-reasoning/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: Change w, the tilt of the lower line. / Stop when c and f are equal. For students who need more support: Allow the Run board for checking each step: set the given angle, pick pairs and read the names. For early finishers: Write a three-step puzzle for a partner: give one angle, ask for an angle three rules away, and write the model reasons. When students get stuck: A student links two angles that only look related (Kai's mistake). Ask: Which rule links them? Name it. Fix: If no rule links them directly, find an angle in between that does. / A student gets the right angle with no reasons. Ask: How would you convince someone who did not see the diagram? Fix: Write one reason per line: the angle, then why. / A student cannot find a first step. Ask: Which angles can you find straight from the given one? Fix: List every angle one rule away from the given one; the target is usually among those, or one more step on. -->

<!-- Speaker note: Teacher: collect two different routes to f and ask which is shorter. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Down the page

**Find $a$, with a reason for each step.**

* Answer: $c = 75^\circ$ (corresponding to $g$); $a = 180^\circ - 75^\circ = 105^\circ$ ($a$ and $c$ are on a straight line)

<!-- Speaker note: c = 75° (corresponding to g), a = 105° (a straight line with c). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Through the middle

**Find $h$, with a reason for each step.**

* Answer: $e = 180^\circ - 80^\circ = 100^\circ$ (co-interior with $c$); $h = 100^\circ$ (vertically opposite $e$)

<!-- Speaker note: e = 100° (co-interior with c), h = 100° (vertically opposite e). Or g = 80° (corresponding to c), h = 100° (a straight line with g). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: one crossing

**Review: Tom says "$b$ and $c$ are alternate angles". Which is right?**

- They are vertically opposite: both at the top crossing
- They are corresponding
- Tom is right

* Answer: They are vertically opposite: both at the top crossing

<!-- Speaker note: b and c are at the same crossing, opposite each other: vertically opposite (and equal for that reason). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: set it up

Write your two steps first, then turn the line.

Stretch: write your own two-step puzzle for a partner: give one angle, ask for one that is not a named pair with it, and write the model reasons.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/angle-reasoning/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Working backwards from a target angle two steps away, then checking on the board, reverses the chain: reasoning first, the board only confirms. Hints, in order: h and f are on a straight line, so f = 180° − 125°. / b is corresponding to f. -->

<!-- Speaker note: Teacher: ask students to say their two steps before they turn the line. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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°).

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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. -->