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# Finding angles with parallel lines

Angles and shapes · Parallel lines

<!-- Speaker note: Students use the parallel-line rules (corresponding and alternate equal, co-interior add to 180°) with straight-line and vertically opposite angles to find unknown angles from one given angle, naming a reason for each (AC9M7M04, node 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** when the lines are parallel, corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°. From one angle you can find all eight, giving a reason for each.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A carpenter measures one angle where a brace crosses two parallel shelves. Without measuring again, she knows every other angle. How?

<!-- Speaker note: In Angles at parallel lines you named the pairs: corresponding, alternate and co-interior. Now the useful part: when the lines are parallel, each pair has a rule, so one angle tells you the rest. You also learn to give a **reason** for each angle you find, such as "alternate angles are equal". -->

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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. $c = 65^\circ$, and $e$ is co-interior with $c$. What is $e$?**

- $65^\circ$
- $295^\circ$
- $25^\circ$
- $115^\circ$

<!-- Speaker note: What happens: 115°. Co-interior angles are not equal: they add to 180°, so e = 180° − 65° = 115°. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 65° are using 'equal' for every pair. Do not correct it yet. -->

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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%2Fparallel-angles%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/parallel-angles/?full=step-2#step-2)

<!-- Speaker note: Then turn the crossing line again. Do $c$ and $e$ always add to $180^\circ$? -->

<!-- Speaker note: Teacher: after setting c = 65°, ask students to turn the line and say what stays true (c + e = 180°). -->

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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?

* 115°. Co-interior angles are not equal: they add to 180°, so e = 180° − 65° = 115°.

<!-- 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>

## Equal pairs

Here is how one angle gives all eight, with a reason for each.

Then find $f$ in the picture.

$$b = 70^\circ$$

1) $c = 70^\circ$ — Vertically opposite $b$: equal.
2) $a = 110^\circ$ — On a straight line with $b$: $180 - 70$.
3) $f = 70^\circ$ — Corresponding to $b$: equal.
4) $e = 110^\circ$ — Co-interior with $c$: they add to $180^\circ$.
5) $g = 70^\circ,\; h = 110^\circ$ — Corresponding to $c$ and to $d$ (or vertically opposite $f$ and $e$).

<!-- 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 "Given." Line 1: write c = 70^ and say "Vertically opposite b: equal." Line 2: write a = 110^ and say "On a straight line with b: 180 - 70." Line 3: write f = 70^ and say "Corresponding to b: equal." Line 4: write e = 110^ and say "Co-interior with c: they add to 180^." Line 5: write g = 70^, h = 110^ and say "Corresponding to c and to d (or vertically opposite f and e)." -->

<!-- Speaker note: Teacher: model the worked example live, writing the reason beside each angle. Then ask: could you have found e another way? -->

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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>

## Equal pairs

**Find $f$.**

* Answer: 72

<!-- Speaker note: b and f are corresponding angles, so f = 72°. -->

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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

**Leo finds $f$ when $d = 110^\circ$. Click the line where his working goes wrong.**

```
 1  d = 110° (given).
 2  d and f are between the parallel lines, on the same side.
 3  So d and f are co-interior angles.
 4  Co-interior angles are equal, so f = 110°.
```

* Answer: line 4

<!-- Speaker note: Line 4: co-interior angles add to 180°, so f = 180° − 110° = 70°. Only corresponding and alternate angles are equal. -->

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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>

## Co-interior

**Find $f$ correctly.**

* Answer: 70

<!-- Speaker note: Co-interior angles add to 180°: f = 180° − 110° = 70°. -->

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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



- **Alternate:** Each task changes one thing. Give the reason as well as the size.
- **Co-interior from below:** 
- **Two steps:** 
- **Check a chain:** 
- **Not parallel:** 

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

<!-- Speaker note: What this step is for: 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. Hints, in order: a and e are corresponding. / Then e and g are on a straight line. / h = 180° − t, so t = 50°. / b is corresponding to f, and f is on a straight line with h. For students who need more support: Use the Run board to check each answer: set the given angle, pick the pair and read the size. For early finishers: 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.) When students get stuck: A student makes co-interior angles equal (Leo's mistake). Ask: On the board, pick c and e. Are they the same size? Fix: Co-interior angles add to 180°: one is acute, the other obtuse, unless both are 90°. / A student finds the right size but cannot name the reason. Ask: Where are the two angles: same place, between and opposite, between and same side, or one crossing? Fix: Name the pair first, then its rule. / A student uses the rules on lines that are not parallel. Ask: Where are the arrows that say the lines are parallel? Fix: No arrows (or a tilted line): the pair rules do not apply. -->

<!-- Speaker note: Teacher: in the two-step task, collect different routes to g and the reasons each uses. -->

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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>

## Corresponding

**Find $c$.**

* Answer: 80

<!-- Speaker note: Corresponding angles are equal: c = 80°. -->

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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>

## Co-interior

**Find $d$.**

* Answer: 118

<!-- Speaker note: Co-interior angles add to 180°: d = 180° − 62° = 118°. -->

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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>

## The reason

**$e = 120^\circ$ and $c = 60^\circ$. Which reason links them?**

- Alternate angles are equal
- Co-interior angles add to 180°
- Corresponding angles are equal

* Answer: Co-interior angles add to 180°

<!-- Speaker note: c and e are between the lines on the same side, and 60° + 120° = 180°. -->

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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: names

**Review: which pair is alternate?**

- $d$ and $e$
- $a$ and $d$
- $d$ and $h$
- $d$ and $f$

* Answer: $d$ and $e$

<!-- Speaker note: d and e: between the lines, on opposite sides. -->

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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: equal co-interior angles

Then: can alternate angles ever add to $180^\circ$? When?

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

<!-- Speaker note: What this step is for: Making a co-interior pair equal reverses the rule: two angles that add to 180° are equal only when both are 90°. Hints, in order: c and e add to 180°. If they are equal, what is each? / Turn until c = 90°. -->

<!-- Speaker note: Teacher: ask why c and e can only be equal at 90°. -->

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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. 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).

<!-- 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 (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). -->