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# Angles at parallel lines

Angles and shapes · Parallel lines

<!-- Speaker note: Students turn a transversal across two parallel lines, see that the same place at each crossing gives the same angle, and learn to name corresponding, alternate and co-interior pairs in any orientation, keeping them apart from vertically opposite angles (AC9M7M04, node 1). 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** a line crossing two parallel lines makes eight angles. Pairs in the same place at each crossing are corresponding; between the lines on opposite sides, alternate; between the lines on the same side, co-interior.

- 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°.

<!-- Speaker note: 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. -->

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## Why it matters

> Railway tracks, the lines on a car park, the rungs of a ladder: parallel lines crossed by another line. The angles they make come in pairs.

<!-- Speaker note: Two lines are **parallel** when they go the same way and never meet, like railway tracks. Arrows (>>) on the lines mark them as parallel. A third line that crosses them is a **transversal**. The crossings make eight angles, named $a$ to $h$ here. In this lesson you learn the names of the pairs and how to spot them in any diagram. -->

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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

**$a$ 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 $e$?**

- One grows while the other shrinks
- Nothing links them: they change separately
- They stay the same size as each other

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

<!-- Speaker note: Teacher: take a show of hands. Students who chose 'one grows, the other shrinks' may be thinking of a and b (a straight line). -->

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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-lines%2Fstep-2&code=widget%3A+parallel%0Aslider+t%3A+30%2C+150%2C+60%0Apair%3A+a%2C+e%0A&fresh=1}}

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

<!-- Speaker note: $t$ is the angle the crossing line makes with the parallel lines. Turn it both ways. You can pick other angles too: click them, or use the letters. -->

<!-- Speaker note: Teacher: ask students to say why the bottom crossing looks like the top one (the lines go the same 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>

## What happened?

* 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.

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

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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>

## The three pairs

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Fparallel-lines%2Fstep-3&code=widget%3A+parallel%0At%3A+60%0Aangles%3A+hidden%0Anames%3A+off%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/parallel-lines/?full=step-3#step-3)

<!-- Speaker note: What this step is for: 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. -->

<!-- Speaker note: Teacher: draw the eight angles on the board and colour each pair as you name it. Ask: does each pair use both crossings? -->

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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

**Mia names a pair of angles. Click the line where her reasoning goes wrong.**

```
 1  Mia looks at angles b and c.
 2  b and c are on opposite sides of the crossing line.
 3  So b and c are alternate angles.
 4  Alternate angles are equal, so b = c.
```

* Answer: line 3

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

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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

**Which pair is co-interior?**

- $c$ and $e$
- $c$ and $g$
- $c$ and $f$
- $b$ and $c$

* Answer: $c$ and $e$

<!-- Speaker note: c and e are both between the lines and both on the left of the crossing line: co-interior. -->

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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



- **Corresponding:** Each task changes one thing. This board does not name the pairs, so the check tells you.
- **Co-interior:** 
- **Vertically opposite:** 
- **Not parallel:** 
- **A right angle:** 

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

<!-- Speaker note: What this step is for: 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. Hints, in order: The same place at each crossing: above the line and left, at the top and at the bottom. / Between the lines, on the same side of the crossing line. / Both at one crossing, opposite each other. / Move w back to 0, then away again. Watch c and f. / What angle does a crossing line make when it is square to the lines? For students who need more support: Use the board with names on (the Run board): pick pairs and read the names before trying the boards without them. For early finishers: 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.) When students get stuck: A student names b and c as alternate (Mia's mistake). Ask: Which crossing is each angle at? Fix: The parallel-line pairs take one angle from each crossing. b and c are vertically opposite. / A student mixes up alternate and co-interior. Ask: Are both angles on the same side of the crossing line? Fix: Same side: co-interior (C). Opposite sides: alternate (Z). / A student cannot find pairs when the diagram is turned. Ask: Where are the parallel lines? Which angles are between them? Fix: Shade the band between the parallel lines first; alternate and co-interior angles are inside it. -->

<!-- Speaker note: Teacher: before the tilt task, ask what changes if the lines are not parallel. Let the board show it. -->

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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>

## Name it

**$d$ and $e$ are**

- corresponding
- alternate
- vertically opposite
- co-interior

* Answer: alternate

<!-- Speaker note: Between the lines, on opposite sides: alternate. -->

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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>

## Name it again

**$b$ and $f$ are**

- corresponding
- co-interior
- alternate
- vertically opposite

* Answer: corresponding

<!-- Speaker note: Both above their line, on the right: the same place at each crossing, so corresponding. -->

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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>

## One crossing

**$a$ and $d$ are**

- corresponding
- co-interior
- vertically opposite
- alternate

* Answer: vertically opposite

<!-- Speaker note: At one crossing, opposite each other: vertically opposite. -->

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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: a straight line

**Review: $a$ and $b$ are on a straight line. $a = 115^\circ$. What is $b$?**

* Answer: 65

<!-- Speaker note: Angles on a straight line add to 180°: 180 − 115 = 65°. -->

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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: a pair that adds to 180°

Find a pair, then turn the crossing line. Do they still add to $180^\circ$?

Stretch: which pairs add to $180^\circ$ but are not between the lines?

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

<!-- Speaker note: What this step is for: Finding a pair between the lines that adds to 180° reverses the lesson: from what the angles do back to where they are. Hints, in order: Look only between the parallel lines: c, d, e and f. / Which two of those add to 180°? Check with the sizes. -->

<!-- Speaker note: Teacher: collect the pairs students found and ask why only the pairs between the lines always work. -->

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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. 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°.

<!-- 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: 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. -->