Unknown angles in shapes

Classroom resources

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Students will be able to

• I can find an unknown angle in a triangle or polygon using its angle sum.
• I can combine angle sums with straight-line and parallel-line angles, giving a reason for each step.
• I can explain why an exterior angle of a triangle equals the sum of the two interior opposite angles.

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, then compare with its model. Students need Angles in a triangle, Angles in a polygon and Reasoning with angles.

The one idea: choose the right angle sum for the shape, then chain steps with reasons. The common mistake is using 180° for every shape; it is planted in Zoe's working, where a negative angle is the clue.

As in Reasoning with angles, the free working is not marked: students write it their way. The answer is marked, and at the key step they choose the reason from a list. Then they compare with a model and tick a self-check. The isosceles task is wider than the minimum: the equal angles are stated in the question.

Running the lesson

1. Predict: take a show of hands, then ask students who chose 110° how they got it. Collect both routes (two steps, or a + b).
2. Run: ask why x = a + b must be true (a + b + c = 180° and x + c = 180°).
3. Investigate: model know, choose, say why for the triangle. Ask: what does a negative answer tell you (Zoe)?
4. Modify: before the pentagon, ask students for its angle sum and how they know.
5. Make: ask students what c must be before they move anything.

When students get stuck

Answers to the checks

1. the outside angle: A triangle has $a = 50^\circ$ and $b = 60^\circ$. The side through $C$ is extended past $C$, making an outside angle $x$ with the third side. What is $x$?
   110°. c = 180° − 50° − 60° = 70° (angles in a triangle), then x = 180° − 70° = 110° (a straight line). Notice: 110° = 50° + 60°.
2. drag the triangle: Drag the corners (or use Move) until $x$ is more than $120^\circ$. Each time, compare $x$ with $a + b$.
   
3. Triangle, then a straight line: Find $x$. Write each step with its reason.
   $c = 180^\circ - 59^\circ - 45^\circ = 76^\circ$ (angles in a triangle add to $180^\circ$); $x = 180^\circ - 76^\circ = 104^\circ$ ($c$ and $x$ are on a straight line)
   c = 76° (angles in a triangle), x = 104° (a straight line with c). Check: 59° + 45° = 104°.
4. Find the mistake: Zoe finds the fourth angle of a quadrilateral. Click the line where her working goes wrong.
   line 2
   Line 2: a quadrilateral's angles add to (4 − 2) × 180° = 360°, not 180°. d = 360° − 76° − 68° − 102° = 114°. A negative angle on line 4 is the clue that something earlier went wrong.
5. A quadrilateral: Find $d$, with your reason.
   $a + b + c + d = 360^\circ$ (angles in a quadrilateral: $(4 - 2) \times 180^\circ$); $d = 360^\circ - 76^\circ - 68^\circ - 102^\circ = 114^\circ$ (subtract the angles you know)
   (4 − 2) × 180° = 360°, so d = 360° − 246° = 114°.
6. Isosceles: $CA = CB$, so the angles at $A$ and $B$ are equal. Find $c$, with a reason for each step.
   $b = 56^\circ$ (isosceles triangle: the angles opposite the equal sides are equal); $c = 180^\circ - 56^\circ - 56^\circ = 68^\circ$ (angles in a triangle add to $180^\circ$)
   b = 56° (isosceles), c = 68° (angles in a triangle).
7. A pentagon: Find $e$.
   98
   (5 − 2) × 180° = 540°, so e = 540° − 442° = 98°.
8. Exterior at another corner: Find the exterior angle $x$ at $B$.
   $b = 180^\circ - 63^\circ - 87^\circ = 30^\circ$ (angles in a triangle add to $180^\circ$); $x = 180^\circ - 30^\circ = 150^\circ$ ($b$ and $x$ are on a straight line)
   b = 30°, x = 150°. Or straight away: x = a + c = 63° + 87° = 150° (the exterior angle equals the sum of the two interior opposite angles).
9. A right angle: Find $c$.
   100
   360° − 68° − 90° − 102° = 100°.
10. Impossible triangles: Edge case: can a triangle have two angles of $100^\circ$?
    No: they already add to more than 180°
    100° + 100° = 200° > 180°, so no. A triangle has at most one obtuse angle.
11. Triangle: A triangle has angles of $35^\circ$ and $85^\circ$. Find the third angle.
    60
    180° − 35° − 85° = 60°.
12. Quadrilateral: Three angles of a quadrilateral are $90^\circ$, $85^\circ$ and $120^\circ$. Find the fourth.
    65
    360° − 295° = 65°.
13. Exterior angle backwards: A triangle has an exterior angle of $130^\circ$ at $C$, and $a = 70^\circ$. Find $b$, with your reasons.
    $c = 180^\circ - 130^\circ = 50^\circ$ ($c$ and the exterior angle are on a straight line); $b = 180^\circ - 70^\circ - 50^\circ = 60^\circ$ (angles in a triangle add to $180^\circ$)
    c = 50°, b = 60°. Or: 130° = a + b, so b = 60°.
14. Review: parallel lines: Review: parallel lines, $c = 70^\circ$. Which is a correct first step to find $e$?
    $e = 110^\circ$: co-interior with $c$
    Co-interior angles add to 180°: e = 110°.
15. a given exterior angle: Move the corners so that the exterior angle $x$ is $135^\circ$.
    
    c = 45°, so a + b = 135°: for example a = 90° and b = 45°.

Solutions

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

one change at a time › Isosceles

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

one change at a time › A pentagon

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

one change at a time › Exterior at another corner

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

one change at a time › A right angle

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

one change at a time › Impossible triangles

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

a given exterior angle

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

Stretch and support

Stretch: Prove that the exterior angles of any convex polygon add to 360°, starting from a triangle (each exterior + interior = 180°).

Support: Use the Run board to check: set a triangle, read its angles, and compare with your working.

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 "Unknown angles in shapes".

The idea: Finding unknown angles using angle sums: I can Students use triangle and polygon angle sum properties to find unknown angles, giving reasons.

What understanding looks like:
- Correctly finds unknown angles using angle sum properties.
- States the property used as the reason for each step.
- Applies the correct angle sum for the polygon type.

Mistakes students often make here:
- A student uses 180° for a quadrilateral (Zoe's mistake).
- A student stops at the interior angle when asked for the exterior one.
- A student writes numbers with no reasons.

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 (Interior angle sum of polygons by decomposition), 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 "Unknown angles in shapes".

Exit ticket

A triangle has angles of 48° and 67°. Find the third angle, and the exterior angle next to it, with reasons.

180° − 48° − 67° = 65° (angles in a triangle). Exterior angle 180° − 65° = 115° (a straight line). Check: 48° + 67° = 115°.

What each step is for

• the outside angle: Committing to the exterior angle before working it out shows whether they see the two-step chain (angle sum, then straight line), and sets up x = a + b.
• drag the triangle: Dragging the triangle while watching x shows the exterior angle always equals the two opposite angles added: the two-step chain gives a shortcut.
• angle sums in steps: Choose the angle sum for the shape, then any straight-line or parallel-line step, with a reason each time. Zoe's mistake uses the triangle's 180° for a quadrilateral.
• one change at a time: One change at a time: an isosceles triangle, a pentagon's sum, the exterior angle at another corner, a right angle in a quadrilateral, then the edge case of a triangle that cannot exist.
• unknown angles: Angle sums and exterior angles from words, mixed, with the polygon sum planted; the review keeps the parallel-line chains fresh.
• a given exterior angle: Making a triangle with a given exterior angle reverses the reasoning: x = 135° needs c = 45°, from a + b = 135°.

Where it leads

This completes the Year 7 angle work: parallel lines (AC9M7M04) and angle sums (AC9M7M05).

Where it leads: congruence and similarity (Years 8 and 9) and geometric proof, where chains of reasons are the whole point.