← Angles and shapes

Unknown angles in shapes

Find unknown angles in triangles and polygons in steps, with a reason for each: the angle sum of the shape, angles on a straight line, equal angles in an isosceles triangle. An exterior angle of a triangle equals the sum of the two interior opposite angles.

A roof truss, a kite, a bridge brace: builders work out the angle they cannot measure from the ones they can. Every step needs a reason.

You know the angle sums: a triangle's angles add to 180∘180^\circ, and a polygon with nn sides has (n−2)×180∘(n - 2) \times 180^\circ. With straight-line angles and the parallel-line rules, you can find angles you cannot measure.

As in Reasoning with angles, each step needs a reason. Write your working your own way; the checker marks your answer and a reason at the key step, then shows a model to compare.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

The outside angle

Two steps: which angle can you find first?

A triangle has a=50∘a = 50^\circ and b=60∘b = 60^\circ. The side through CC is extended past CC, making an outside angle xx with the third side. What is xx?

TryTry it on the board and compare the result with your prediction.

Drag the triangle

Was your prediction right? Is xx always a+ba + b? Why must it be?

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Drag the corners (or use Move) until xx is more than 120∘120^\circ. Each time, compare xx with a+ba + b.

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Angle sums in steps

Here is the routine: know, choose the rule, say why.

Worked exampleBefore each step, say what you think comes next.
  1. a=59∘,  b=45∘a = 59^\circ,\; b = 45^\circKnow: the given angles.

Now find xx yourself, your own way.

a = 59^\circ, b = 45^\circ. The side BC is extended past C.
a=59∘a = 59^\circ, b=45∘b = 45^\circ. The side BCBC is extended past CC.

Read Zoe's working. Which line uses the wrong fact?

A quadrilateral: a = 76^\circ, b = 68^\circ, c = 102^\circ.
A quadrilateral: a=76∘a = 76^\circ, b=68∘b = 68^\circ, c=102∘c = 102^\circ.

Find xx. Write each step with its reason.

Why is c=180∘−59∘−45∘=76∘c = 180^\circ - 59^\circ - 45^\circ = 76^\circ?

Zoe finds the fourth angle of a quadrilateral. Click the line where her working goes wrong.

Find dd, with your reason.

Why is a+b+c+d=360∘a + b + c + d = 360^\circ?

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

CA = CB (isosceles). a = 56^\circ.
CA=CBCA = CB (isosceles). a=56∘a = 56^\circ.
A pentagon: a = 104^\circ, b = 117^\circ, c = 85^\circ, d = 136^\circ.
A pentagon: a=104∘a = 104^\circ, b=117∘b = 117^\circ, c=85∘c = 85^\circ, d=136∘d = 136^\circ.
a = 63^\circ, c = 87^\circ. The side AB is extended past B.
a=63∘a = 63^\circ, c=87∘c = 87^\circ. The side ABAB is extended past BB.
A quadrilateral with a right angle at B: a = 68^\circ, d = 102^\circ.
A quadrilateral with a right angle at BB: a=68∘a = 68^\circ, d=102∘d = 102^\circ.
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CA=CBCA = CB, so the angles at AA and BB are equal. Find cc, with a reason for each step.

Why is c=180∘−56∘−56∘=68∘c = 180^\circ - 56^\circ - 56^\circ = 68^\circ?

Find ee.

Find the exterior angle xx at BB.

Find cc.

Edge case: can a triangle have two angles of 100∘100^\circ?

PractiseA few on your own, to check you have it.

Unknown angles

A triangle has angles of 35∘35^\circ and 85∘85^\circ. Find the third angle.

Three angles of a quadrilateral are 90∘90^\circ, 85∘85^\circ and 120∘120^\circ. Find the fourth.

A triangle has an exterior angle of 130∘130^\circ at CC, and a=70∘a = 70^\circ. Find bb, with your reasons.

Review: parallel lines, c=70∘c = 70^\circ. Which is a correct first step to find ee?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A given exterior angle

First work out what cc must be. Then find a triangle that has it.

Stretch: design an angle puzzle for a partner with two unknown angles, and write the model reasons.

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Need a hint?

x = 135° means c = 45°.

Another hint

A right angle at A and equal legs give 45° at C: try A (1, 1), B (7, 1), C (1, 7).

Move the corners so that the exterior angle xx is 135∘135^\circ.

Use the board. This ticks when you get there.

For teachers

Students find unknown angles in triangles, quadrilaterals and pentagons in steps, with a reason for each (angle sum, straight line, isosceles), and explain why an exterior angle equals the sum of the two interior opposite angles; working is their own, the answer and a key reason are marked (AC9M7M05, node 3). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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

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.

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.

What each step is for

Now you can
Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson7. Classifying trianglesName a triangle twice: by its sides (scalene, isosceles, equilateral) and by its biggest angle (acute, right, obtuse). Equal sides sit opposite equal angles, so some pairs are impossible: no right-angled equilateral triangle.