← Angles and shapes

Reasoning with angles

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.

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.

You know the rules: corresponding and alternate angles are equal, co-interior angles add to 180∘180^\circ, and from Year 6, vertically opposite angles are equal and angles on a straight line add to 180∘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.

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

Two steps

bb and hh are not one of the pairs. Which angle could you find first?

Two parallel lines are crossed by a transversal. b=70∘b = 70^\circ. What is hh?

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

Check the chain

Then turn the crossing line. Is hh always 180∘−b180^\circ - b? Which two rules make that true?

Try it yourselfYour edits are saved in this browser.

Turn the crossing line until b=70∘b = 70^\circ, then pick bb and hh. Was your prediction right?

Use the board. This ticks when you get there.

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

Reasons for every step

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

Worked exampleBefore each step, say what you think comes next.
  1. b=70∘b = 70^\circKnow: the given angle.

Now find hh 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.

Two parallel lines. b = 70^\circ.
Two parallel lines. b=70∘b = 70^\circ.

Read Kai's reasoning one line at a time. Which step has no real rule behind it?

Two parallel lines. d = 115^\circ.
Two parallel lines. d=115∘d = 115^\circ.

Find hh. Write each step with its reason.

Why is f=70∘f = 70^\circ?

Kai finds ee when b=70∘b = 70^\circ. Click the line where his reasoning goes wrong.

Find gg. Write each step with its reason.

Why is e=115∘e = 115^\circ?

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing. Your working is yours; the marked parts are the answer and one reason.

Two parallel lines. a = 130^\circ.
Two parallel lines. a=130∘a = 130^\circ.
Two parallel lines. e = 140^\circ.
Two parallel lines. e=140∘e = 140^\circ.
Two parallel lines. h = 125^\circ.
Two parallel lines. h=125∘h = 125^\circ.
c = 60^\circ, f = 45^\circ. No arrows on the lines.
c=60∘c = 60^\circ, f=45∘f = 45^\circ. No arrows on the lines.
Need a hint?

Change w, the tilt of the lower line.

Another hint

Stop when c and f are equal.

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Find ff, with a reason for each step.

Why is f=180∘−130∘=50∘f = 180^\circ - 130^\circ = 50^\circ?

Find bb, going up from the bottom line.

Why is c=180∘−140∘=40∘c = 180^\circ - 140^\circ = 40^\circ?

Find cc. (Use your own steps; two are enough.)

Are these two lines parallel?

Backwards: tilt the lower line until the alternate angles cc and ff are equal. What does that tell you about the lines?

Use the board. This ticks when you get there.

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

Angle chains

Two parallel lines. g = 75^\circ.
Two parallel lines. g=75∘g = 75^\circ.
Two parallel lines. c = 80^\circ.
Two parallel lines. c=80∘c = 80^\circ.
Two parallel lines. b = 70^\circ.
Two parallel lines. b=70∘b = 70^\circ.

Find aa, with a reason for each step.

Find hh, with a reason for each step.

Why is e=180∘−80∘=100∘e = 180^\circ - 80^\circ = 100^\circ?

Review: Tom says "bb and cc are alternate angles". Which is right?

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

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.

Try it yourselfYour edits are saved in this browser.
Need a hint?

h and f are on a straight line, so f = 180° − 125°.

Another hint

b is corresponding to f.

Work out what bb must be for h=125∘h = 125^\circ. Then turn the crossing line until bb is that size: only bb is shown, and this ticks when h=125∘h = 125^\circ.

Use the board. This ticks when you get there.

For teachers

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.

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

Students will be able to

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.

Running the lesson

  1. Predict: take a show of hands. Students who chose 70° are assuming b and h are linked directly.
  2. Run: ask students to say the two rules that make h = 180° − b true for every crossing angle.
  3. Investigate: model know, choose, say why on the board, writing the reason in short form. Accept any correct route.
  4. Modify: collect two different routes to f and ask which is shorter.
  5. Make: ask students to say their two steps before they turn the line.

When students get stuck

Stretch and support

Stretch: Write a three-step puzzle for a partner: give one angle, ask for an angle three rules away, and write the model reasons.

Support: Allow the Run board for checking each step: set the given angle, pick pairs and read the names.

Where it 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.

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 lesson4. Angles in a triangleHowever you change a triangle, its three angles add to 180°. Tear off the corners and they make a straight line.