← Angles and shapes

Finding angles with parallel lines

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.

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

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".

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

A co-interior pair

Corresponding and alternate angles are equal. Is a co-interior pair the same?

Two parallel lines are crossed by a transversal. c=65∘c = 65^\circ, and ee is co-interior with cc. What is ee?

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

Test it

Then turn the crossing line again. Do cc and ee always add to 180∘180^\circ?

Try it yourselfYour edits are saved in this browser.

Turn the crossing line until c=65∘c = 65^\circ, then pick cc and ee. Was your prediction right?

Use the board. This ticks when you get there.

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

From one angle to all eight

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

Worked exampleBefore each step, say what you think comes next.
  1. b=70∘b = 70^\circGiven.

Then find ff in the picture.

Two parallel lines. b = 72^\circ.
Two parallel lines. b=72∘b = 72^\circ.

Read Leo's working one line at a time. Which rule did he use?

Two parallel lines. d = 110^\circ.
Two parallel lines. d=110∘d = 110^\circ.

Find ff.

Leo finds ff when d=110∘d = 110^\circ. Click the line where his working goes wrong.

Find ff correctly.

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing. Give the reason as well as the size.

Two parallel lines. c = 50^\circ.
Two parallel lines. c=50∘c = 50^\circ.
Two parallel lines. e = 125^\circ.
Two parallel lines. e=125∘e = 125^\circ.
Two parallel lines. a = 140^\circ.
Two parallel lines. a=140∘a = 140^\circ.
Need a hint?

a and e are corresponding.

Another hint

Then e and g are on a straight line.

Need a hint?

h = 180° − t, so t = 50°.

Another hint

b is corresponding to f, and f is on a straight line with h.

The lower line is tilted: the lines are not parallel. c = 60^\circ.
The lower line is tilted: the lines are not parallel. c=60∘c = 60^\circ.
Try it yourselfYour edits are saved in this browser.

Alternate: find ff.

Co-interior, starting at the bottom: find cc.

Two steps: find gg.

Turn the crossing line until h=130∘h = 130^\circ. Work out bb first, then pick hh and bb to check.

Use the board. This ticks when you get there.

Edge case: can you use "alternate angles are equal" to find ff here?

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

Angle and reason

Two parallel lines. g = 80^\circ.
Two parallel lines. g=80∘g = 80^\circ.
Two parallel lines. f = 62^\circ.
Two parallel lines. f=62∘f = 62^\circ.
The angles a to h.
The angles aa to hh.

Find cc.

Find dd.

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

Review: which pair is alternate?

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

Equal co-interior angles

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

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

c and e add to 180°. If they are equal, what is each?

Another hint

Turn until c = 90°.

Turn the crossing line so that the co-interior angles cc and ee are equal.

Use the board. This ticks when you get there.

For teachers

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.

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

Students will be able to

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.

Running the lesson

  1. Predict: take a show of hands. Students who chose 65° are using 'equal' for every pair. Do not correct it yet.
  2. Run: after setting c = 65°, ask students to turn the line and say what stays true (c + e = 180°).
  3. Investigate: model the worked example live, writing the reason beside each angle. Then ask: could you have found e another way?
  4. Modify: in the two-step task, collect different routes to g and the reasons each uses.
  5. Make: ask why c and e can only be equal at 90°.

When students get stuck

Stretch and support

Stretch: 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.)

Support: Use the Run board to check each answer: set the given angle, pick the pair and read the size.

Where it 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).

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 lesson3. Reasoning with anglesFind 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.