Angles at parallel lines

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.

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

h

here. In this lesson you learn the names of the pairs and how to spot them in any diagram.

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

Turning the crossing line

Picture the two crossings. Is one a copy of the other?

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 shrinksNothing links them: they change separatelyThey stay the same size as each other

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

Turn the crossing line

$t$ 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.

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$a$ and $e$ are picked. Turn the crossing line ( $t$ ) to $120^\circ$ and watch both. Was your prediction right?

Use the board. This ticks when you get there.

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

The three pairs

Here are the pairs. Each takes one angle from each crossing.

Worked exampleBefore each step, say what you think comes next.

$\text{corresponding: } a \text{ and } e$ The same place at each crossing: both above their line, on the left. (An F shape.)
$\text{alternate: } c \text{ and } f$ Between the parallel lines, on opposite sides of the crossing line. (A Z shape.)
$\text{co-interior: } c \text{ and } e$ Between the parallel lines, on the same side. (A C shape.)
$\text{vertically opposite: } b \text{ and } c$ At one crossing, opposite each other. Not a parallel-lines pair: both are at the same crossing.

Then pick two alternate angles on the board.

Need a hint?

Alternate angles are between the parallel lines.

Another hint

One on each side of the crossing line: c and f, or d and e.

Read Mia's reasoning one line at a time. Where are b and c?

Two parallel lines crossed by a transversal, with the angles a to h. — Two parallel lines crossed by a transversal, with the angles $a$ to $h$ .

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This board does not name the pairs. Pick two alternate angles.

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

Which pair is co-interior?

c

and

e

c

and

g

c

and

f

b

and

c

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing. This board does not name the pairs, so the check tells you.

Need a hint?

The same place at each crossing: above the line and left, at the top and at the bottom.

Need a hint?

Between the lines, on the same side of the crossing line.

Need a hint?

Both at one crossing, opposite each other.

Need a hint?

Move w back to 0, then away again. Watch c and f.

Need a hint?

What angle does a crossing line make when it is square to the lines?

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The crossing line leans the other way now. Pick two corresponding angles.

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Pick two co-interior angles.

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Pick two vertically opposite angles.

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The lower line is tilted by

w

, so the lines are not parallel. Are

c

and

f

still alternate angles?Yes: they are in alternate places, but they need not be equalYes, and they are still equalNo: alternate angles need parallel lines

Edge case: turn the crossing line until all eight angles are the same size.

Use the board. This ticks when you get there.

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

Name the pair

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

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?

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Need a hint?

Look only between the parallel lines: c, d, e and f.

Another hint

Which two of those add to 180°? Check with the sizes.

Pick two angles between the parallel lines, at different crossings, that add to $180^\circ$ .

Use the board. This ticks when you get there.

Angles at parallel lines

Turning the crossing line

Turn the crossing line

The three pairs

One change at a time

Name the pair

A pair that adds to 180°

Students will be able to

Before class

Running the lesson

When students get stuck

Stretch and support

Where it leads

What each step is for