Welcome back, Florence. Lesson 1 gave you three facts: angles on a straight line make 180°, angles around a point make 360°, and where two lines cross, opposite angles are equal. This lesson builds straight on top of them.
Today is about parallel lines — two lines that run alongside each other and never meet — and the angles made when a third line cuts across them. There are three pairs to learn, and each one has a shape you can spot.
It works just like last time: small steps, tap an answer or type one in, and a quiet nudge if it isn't right. There is a part you can drag with your finger, too. Tap Continue when you're ready.
Two lines are parallel when they run alongside each other, always the same distance apart, never meeting — like the two rails of a train track. A line that cuts across both is called a transversal.
Where the transversal crosses, it makes angles. An angle at one crossing and the angle in the same position at the other crossing are called corresponding angles — and when the two lines are parallel, corresponding angles are always equal. Some people spot them as an F shape.
Have a look for yourself. Drag the orange dot to tilt the crossing line, and watch the two marked angles — same position at each crossing, and always the same size.
Angle x is in the same position at its crossing as the 70° angle is at its crossing — both below the line, both on the same side of the transversal. Angles in matching positions like this are corresponding angles.
Because the two lines are parallel, corresponding angles are always equal.
x = 70°
Corresponding angles — angles in the same position at each crossing — are equal.
So the corresponding angle is also 120°.
Corresponding angles are equal.
So the corresponding angle is 48°.
Corresponding angles are equal.
So y = 75°.
The next pair sits between the two parallel lines — one angle on each side of the transversal.
If you trace from one angle, along the transversal, to the other, your pencil makes a Z shape. Angles in this position are called alternate angles, and they too are equal when the lines are parallel.
These two angles sit inside the shape the lines make — trace between them and you get a Z. Angles in this Z position are alternate angles.
When the lines are parallel, alternate angles are equal.
x = 58°
Alternate angles — the two angles inside a ‘Z’ shape — are equal.
So the alternate angle is 50°.
Alternate angles are equal.
So the alternate angle is 133°.
Alternate angles are equal.
So z = 88°.
The last pair also sits between the parallel lines — but this time both angles are on the same side of the transversal.
Tracing between them makes a C shape. These are co-interior angles, and they behave differently from the other two: they do not match. Instead, they add up to 180°.
It is worth knowing all three by their shapes — the F of corresponding, the Z of alternate, the C of co-interior. The shape tells you which rule to reach for.
Two angles between the parallel lines and on the same side of the transversal are co-interior angles — they make a C shape.
Co-interior angles do not match. Instead, they add up to 180°.
112° + x = 180°
Take 112° away from 180° to find x.
x = 180° − 112° = 68°
Co-interior angles add up to 180°.
So the other is 180° − 115° = 65°.
Co-interior angles add up to 180°.
So w = 180° − 72° = 108°.
This last part takes two steps, and it brings back something from Lesson 1 — so it is a chance to use both lessons at once. Here is the first step.
The 105° angle and a sit together on a straight line.
Angles on a straight line add up to 180°, so a = 180° − 105° = 75°.
You found that a = 75°. Now take it down to the other crossing.
Corresponding angles — in the matching position at each crossing — are equal.
a is 75°, so b = 75°.
That's Lesson 2 finished, Florence.
You now know the three pairs of angles that parallel lines make: corresponding angles match, alternate angles match, and co-interior angles add up to 180°. Together with the three facts from Lesson 1, that is six rules — and almost every angle puzzle you'll meet is built from those six.
Two lessons in, and all of it your own work. Lesson 3 will be here whenever you want it.
You can close this page now — or step back through any part you'd like to see again.