Chapter 2 of 8

All 8 chapters
  1. 1Angles on a line and round a point, readunderstood
  2. 2Angles and parallel lines, readunderstood
  3. 3Triangles, readunderstood
  4. 4Angles in polygons, readunderstood
  5. 5Drawing with ruler and compasses, readunderstood
  6. 6Line symmetry and reflection, readunderstood
  7. 7Turning symmetry, readunderstood
  8. 8Solids, nets and views, readunderstood

M3.5.2 Chapter 2

Angles and parallel lines

A road crosses two parallel railway tracks. I know one angle. Can I find the other seven?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find equal angles where a line crosses two parallel lines
  • Name alternate and corresponding angles

Parallel lines run side by side. They stay the same distance apart and never meet. We mark them with arrows that point the same way.

A line that crosses two parallel lines is a transversal. It leans the same way at both crossings. So the angles at the top crossing match the angles at the bottom crossing.

Draw it first

The same place at each crossing

  1. Two parallel lines, each with an arrow. A line crosses both of them. At the top crossing the angle on the upper right is 65° and is shaded with hatching. At the bottom crossing the angle in the same place is x and is also shaded. Under the picture it says x = 65°.

    1 Draw two parallel lines. Mark each with an arrow.

  2. Two parallel lines, each with an arrow. A line crosses both of them. At the top crossing the angle on the upper right is 65° and is shaded with hatching. At the bottom crossing the angle in the same place is x and is also shaded. Under the picture it says x = 65°.

    2 Draw a line across both of them.

  3. Two parallel lines, each with an arrow. A line crosses both of them. At the top crossing the angle on the upper right is 65° and is shaded with hatching. At the bottom crossing the angle in the same place is x and is also shaded. Under the picture it says x = 65°.65°xx = 65°

    3 The angle in the same place at each crossing is the same size.

The two shaded angles are in the same place at each crossing. They are corresponding angles. Look for the letter F. Corresponding angles between parallel lines are equal.

Two parallel lines with arrows and a crossing line. Shaded with hatching are the angle at the upper left of the top crossing and the angle at the upper left of the bottom crossing. Both are 115°.115°115°
These corresponding angles make an F shape. Both are 115°.

Some angles sit between the parallel lines, on opposite sides of the transversal. They are alternate angles. Look for the letter Z. Alternate angles between parallel lines are equal.

Worked example

The two lines are parallel. One angle is 72°. What is the alternate angle x?

  1. Both angles are between the parallel lines.
  2. They are on opposite sides of the transversal.
  3. They make a Z shape, so they are alternate angles.
  4. Alternate angles are equal, so x = 72°.

Answer: 72°

Two parallel lines with arrows and a crossing line. Between the lines, the angle at the lower left of the top crossing is 72° and the angle at the upper right of the bottom crossing is x. Both are shaded with hatching. They make a Z shape.72°x
The angle x is 72°. The two angles lie in a Z shape.

Worked example

The two lines are parallel. The angle at the top is 65°. What is the angle y at the bottom?

  1. The angle next to 65° on the top line is 180° − 65° = 115°.
  2. The angle y is in the same place at the bottom crossing.
  3. So y is a corresponding angle, and y = 115°.

Answer: 115°

Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 65°. The angle at the upper left of the bottom crossing is y.65°y
The angle y is 115°. It sits next to the angle that corresponds to 65°, so y = 180° − 65°.

See it three ways

With things

Draw a line along each long edge of a ruler. The two lines are parallel. Draw a third line across both.

As a picture

Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 70°. The angle in the same place at the bottom crossing is also 70°. Both are shaded with hatching.70°70°

In symbols

The angle at the top = the angle in the same place at the bottom. 70° = 70°.

Try it

Measure the angles between parallel lines

You need: a sheet of paper, a ruler, a protractor, a pencil.

  1. Lay the ruler on the paper. Draw a line along each long edge.
  2. Draw a straight line across both lines.
  3. Measure the four angles at the top crossing. Then measure the four at the bottom.
  4. Compare the angles that are in the same place at each crossing.

What you will see: Each angle matches the one in the same place at the other crossing. Only two sizes appear. A degree or two of difference is normal.

If you cannot try it, this is what happens.
Two parallel lines with arrows and a crossing line. At each crossing the four angles are marked. The upper right and lower left angles are 65°. The other two are 115°. The angles in the same place at the two crossings match.65°115°65°115°65°115°65°115°

A wrong idea many people have

All eight angles are equal.

Why it feels right: The lines are parallel, so everything looks the same.

The test that breaks it: Measure two angles side by side at one crossing. One is small and one is big. They fill a straight line, so they add to 180°.

What is true: Only two sizes appear. Corresponding angles are equal and alternate angles are equal. Angles side by side add to 180°.

Key words

parallel lines
straight lines that stay the same distance apart and never meet.
transversal
a straight line that crosses two or more other lines.
corresponding angles
angles in the same place at each crossing, in an F shape. Between parallel lines they are equal.
alternate angles
angles between the lines on opposite sides of the transversal, in a Z shape. Between parallel lines they are equal.

Back to your guess

The answer is B: Yes. Each angle is the one I know, or 180° minus it.

The tracks are parallel, so the road meets both of them at the same slant. Only two sizes appear. Four angles are the one you know. The other four are 180° minus it.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The lines are parallel. The angle x is in the same place as the 70° angle. What is x?

    Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 70°. The angle in the same place at the bottom crossing is x.70°x
    Tap your answer.

    1. Yes, that is right. The angle x is in the same place at the other crossing. Corresponding angles between parallel lines are equal, so x = 70°.

    2. Not this one. That is 180° − 70°, the angle next to it. The angle in the same place is a corresponding angle, so it is equal to 70°. Go to M3.5.2. Try another.

    3. Not this one. You took 70° from 90°. Corresponding angles between parallel lines are equal. Go to M3.5.2. Try another.

    See the answer

    70°

    The angle x is in the same place at the other crossing. Corresponding angles between parallel lines are equal, so x = 70°.

    • If you chose B: That is 180° − 70°, the angle next to it. The angle in the same place is a corresponding angle, so it is equal to 70°. Go to M3.5.2.
    • If you chose C: You took 70° from 90°. Corresponding angles between parallel lines are equal. Go to M3.5.2.

  2. The lines are parallel. What are the two shaded angles called?

    Two parallel lines with arrows and a crossing line. Two angles between the lines are shaded with hatching. One is at the lower left of the top crossing and one is at the upper right of the bottom crossing. They make a Z shape.
    Tap your answer.

    1. Not this one. Corresponding angles are in the same place at each crossing, like an F. These are between the lines on opposite sides, like a Z. Go to M3.5.2. Try another.

    2. Not this one. Opposite angles face each other at one crossing. These two are at different crossings. Go to M3.5.1. Try another.

    3. Yes, that is right. Both angles are between the parallel lines, on opposite sides of the transversal. They make a Z shape, so they are alternate angles.

    See the answer

    Alternate angles

    Both angles are between the parallel lines, on opposite sides of the transversal. They make a Z shape, so they are alternate angles.

    • If you chose A: Corresponding angles are in the same place at each crossing, like an F. These are between the lines on opposite sides, like a Z. Go to M3.5.2.
    • If you chose B: Opposite angles face each other at one crossing. These two are at different crossings. Go to M3.5.1.

  3. The lines are parallel. One angle is 125°. The angle y is its alternate angle. What is y?

    Two parallel lines with arrows and a crossing line. Between the lines, the angle at the lower right of the top crossing is 125° and the angle at the upper left of the bottom crossing is y. Both are shaded with hatching.125°y
    Tap your answer.

    1. Not this one. That is the angle next to 125° on a straight line. The alternate angle is equal to 125°. Go to M3.5.2. Try another.

    2. Yes, that is right. The two angles make a Z shape between the parallel lines. Alternate angles are equal, so y = 125°.

    3. Not this one. You took 125° from 360°. Alternate angles are equal, so y is 125°. Go to M3.5.2. Try another.

    See the answer

    125°

    The two angles make a Z shape between the parallel lines. Alternate angles are equal, so y = 125°.

    • If you chose A: That is the angle next to 125° on a straight line. The alternate angle is equal to 125°. Go to M3.5.2.
    • If you chose C: You took 125° from 360°. Alternate angles are equal, so y is 125°. Go to M3.5.2.

  4. The lines are parallel. One angle is 64°. What is the angle x?

    Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 64°. The angle at the lower right of the bottom crossing is x.64°x
    Tap your answer.

    1. Yes, that is right. At the top, the lower right angle is next to 64° on a straight line. So it is 180° − 64° = 116°. The angle x is in the same place at the bottom, so x = 116°.

    2. Not this one. The angle x is not in the same place as 64°. It is next to the corresponding angle, so it fills a straight line with it. Go to M3.5.2. Try another.

    3. Not this one. You took 64° from 90°. Find the angle in the same place as x at the top, then use 180°. Go to M3.5.1. Try another.

    See the answer

    116°

    At the top, the lower right angle is next to 64° on a straight line. So it is 180° − 64° = 116°. The angle x is in the same place at the bottom, so x = 116°.

    • If you chose B: The angle x is not in the same place as 64°. It is next to the corresponding angle, so it fills a straight line with it. Go to M3.5.2.
    • If you chose C: You took 64° from 90°. Find the angle in the same place as x at the top, then use 180°. Go to M3.5.1.

  5. Which sentence about two parallel lines and a transversal is true?

    Tap your answer.

    1. Not this one. Alternate angles are equal. The angles that add to 180° are side by side on a straight line. Go to M3.5.2. Try another.

    2. Not this one. Only two sizes appear. Angles side by side add to 180°, so they are equal only when both are 90°. Go to M3.5.2. Try another.

    3. Yes, that is right. Corresponding angles are in the same place at each crossing. Between parallel lines they are equal.

    See the answer

    Corresponding angles are equal.

    Corresponding angles are in the same place at each crossing. Between parallel lines they are equal.

    • If you chose A: Alternate angles are equal. The angles that add to 180° are side by side on a straight line. Go to M3.5.2.
    • If you chose B: Only two sizes appear. Angles side by side add to 180°, so they are equal only when both are 90°. Go to M3.5.2.

  6. The lines are parallel. An angle at the top is 58°. The angle in the same place at the bottom is x. What is the angle next to x on the straight line?

    See the answer

    122°

    The corresponding angle x is 58°. The angle next to x fills a straight line with it, so it is 180° − 58° = 122°.

  7. From an earlier chapter

    Three angles fill a straight line. Two of them are 38° and 52°. What is the third angle?

    See the answer

    90°

    38° + 52° = 90°. Then 180° − 90° = 90°.

Keep it

Copy this into your exercise book.

A line across parallel lines makes two sizes of angle. Corresponding angles (F) are equal. Alternate angles (Z) are equal. Angles side by side add to 180°.

Draw: Two parallel lines with arrows and a line across them. Shade a pair of corresponding angles and a pair of alternate angles.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (801 KB). It opens on any phone.