Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 4, lesson 2 (angles on a line and at a point): two lines cross and one angle is 70°. What is the angle next to it?
Answer
110°. Angles on a straight line add up to 180°, and 180 − 70 = 110. - From Unit 1, lesson 6 (order of operations): what is 180 − 2 × 65?
Answer
50. Multiply first: 2 × 65 = 130, then 180 − 130 = 50. - From Unit 3 (patterns, expressions and equations): solve 2x + 10 = 70.
Answer
x = 30. Subtract 10 from both sides to get 2x = 60, then divide by 2.
Words used in this lesson
- transversal
- a line that crosses two or more other lines. For example: The diagonal band across two stripes.
- corresponding angles
- angles in the same position at the two crossings. For example: The F shape; equal for parallel lines.
- alternate angles
- angles inside the lines on opposite sides of the transversal. For example: The Z shape; equal for parallel lines.
- co-interior angles
- angles inside the lines on the same side of the transversal. For example: The C shape; add to 180° for parallel lines.
- parallel
- in the same direction and never meeting. For example: Two rails of a track.
- converse
- the same statement the other way round. For example: If angles are equal, the lines are parallel.
One crossing, copied
A weaver in Cusco sets two parallel stripes in a belt and crosses both with a diagonal band of another colour. The band meets each stripe at the same slant, so the same angles show up twice. A line that crosses two other lines is called a transversal. When the other two lines are parallel, the angles at one crossing are copies of the angles at the other.
Predict first: at the top crossing the angle is 65°. Which angle at the bottom crossing do you think is also 65°? Press the buttons to test your guess, then tilt the transversal.
Corresponding angles sit in the same position at the two crossings. Both are 65°, so they are equal.
Show me the steps
- Corresponding at 65°: the angle at the top crossing and the one in the same position at the bottom crossing are both 65°. They are equal.
- Press Alternate. The angles inside the parallels, on opposite sides of the transversal, are both 65°. They are equal too.
- Press Co-interior. The angles inside the parallels on the same side are 115° and 65°, which add up to 180°.
- Move the tilt to 110° with Corresponding. Both are 110°: the pair stays equal whatever the tilt.
Parallel lines cut by a transversal make three kinds of pair.
- Corresponding angles are in the same position at the two crossings (an F shape). They are equal.
- Alternate angles are inside the parallels on opposite sides of the transversal (a Z shape). They are equal.
- Co-interior angles are inside the parallels on the same side (a C shape). They add up to 180°.
Why: slide one crossing along the transversal to the other, and the angle comes with it, so corresponding angles are equal. Vertically opposite angles are equal too, which turns a corresponding pair into an alternate pair. A straight line turns an equal pair into a co-interior pair. These facts only hold when the lines are parallel, shown by arrows.
Write the name, not the letter: “alternate angles on parallel lines are equal”, never “Z angles”. Many mark schemes do not accept the letter.
The test for parallel lines
The facts also run the other way: if a pair of corresponding angles is equal, the lines must be parallel. Tilt the lower line and watch what happens when the angles stop matching.
The corresponding angles are 60° and 50°. They are not equal, so the lines are not parallel: they would meet far to the right.
Show me the steps
- Transversal 60°, tilt 10°. The top corresponding angle is 60° and the bottom one is 60 − 10 = 50°. They differ, so the lines are not parallel.
- Set the tilt to 0°. Both angles are 60°, they are equal, and the lines are parallel.
- Set the tilt to −10°. The top angle is still 60° and the bottom angle is 60 + 10 = 70°. They differ, and the lines would meet on the left.
- Set the transversal to 110° with tilt 0°. Both angles are 110°: equal again, so parallel for any transversal.
A common mistake: “These two lines look parallel, so the angles are equal.” Looks prove nothing: a 2° tilt is almost invisible. Only arrows on the lines, the symbol AB ∥ CD, or equal corresponding angles, say that two lines are parallel. If there is no such evidence, you may not use the F, Z and C facts.
Worked example: a chain of angles with reasons
Two parallel lines are cut by a transversal. One angle between the parallel lines is 52°. Find (a) the corresponding angle, (b) the alternate angle and (c) the co-interior angle, giving a reason each time.
- (a) The corresponding angle is 52°, because corresponding angles on parallel lines are equal.
- (b) The alternate angle is 52°, because alternate angles on parallel lines are equal.
- (c) The co-interior angle is 180 − 52 = 128°, because co-interior angles on parallel lines add up to 180°.
- Check with the picture: the 52° is acute and 128° is obtuse. The angle next to 52° on the same line is also 128°, which agrees. 52°, 52°, 128°
Your go. The first two steps are given. An angle between two parallel lines is 117°. Find the corresponding angle, the alternate angle and the co-interior angle, with a reason each.
- The corresponding angle is 117°, because corresponding angles on parallel lines are equal.
- The alternate angle is 117°, because alternate angles on parallel lines are equal.
- Now find the co-interior angle yourself and say why, then check.
Check the full answer
The co-interior angle is 180 − 117 = 63°, because co-interior angles on parallel lines add up to 180°. The set is 117°, 117° and 63°.
Check before you go on
1. Two parallel lines are cut by a transversal. One angle is 70°. Which angle must also be 70°?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Two parallel lines are cut by a transversal. One angle is 112°. The corresponding angle is x°. What is x?
3. Two parallel lines are cut by a transversal. One angle between the parallel lines is 75°. The co-interior angle on the same side is x°. What is x?
Level 2 · Building
4. Two lines are parallel. Angle a is 48° and it is next to angle b on a straight line. Angle c is the alternate angle to b. What is c, in degrees?
5. Two parallel lines are cut by a transversal. Two alternate angles are (2x + 10)° and 70°. Find x.
6. Lines AB and CD are parallel, and a transversal makes two alternate angles, a and b. A student writes a = b. Which reason earns the mark?
Level 3 · Stretch
7. Two parallel lines are cut by a transversal. Two co-interior angles are (4x + 10)° and (2x + 20)°. Find x.
8. Two parallel lines are cut by a transversal. Two alternate angles are (3x + 20)° and (5x − 40)°. How big is each of these angles, in degrees?
Something to think about
Set the transversal at exactly 90° to two parallel lines. What do the corresponding, alternate and co-interior pairs become? Does a rule break, or is this one more case of the same rule?
One way to think about it
All eight angles are 90°. Corresponding and alternate pairs are 90° and 90°, which are equal, and the co-interior pair is 90° + 90° = 180°. No rule breaks: this is one more position of the transversal. A transversal at 90° is perpendicular to both lines, which are parallel to each other.
Exit ticket
9. Two lines that are not parallel are crossed by a transversal. Which pair of angles is still always equal?