E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Angles Formed by Parallel Lines and a Transversal

E2.2 A1About 50 minutes

Learning goals

I can name corresponding, alternate and co-interior angles when a transversal crosses two lines.
I can use the angle properties of intersecting and parallel lines to find unknown angles, and give a reason for each step.
I can use angles to decide whether two lines are parallel.

Words to know

parallel lines
Lines in the same flat surface that never meet. They stay the same distance apart. Matching arrow marks (>) on a diagram show that lines are parallel.

transversal
A line that crosses two or more other lines.

opposite angles
The angles across from each other where two lines cross. They are always equal. (Also called vertically opposite angles.)

corresponding angles
Angles in the same position at each of the two crossing points. When the lines are parallel, corresponding angles are equal.

alternate angles
Angles on opposite sides of the transversal and between the two lines (alternate interior angles). When the lines are parallel, alternate angles are equal.

co-interior angles
Angles on the same side of the transversal and between the two lines. When the lines are parallel, co-interior angles add to 180°.

supplementary angles
Two angles that add to 180°, like the two angles on a straight line.

Remember from Grade 6

In Grade 6 you learned that angles on a straight line add to 180° (they are supplementary), angles that make a right angle add to 90° (they are complementary), angles all the way around a point add to 360°, and when two lines cross, the opposite angles are equal. In Grade 8 you will use those facts together with a new idea: what happens when one line crosses two parallel lines.

Let's start with a road map

Picture two parallel streets running east to west, and one long road cutting across both of them on a slant. Stand at the first corner and look at the angles where the roads meet. Now walk to the second corner. The angles look exactly the same! That is not a coincidence. When a line crosses two parallel lines, the angles at the two crossing points match. Today you'll learn which angles match and why.

Eight angles, two crossing points

A transversal is a line that crosses two or more lines. When a transversal crosses two lines, it makes 8 angles: 4 at each crossing point. In the diagram, the angles are labelled a to h. The arrow marks (>) show that the two horizontal lines are parallel.

Two parallel lines (arrow marks) cut by a transversal. Angles a, b, c, d are at the top crossing point; e, f, g, h are in the same positions at the bottom crossing point.

Pair name	Where to find them	Pairs in the diagram	When the lines are parallel
Opposite angles	Across from each other at one crossing point	a and d, b and c, e and h, f and g	Equal (true for any two crossing lines)
Corresponding angles	Same position at each crossing point (an F shape)	a and e, b and f, c and g, d and h	Equal
Alternate angles	Between the lines, on opposite sides of the transversal (a Z shape)	c and f, d and e	Equal
Alternate exterior angles	Outside the lines, on opposite sides of the transversal	a and h, b and g	Equal
Co-interior angles	Between the lines, on the same side of the transversal (a C or U shape)	c and e, d and f	Add to 180°

The angle pairs to know. Only opposite angles are equal for any two crossing lines. The others need the lines to be parallel.

The same diagram with the angles measured. The transversal makes a 60° angle with each parallel line, so every angle is either 60° or 120°, and 60° + 120° = 180°. The angles are drawn to their true size.

Why do these rules work?

Think of sliding the top crossing point down the transversal until it sits on top of the bottom one. Because the lines are parallel, the top line lands exactly on the bottom line, so every angle lands on its corresponding angle. That is why corresponding angles are equal.

The other rules follow from that one:

Alternate angles: c = b (opposite angles) and b = f (corresponding). So c = f.
Co-interior angles: e and f sit on a straight line, so e + f = 180°. And c = f. So c + e = 180°.

It works both ways

If the lines are parallel, then corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°.
If you measure and find that corresponding angles are equal (or alternate angles are equal, or co-interior angles add to 180°), then the lines must be parallel.
If the angles don't follow the rules, the lines are not parallel, and they will meet somewhere if you extend them far enough.

Worked example

Example 1: Find all eight angles

In the measured diagram, angle b = 60°. Find the other seven angles, with a reason for each.

a = 180° − 60° = 120°, because a and b are on a straight line (supplementary).
c = 60°, opposite to b. d = 120°, opposite to a.
Now move to the bottom crossing point. f = 60°, corresponding to b.
e = 120°, corresponding to a. g = 60°, corresponding to c. h = 120°, corresponding to d.
Check one pair a different way: c and e are co-interior, and 60° + 120° = 180° ✓

a = 120°, c = 60°, d = 120°, e = 120°, f = 60°, g = 60°, h = 120°. Every angle is either 60° or its supplement, 120°.

Worked example

Example 2: Co-interior angles

Two parallel lines are cut by a transversal. One of the co-interior angles is 115°. Find the other one.

Co-interior angles add to 180° when the lines are parallel.
180° − 115° = 65°.

The other co-interior angle is 65° (reason: co-interior angles between parallel lines add to 180°).

Worked example

Example 3: Are the lines parallel?

A carpenter is checking two boards that are supposed to be parallel. A straight brace crosses both boards. The two co-interior angles measure 82° and 96°. Are the boards parallel?

If the boards were parallel, the co-interior angles would add to 180°.
82° + 96° = 178°, not 180°.
So the boards are not parallel. They are slightly closer together on one side and would meet if they were extended far enough.

No. The co-interior angles add to 178°, not 180°, so the boards are not parallel.

Think like a mathematician

Write a reason beside every angle you find: opposite angles, corresponding angles, alternate angles, co-interior angles, straight line. Reasons let someone else follow your thinking. They also help you catch a mistake, because every step must match a rule. If you get stuck, look for the letter shapes: F for corresponding, Z for alternate, C for co-interior.

Hands-on activity

Angle hunt on lined paper

You need: a sheet of lined paper (the lines are parallel), a ruler, a pencil, and a protractor. No protractor? Use a piece of tracing paper or parchment paper instead.

Pick two lines on the paper that are a few lines apart. Use the ruler to draw a slanted transversal across both.
Label the eight angles a to h, in the same positions as the diagram in this lesson.
Measure all eight angles with the protractor. Which ones are equal? Which pairs add to 180°?
No protractor? Trace angle c onto tracing paper, then slide the tracing down the transversal. Does it fit exactly over angle f? Try c on g and b on f too.
Now draw two lines that are clearly not parallel (like a narrow V shape) and a transversal across them. Measure again. Do the rules still work?

What to look for: with the parallel lines, corresponding and alternate angles match (within a degree or two, because measuring is never perfect) and co-interior angles add to about 180°. With the non-parallel lines, they don't. Only the opposite angles stay equal.

Math talk: "Why do the opposite angles stay equal even when the lines are not parallel?"

Just for fun

Why are parallel lines so sad?

They have so much in common, but they're never going to meet.

Before you practise

In the practice questions, the letters a to h are in the same positions as in the lettered diagram, but the angle sizes are different from the drawing. The lines are parallel unless the question says otherwise.

Your turn

Which angle corresponds to angle c?

Which angle is alternate to angle d?

Angle a = 110°. Find angle d.

Angle a = 110°. Find angle e.

Angle a = 110°. Find angle f.

Angle c = 72°. Find angle e.

A transversal crosses two lines. A pair of alternate angles measures 64° and 66°. Are the lines parallel?

More practice: warm-up

What is the supplement of 35°?

Two lines cross. One angle is 48°. What is the angle opposite to it?

Two parallel lines are cut by a transversal. One angle is 125°. What is its corresponding angle?

When the lines are parallel, co-interior angles add to what?

What kind of pair are angles c and f?

More practice: on your own

Angle b = 57°. Find angles a, c and d.

Angle h = 131°. Find angle b.

The rungs of a ladder are parallel. One side rail is slanted and crosses two rungs. At the top rung, the angle above the rung and to the left of the rail is 75°. What is the angle in the same position at the next rung down?

Two parallel lines are cut by a transversal. One co-interior angle is 40° more than the other. Find both angles.

Angle e is twice angle f. Find both.

For Challenge question 2. The two horizontal lines are parallel. Point P is between them. The angles are drawn to their true size.

Challenge

Two parallel lines are cut by a transversal. A pair of corresponding angles measure (3x + 15)° and (5x − 25)°. Find x and the size of the angles.

Look at the diagram above. Find the angle at P.

Explain, using corresponding angles and a straight line, why co-interior angles add to 180° when the lines are parallel. Use the letters from the lettered diagram.

Check your understanding

Quiz

Two parallel lines are cut by a transversal. Corresponding angles are:

Two parallel lines are cut by a transversal. One co-interior angle is 72°. What is the other?

What is a line that crosses two or more other lines called?

Which letter shape helps you find alternate angles?

Angle a = 140° (lettered diagram, parallel lines). What is angle h?

A transversal crosses two lines. The alternate angles measure 50° and 55°. What can you say about the two lines?

For parents and teachers

Big idea: The Ontario curriculum (E2.2) asks Grade 8 students to solve problems involving angle properties, including the properties of intersecting and parallel lines. Your child already knows (from Grade 6) that opposite angles are equal and angles on a straight line add to 180°. This lesson adds the angle pairs made when a transversal crosses parallel lines.

Common difficulties:

Using the parallel-line rules when the lines are not parallel. The rules only work for parallel lines (look for the arrow marks). Only opposite angles are equal for any crossing lines.
Mixing up which pairs are equal and which add to 180°. A quick check: in these diagrams, every angle is either the "small" one or the "big" one. Two small angles or two big angles are equal; a small and a big add to 180°.
Giving answers without reasons. Encourage your child to write the reason beside every angle, the way they will in high school geometry.

How to help: Look for parallel lines with a transversal around you: railway tracks and ties, ladder rungs, a fence with a diagonal brace, lined paper. Ask, "Which angles here must be equal?" This lesson also builds A1 skills: reasoning, proving and communicating.