Geometry of straight lines (Grade 8) — Mathematics Grades 8–11

Naming and calculating angles at intersecting and parallel lines.

Gr 8 Term 2 — Geometry of straight linesTerm 2

Practise Geometry of straight lines in the app →

What gets asked

Name a pair of angles formed by intersecting or parallel lines.
State whether a named pair of angles is equal or adds to 180°.
Calculate an unknown angle at a point or on a straight line.
Calculate an unknown angle where a transversal crosses parallel lines.

You must be able to

Tell corresponding, alternate and co-interior angles apart.
Know which angle pairs are equal and which add up to 180°.
Use angles on a straight line to find an unknown angle.
Set up and solve an equation for an angle problem, with reasons.

Traps that cost marks

Saying co-interior angles are equal instead of adding to $180°$ . Co-interior angles are supplementary: they add up to $180°$ , they are not equal.
Using $360°$ for angles that lie on a straight line. Angles on a straight line add up to $180°$ .
Setting two co-interior angles equal to each other instead of adding them to $180°$ . Co-interior angles between parallel lines always add up to $180°$ .
Assuming two lines are parallel just because they look parallel in the sketch. Only use the parallel-line angle rules when the lines are marked or stated as parallel.

Worked example

Two parallel lines are cut by a transversal. One co-interior angle is

70°

. Find the other one.

Co-interior angles between parallel lines add up to $180°$ .
$180 - 70 = 110$ .
The other angle is $110°$ .

Naming angle pairs

Every angle pair at a crossing has a name, and the name tells you the rule.

supplementary: Two angles that add up to $18 0^{\circ}$ , a straight line's worth.
vertically opposite angles: The two angles facing each other where two straight lines cross.
transversal: A line that cuts across two other lines.
corresponding angles: One angle at each crossing, both in the same position there.
alternate angles: A pair between the two lines, on opposite sides of the transversal.
co-interior angles: A pair between the two lines, on the same side of the transversal.

Where two straight lines cross, four angles are made. The two facing each other are vertically opposite angles, and they are equal. Two angles side by side on one line are supplementary. If the lines cross at $9 0^{\circ}$ they are perpendicular.

A transversal cuts across two lines, making eight angles in all. Name the pair before you use any rule. The name decides whether they are equal or supplementary.

Corresponding angles sit in the same position at each crossing. Alternate angles sit between the two lines, on opposite sides of the transversal. Co-interior angles also sit between the lines, but on the same side.

These rules only start once the lines are parallel. Then corresponding and alternate angles are equal, while co-interior angles are supplementary. Look for the arrow marks, or words saying the lines are parallel. Without those, two angles can look equal and still not be.

Your textbook may write this angle as $\hat{O}_{1}$ — the vertex, then a number read off the diagram. This site writes $A \hat{O} C$ . Same angle.

Rules to remember

Vertically opposite angles are equal, whether the lines are parallel or not.
Corresponding and alternate angles are equal only on parallel lines.

Examples

Two straight lines cross, making an angle of

6 5^{\circ}

. Name and size the other three angles.

Worked answer

The angle facing it is vertically opposite, so it is $6 5^{\circ}$ too.
The other two each sit on a straight line with it.
That makes them supplementary: $180 - 65 = 115$ .
So those two are $11 5^{\circ}$ each.

Answer: $6 5^{\circ}$ , $11 5^{\circ}$ and $11 5^{\circ}$

Facing angles are equal, and side-by-side angles fill a straight line.

A transversal cuts two parallel lines. Two angles lie between the lines, on the same side of it. One is

7 0^{\circ}

. Find the other.

Worked answer

Between the lines and on the same side names them co-interior angles.
The lines are parallel, so co-interior angles are supplementary.
$180 - 70 = 110$ .

Answer: $11 0^{\circ}$

The name came first, and the name is what picked the rule.

A transversal cuts two parallel lines. Two angles lie between the lines, on opposite sides of it. One is

11 8^{\circ}

. Find the other.

Worked answer

Between the lines and on opposite sides names them alternate angles.
Co-interior angles would be on the same side, so this is a different pair.
On parallel lines, alternate angles are equal.
So the other angle is $11 8^{\circ}$ as well.

Answer: $11 8^{\circ}$

Both pairs lie between the lines, so the side of the transversal separates them.

Traps

Calling two angles that sit side by side vertically opposite, because they meet at one point. Vertically opposite angles face each other across the crossing. Side-by-side angles are supplementary.
Naming a pair corresponding just because the two angles look equal. Check the position of each angle against the transversal before you name the pair.
Saying co-interior angles are equal instead of adding to $18 0^{\circ}$ . Co-interior angles are supplementary: they add up to $18 0^{\circ}$ , they are not equal.
Assuming two lines are parallel just because they look parallel in the sketch. Only use the parallel-line angle rules when the lines are marked or stated as parallel.

Learn and practise “Naming angle pairs” in the app →

Angles on lines and parallels

Working out the angle you were not given, and writing down the rule that let you do it.

adjacent angles: Two angles side by side, sharing one arm and one corner.
revolution: One full turn right around a point, worth $36 0^{\circ}$ .
reason: The rule you name to show why a line of your working is true.

Adjacent angles on a straight line add up to $18 0^{\circ}$ . Adjacent angles right around a point add up to $36 0^{\circ}$ , one full revolution. Pick the right total first. Using $36 0^{\circ}$ where a straight line needs $18 0^{\circ}$ doubles your answer.

Adding the angles you were given is only half the job. If two angles on a line are $x$ and $5 0^{\circ}$ , then $x = 180 - 50 = 130$ . The subtraction is what finds the unknown.

On parallel lines, let the name of the pair choose the rule. Corresponding and alternate angles are equal, so you copy the size across. Co-interior angles are supplementary, so you subtract from $18 0^{\circ}$ .

In a longer problem, give the unknown angle a letter and write an equation. Solve it, then read the sizes back to check they are sensible. Never treat lines as parallel unless the diagram or the words say so.

Every statement needs a reason beside it. Write the working, then name the rule, such as 'angles on a straight line'. In an exam, a correct number with no reason still loses marks.

Rules to remember

Angles on a straight line add up to $18 0^{\circ}$ .
Angles around a point add up to $36 0^{\circ}$ .

Examples

Three angles sit side by side on a straight line. They are

x

4 0^{\circ}

and

7 5^{\circ}

. Find

x

Worked answer

They lie on a straight line, so the total is $18 0^{\circ}$ , not $36 0^{\circ}$ .
Add the two known angles: $40 + 75 = 115$ .
Take that from the total: $180 - 115 = 65$ .
So $x = 6 5^{\circ}$ , and the reason is: angles on a straight line.

Answer: $x = 6 5^{\circ}$

Adding was only half of it, and the subtraction finished the job.

Parallel lines

P Q

and

R S

are cut by a transversal. A co-interior pair measures

3 x

and

6 0^{\circ}

. Find

x

Worked answer

The lines are parallel, so co-interior angles are supplementary.
That gives the equation $3 x + 60 = 180$ .
Take $60$ from both sides: $3 x = 120$ .
Divide both sides by $3$ : $x = 40$ .
The two angles are then $12 0^{\circ}$ and $6 0^{\circ}$ .

Answer: $x = 40$

The equation came from the name of the pair, not from how the angles looked.

Two straight lines cross. One angle is

2 x

and the angle facing it is

7 0^{\circ}

. Find

x

and the angle beside it.

Worked answer

Vertically opposite angles are equal, so $2 x = 70$ .
Divide both sides by $2$ : $x = 35$ .
The angle beside it lies on a straight line with the $7 0^{\circ}$ .
So that angle is $180 - 70 = 110$ .

Answer: $x = 35$ , and the angle beside it is $11 0^{\circ}$

Two rules were used here, so two reasons have to be written down.

Traps

Using $36 0^{\circ}$ for angles that lie on a straight line. Angles on a straight line add up to $18 0^{\circ}$ .
Adding the known angles and stopping there. Adding is the first step. Taking that total from $18 0^{\circ}$ is what gives the unknown angle.
Setting two co-interior angles equal to each other instead of adding them to $18 0^{\circ}$ . Co-interior angles between parallel lines always add up to $18 0^{\circ}$ .
Writing the answer down with no reason next to it. Name the rule you used for every line, such as 'vertically opposite angles'.

Learn and practise “Angles on lines and parallels” in the app →

Grade 8 Geometry of straight lines

What gets asked

You must be able to

Traps that cost marks

Worked example

Naming angle pairs

Rules to remember

Examples

Traps

Angles on lines and parallels

Rules to remember

Examples

Traps

About this material