Geometry of straight lines (Grade 9) — Mathematics Grades 8–11

Naming angle pairs at intersecting and parallel lines, and calculating unknown angles with reasons.

Gr 9 Term 2 — Geometry of straight linesTerm 2

Practise Geometry of straight lines in the app →

What gets asked

Name a pair of angles, such as alternate, co-interior or vertically opposite.
Calculate an unknown angle on a straight line or at intersecting lines.
Calculate an unknown angle at parallel lines cut by a transversal, with a reason.
Set up and solve an equation for an angle given as an algebraic expression.

You must be able to

Tell alternate, corresponding and co-interior angles apart on a diagram.
Use the fact that angles on a straight line add up to 180 degrees.
Give the correct geometric reason for an angle calculation at parallel lines.
Write an equation from two angle expressions and solve for the unknown.

Traps that cost marks

Mixing up alternate angles and co-interior angles, since both sit between the parallel lines. Alternate angles are equal and form a Z-shape. Co-interior angles are supplementary and form a C-shape.
Treating co-interior angles as equal instead of supplementary. Co-interior angles on parallel lines always add up to 180 degrees.
Getting the correct angle size but citing the wrong reason. Match the reason to the angle pair actually used, such as corresponding, not alternate.

Worked example

Two straight lines cross at point T. One angle at T is (3x + 10)°, and the angle next to it on the same straight line is (2x + 15)°. Solve for x.

Angles on a straight line add up to 180°.
$(3 x + 10) + (2 x + 15) = 180$ .
$5 x + 25 = 180$ , so $x = 31$ .

Angle pairs

The name of each angle pair at a crossing, and whether that pair is equal or adds to a straight line.

transversal: A straight line that cuts across two or more other lines.
vertically opposite angles: The two angles facing each other where two lines cross.
corresponding angles: A pair in the same position at each crossing. They make an F-shape.
alternate angles: A pair on opposite sides of the transversal, between the lines. A Z-shape.
co-interior angles: A pair on the same side of the transversal, between the lines. A C-shape.
supplementary: Two angles that add up to $18 0^{\circ}$ , the angle on a straight line.

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.

A transversal cuts two lines and makes eight angles. Name the pair before you use any rule. The name is what decides whether the angles are equal or supplementary.

Corresponding angles sit in matching positions, one at each crossing. Alternate angles sit on opposite sides of the transversal, between the lines. Co-interior angles sit on the same side, between the lines.

On parallel lines, corresponding angles are equal and alternate angles are equal. Co-interior angles are supplementary instead. These three rules need the arrow marks that show the lines are parallel.

Rules to remember

co-interior angles on parallel lines: $x + y = 18 0^{\circ}$

Examples

Two straight lines cross. One of the four angles is

5 0^{\circ}

. Find the other three.

Worked answer

The angle facing it is vertically opposite, so it is $5 0^{\circ}$ too.
Each angle beside it is supplementary to it.
$180 - 50 = 130$ .
The other two angles are both $13 0^{\circ}$ .

Answer: $5 0^{\circ}$ , $13 0^{\circ}$ and $13 0^{\circ}$

Four angles at a crossing come in two equal pairs.

A transversal cuts two parallel lines. One of the angles between those two lines is

11 5^{\circ}

. Find the co-interior angle that goes with it.

Worked answer

The lines are parallel, so co-interior angles are supplementary.
They add up to $18 0^{\circ}$ .
$180 - 115 = 65$ .
The co-interior angle is $6 5^{\circ}$ .

Answer: $6 5^{\circ}$

Co-interior is the one named pair that adds up instead of matching.

A transversal cuts two lines. An angle of

7 0^{\circ}

at one crossing corresponds to one of

8 0^{\circ}

at the other. Are the lines parallel?

Worked answer

On parallel lines, corresponding angles are equal.
Here they are $7 0^{\circ}$ and $8 0^{\circ}$ .
Those are not equal, so the test fails.
The lines are not parallel.

Answer: no, they are not parallel

The rule works both ways, so unequal corresponding angles rule parallel out.

Traps

Mixing up alternate angles and co-interior angles, since both sit between the parallel lines. Alternate angles are equal and form a Z-shape. Co-interior angles are supplementary and form a C-shape.
Calling a corresponding pair vertically opposite because the diagram is turned. Vertically opposite angles meet at ONE crossing. Corresponding angles sit at two crossings.
Treating co-interior angles as equal instead of supplementary. Co-interior angles on parallel lines always add up to 180 degrees.
Using the alternate rule on lines that are not marked parallel. Check for the arrow marks first. Without them these rules do not hold.

Learn and practise “Angle pairs” in the app →

Find unknown angles

Angles on lines follow fixed rules, and in geometry the reason scores as well as the number.

vertically opposite: The two angles facing each other where two lines cross. They are equal.

Angles on a straight line add up to $18 0^{\circ}$ , not $36 0^{\circ}$ . Angles round a point add up to $36 0^{\circ}$ . Knowing which situation you are in is most of the work.

Where two straight lines cross, the vertically opposite angles are equal, and each pair of angles next to each other on a line is supplementary.

When a transversal cuts two PARALLEL lines, corresponding angles are equal and alternate angles are equal. Co-interior angles, on the same side between the lines, are supplementary.

These three rules need the lines to be parallel. If the diagram does not mark them parallel, you may not use them. Check for the arrows before you start.

An angle can be given as an expression, such as $(2 x + 10)^{\circ}$ . Set up an equation, solve for $x$ , then put $x$ back in to get the ANGLE. The question asked for the angle, not for $x$ .

Rules to remember

angles on a straight line: $a + b = 18 0^{\circ}$
angles round a point: $a + b + c = 36 0^{\circ}$

Examples

Two angles lie next to each other on a straight line. One is

7 0^{\circ}

. Find the other.

Worked answer

Angles on a straight line add up to $18 0^{\circ}$ .
So the second angle is $180 - 70$ .
$180 - 70 = 110$ .
The other angle is $11 0^{\circ}$ .

Answer: $11 0^{\circ}$

Using $36 0^{\circ}$ here would give $29 0^{\circ}$ , which is larger than a straight line.

A transversal cuts two parallel lines. One angle is

6 5^{\circ}

. Find the corresponding angle at the other crossing, and give a reason.

Worked answer

The lines are parallel, so corresponding angles are equal.
The corresponding angle is therefore also $6 5^{\circ}$ .
The reason is: corresponding angles, parallel lines.

Answer: $6 5^{\circ}$ , corresponding angles

The number alone does not score full marks. The reason names the rule you used.

Two angles on a straight line are

(2 x + 10)^{\circ}

and

(3 x)^{\circ}

. Find the larger angle.

Worked answer

They are on a straight line, so they add to $180$ .
$2 x + 10 + 3 x = 180$ , so $5 x + 10 = 180$ .
Subtract $10$ : $5 x = 170$ , so $x = 34$ .
First angle: $2 (34) + 10 = 78$ . Second: $3 (34) = 102$ .
The larger angle is $10 2^{\circ}$ .

Answer: $10 2^{\circ}$

Stopping at $x = 34$ answers a question nobody asked.

Traps

Using $36 0^{\circ}$ for the angles on a straight line. A straight line is $18 0^{\circ}$ . Only the angles round a point add to $36 0^{\circ}$ .
Getting the size right but naming the wrong rule, such as calling an alternate pair corresponding. Corresponding angles sit in matching positions; alternate angles sit on opposite sides between the lines.
Solving for $x$ and offering that as the size of the angle. Substitute $x$ back into the expression. If $x = 34$ , then $(3 x)^{\circ}$ is $10 2^{\circ}$ .

Learn and practise “Find unknown angles” in the app →

Grade 9 Geometry of straight lines

What gets asked

You must be able to

Traps that cost marks

Worked example

Angle pairs

Rules to remember

Examples

Traps

Find unknown angles

Rules to remember

Examples

Traps

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