Parallel Lines and Transversals

When a straight line crosses two parallel lines it makes corresponding, alternate and interior angles. Learn to name each pair in a figure, use the rules to find unknown angles, and test whether two lines are parallel.

Before you start

An angle is named with three letters, the vertex in the middle: $\angle APQ$ is the angle at $P$ between $PA$ and $PQ$ (Lesson 11.1).
Adjacent angles on a straight line add up to $180^{\circ}$ (adj. ∠s on a str. line). If one of them is $65^{\circ}$ , the other is $180^{\circ} - 65^{\circ} = 115^{\circ}$ (Lesson 11.2).
When two straight lines cross, vertically opposite angles are equal (vert. opp. ∠s) (Lesson 11.3).
An equation such as $5x - 25 = 3x + 15$ is solved by doing the same to both sides (Chapter 9).

Parallel Lines and a Transversal

Parallel lines are straight lines on a flat surface that never meet, however far they are extended. They stay the same distance apart all the way along, like the two rails of an MRT track or the printed lines of an exercise book.

In a figure, parallel lines are marked with matching arrows. If $AB$ is parallel to $CD$ , we write $AB \parallel CD$ ; in a reason it is written AB // CD. A second pair of parallel lines in the same figure is marked with double arrows.

Never decide that two lines are parallel because they look parallel. Go by the arrows, or by the words of the question.

Parallel lines and transversals

Parallel lines are straight lines on a flat surface that never meet. A transversal is a straight line that crosses two or more lines at different points.

Example 1: Reading the arrow marks

Write down the pairs of parallel lines in the figure. Is $AB$ parallel to $BC$ ?

Solution

The single arrows are on $AB$ and $DC$ , so $AB \parallel DC$ .

The double arrows are on $AD$ and $BC$ , so $AD \parallel BC$ .

$AB$ and $BC$ meet at $B$ , so they are not parallel. Their marks are different too: single arrows on $AB$ , double arrows on $BC$ .

$AB \parallel DC$ and $AD \parallel BC$ . $AB$ is not parallel to $BC$ : they meet at $B$ .

When a transversal crosses two lines, it makes four angles at each crossing: eight angles in all. In the figure below, $AB \parallel CD$ and the transversal $EF$ crosses both of them.

At one crossing you already know how the angles are related. $a$ and $d$ are vertically opposite angles, so $a = d$ . $a$ and $b$ are adjacent angles on the straight line $AB$ , so $a + b = 180^{\circ}$ .

This lesson is about something new: how an angle at one crossing is related to an angle at the other crossing.

Corresponding, Alternate and Interior Angles

Describe where each angle is in two ways: which side of the transversal it is on (left or right of $EF$ ), and which side of its own line it is on (above or below). The four angles between the parallel lines are $c$ and $d$ (below $AB$ ) and $e$ and $f$ (above $CD$ ).

Each of the three kinds of pair below has one angle at each crossing. The table after the box lists the pairs in the figure of the eight angles.

The three pairs of angles

Corresponding angles are at different crossings and in the same position at each: on the same side of the transversal, and both above their line or both below it.

Alternate angles are at different crossings, both between the two lines, and on opposite sides of the transversal.

Interior angles are at different crossings, both between the two lines, and on the same side of the transversal. They are also called co-interior angles.

If the lines are not drawn across the page, "above" and "below" become the two sides of each line: for upright lines, corresponding angles are both on the left of their line or both on the right.

Pair	Where the two angles are	In the figure
Corresponding angles	Same side of the transversal; same side of their own lines	$a$ and $e$ , $b$ and $f$ , $c$ and $g$ , $d$ and $h$
Alternate angles	Both between the lines; opposite sides of the transversal	$c$ and $f$ , $d$ and $e$
Interior angles	Both between the lines; same side of the transversal	$c$ and $e$ , $d$ and $f$

A letter traced along the lines can help you spot each pair. The letter may be turned round, back to front or slanted.

Corresponding angles

d

and

h

make an F shape.

Alternate angles

c

and

f

make a Z shape.

Interior angles

d

and

f

make a C shape.

Two angles at the same crossing are never a pair

Look at $c$ and $d$ . Both are between the lines and they are on opposite sides of the transversal, but they are at the same crossing, so they are not alternate angles. They are adjacent angles on the straight line $AB$ , and $c + d = 180^{\circ}$ .

Example 2: Naming the pairs

In the figure, $AB \parallel CD$ .

Say whether each pair of angles is corresponding, alternate or interior, or none of these.

(a) $q$ and $u$ (b) $s$ and $u$ (c) $s$ and $t$ (d) $r$ and $s$

(e) Which angle corresponds to $w$ ? (f) Which angle is alternate to $t$ ?

Solution

(a) $q$ is above $AB$ , on the left of $EF$ . $u$ is above $CD$ , on the left of $EF$ . They are in the same position at different crossings: corresponding angles.

(b) $s$ is below $AB$ , so it is between the lines, on the right of $EF$ . $u$ is above $CD$ , between the lines, on the left of $EF$ . Both are between the lines, on opposite sides of the transversal: alternate angles.

(d) $r$ and $s$ are both at the crossing with $AB$ , so they are none of these. They are adjacent angles on the straight line $AB$ .

(e) $w$ is below $CD$ , on the right of $EF$ . At the other crossing, the angle below $AB$ on the right of $EF$ is $s$ .

(f) $t$ is between the lines, on the right of $EF$ . Its alternate angle is between the lines, on the left, at the other crossing: $r$ .

(a) Corresponding (b) Alternate (c) Interior (d) None of these: they are at the same crossing. (e) $s$ (f) $r$

The Angle Rules for Parallel Lines

When the two lines are parallel, the transversal crosses both of them at the same slant. So the crossing with $CD$ is an exact copy of the crossing with $AB$ , moved along the transversal, and each angle at one crossing is equal to the angle in the same position at the other. That is the rule for corresponding angles.

The other two rules follow from it. In the figure of the eight angles $a$ to $h$ : $c = b$ (vert. opp. ∠s) and $b = f$ (corr. ∠s, AB // CD), so $c = f$ , and alternate angles are equal. Also $c + a = 180^{\circ}$ (adj. ∠s on a str. line) and $a = e$ (corr. ∠s, AB // CD), so $c + e = 180^{\circ}$ , and interior angles add up to $180^{\circ}$ .

Angles formed by parallel lines and a transversal

If $AB \parallel CD$ , then:

corresponding angles are equal; the reason is written (corr. ∠s, AB // CD);

alternate angles are equal; the reason is written (alt. ∠s, AB // CD);

interior angles add up to $180^{\circ}$ ; the reason is written (int. ∠s, AB // CD).

Each reason names the parallel lines that it uses.

So when the lines are parallel, the eight angles have only two sizes. Here the transversal makes an angle of $60^{\circ}$ with the parallel lines.

Four of the angles are $60^{\circ}$ and four are $120^{\circ}$ , and $60^{\circ} + 120^{\circ} = 180^{\circ}$ . If you know one of the eight angles, you can find all the others.

Example 3: Corresponding angles

In the figure, $AB \parallel CD$ . Find $x$ .

Solution

The $65^{\circ}$ angle is above $AB$ , on the right of $EF$ . $x^{\circ}$ is above $CD$ , on the right of $EF$ . They are in the same position at the two crossings, so they are corresponding angles.

One of them is outside the parallel lines and the other is between them. That does not matter: only the position at each crossing does.

$x = 65$ (corr. ∠s, AB // CD)

$x = 65$ , so the angle is $65^{\circ}$ .

Example 4: Alternate angles

In the figure, $AB \parallel CD$ . Find $y$ .

Solution

The $72^{\circ}$ angle is below $AB$ and $y^{\circ}$ is above $CD$ , so both are between the lines. The $72^{\circ}$ angle is on the right of $EF$ and $y^{\circ}$ is on the left. They are alternate angles.

$y = 72$ (alt. ∠s, AB // CD)

$y = 72$ , so the angle is $72^{\circ}$ .

Example 5: Interior angles

In the figure, $AB \parallel CD$ . Find $z$ .

Solution

$z^{\circ}$ is below $AB$ and the $70^{\circ}$ angle is above $CD$ , so both are between the lines, and both are on the right of $EF$ . They are interior angles, so they add up to $180^{\circ}$ .

$z + 70 = 180$ (int. ∠s, AB // CD)

$z = 180 - 70 = 110$

Check: $110^{\circ} + 70^{\circ} = 180^{\circ}$ .

$z = 110$ , so the angle is $110^{\circ}$ .

Example 6: A pair and a straight line

In the figure, $AB \parallel CD$ . Find $x$ and $y$ , giving a reason for each.

Solution

$x^{\circ}$ and the $130^{\circ}$ angle are both above their line and on the left of $EF$ : corresponding angles.

$x = 130$ (corr. ∠s, AB // CD)

$x^{\circ}$ and $y^{\circ}$ are adjacent angles on the straight line $CD$ .

$y = 180 - 130 = 50$ (adj. ∠s on a str. line)

Check: $130^{\circ} + 50^{\circ} = 180^{\circ}$ .

$x = 130$ and $y = 50$ , so the angles are $130^{\circ}$ and $50^{\circ}$ .

Using Algebra

When the angles are given as expressions in $x$ , name the pair first, then use its rule to form an equation: the two expressions are equal for corresponding or alternate angles, and they add up to $180$ for interior angles. Solve the equation, work out each angle, and check it.

As everywhere in this chapter, $x$ is a number, not an angle: the angle is $(3x + 15)^{\circ}$ , so when $x = 20$ it is $75^{\circ}$ .

Example 7: Alternate angles with algebra

In the figure, $AB \parallel CD$ . Find $x$ , and the size of each marked angle.

Solution

Both angles are between the lines, on opposite sides of $EF$ : alternate angles, so they are equal.

$5x - 25 = 3x + 15$ (alt. ∠s, AB // CD)

Subtract $3x$ from both sides: $2x - 25 = 15$

Add $25$ to both sides: $2x = 40$

Divide both sides by $2$ : $x = 20$

The angles: $3x + 15 = 3(20) + 15 = 75$ and $5x - 25 = 5(20) - 25 = 75$ .

Check: both angles are $75^{\circ}$ , so they are equal, as alternate angles must be.

$x = 20$ , and each marked angle is $75^{\circ}$ .

Example 8: Interior angles with algebra

In the figure, $AB \parallel CD$ . Find $x$ , and the size of each marked angle.

Solution

Both angles are between the lines and on the right of $EF$ : interior angles, so they add up to $180^{\circ}$ .

$(2x + 30) + (3x - 10) = 180$ (int. ∠s, AB // CD)

Collect like terms: $5x + 20 = 180$

Subtract $20$ from both sides: $5x = 160$

Divide both sides by $5$ : $x = 32$

The angles: $2x + 30 = 2(32) + 30 = 94$ and $3x - 10 = 3(32) - 10 = 86$ .

Check: $94^{\circ} + 86^{\circ} = 180^{\circ}$ .

$x = 32$ ; the marked angles are $94^{\circ}$ and $86^{\circ}$ .

When the Lines Are Not Parallel

The words corresponding, alternate and interior describe positions, so they can be used for any two lines cut by a transversal. The rules cannot: they hold only when the lines are parallel.

Here $AB$ and $CD$ are not parallel: they have no arrows, and they were drawn so that they would meet if they were extended to the right. The $70^{\circ}$ and $58^{\circ}$ angles are corresponding angles, but they are not equal.

Check that the lines are parallel first

Before you write (corr. ∠s), (alt. ∠s) or (int. ∠s), make sure the lines are parallel: marked with arrows, or stated in the question. If they are not, the angles of a pair need not be equal or add up to $180^{\circ}$ , and an unknown angle cannot be found from them.

Testing Whether Two Lines Are Parallel

The rules also work the other way round. If a transversal makes equal corresponding angles with two lines, the lines cannot be closing in or spreading apart: they are parallel. The same is true when alternate angles are equal, and when interior angles add up to $180^{\circ}$ .

Two angles that add up to $180^{\circ}$ are supplementary (Lesson 11.1); in a reason this is shortened to "supp.".

Tests for parallel lines

Two lines cut by a transversal are parallel if any one of these is true:

a pair of corresponding angles are equal; the reason is written (corr. ∠s equal);

a pair of alternate angles are equal; the reason is written (alt. ∠s equal);

a pair of interior angles add up to $180^{\circ}$ ; the reason is written (int. ∠s supp.).

If a pair does not fit its rule (corresponding or alternate angles that are not equal, or interior angles that do not add up to $180^{\circ}$ ), the lines are not parallel.

When you are showing that two lines are parallel, do not write (…, AB // CD): that reason uses the fact you are trying to prove.

Example 9: Are the lines parallel?

In each figure, $AB$ and $CD$ are straight lines. Is $AB \parallel CD$ ? Give a reason.

(a)

(b)

Solution

(a) The $112^{\circ}$ angle is below $AB$ and the $68^{\circ}$ angle is above $CD$ , so both are between the lines, and both are on the right of $EF$ . They are interior angles.

$112^{\circ} + 68^{\circ} = 180^{\circ}$

So $AB \parallel CD$ (int. ∠s supp.).

(b) The $58^{\circ}$ angle is below $AB$ , on the left of $EF$ . The $62^{\circ}$ angle is above $CD$ , on the right of $EF$ . Both are between the lines, on opposite sides of the transversal: alternate angles.

If $AB$ were parallel to $CD$ , these alternate angles would be equal. But $58^{\circ} \neq 62^{\circ}$ , so $AB$ is not parallel to $CD$ .

The lines in (b) look almost parallel, but they are not: $CD$ is turned $4^{\circ}$ away from the direction of $AB$ . That is why you test with angles instead of judging by eye.

(a) Yes: the interior angles add up to $180^{\circ}$ . (b) No: the alternate angles, $58^{\circ}$ and $62^{\circ}$ , are not equal.

Common Mistakes

Using a rule when the lines are not parallel. The rules for corresponding, alternate and interior angles hold only when the lines are marked or stated to be parallel.
Making interior angles equal. Interior angles add up to $180^{\circ}$ ; corresponding and alternate angles are the ones that are equal. In Example 5, $z = 110$ , not $70$ .
Pairing two angles at the same crossing. Corresponding, alternate and interior angles always have one angle at each crossing. Two angles between the lines at the same crossing, on opposite sides of the transversal, add up to $180^{\circ}$ ; they are not alternate angles.
Thinking that corresponding angles must both be above their lines, or both outside the parallel lines. Corresponding angles are in the same position at their crossings: both may be below their lines, and one may be outside the lines while the other is between them, as in Example 3.
Writing a letter shape as the reason. The F, Z and C shapes help you to spot a pair, but the reason you write is (corr. ∠s, AB // CD), (alt. ∠s, AB // CD) or (int. ∠s, AB // CD).
Deciding by eye that lines are parallel. Lines that look parallel may not be, as in Example 9(b). Go by the arrows, the words of the question, or a test with angles.

Practice Questions

Try each question before you open its solution. Name the pair of angles you use, and give a reason for every angle you find.

Question 1

In the figure, $AB \parallel CD$ , and the transversal $EF$ crosses them at $G$ and $H$ .

Say whether each pair of angles is corresponding, alternate or interior, or none of these.

(a) $\angle EGB$ and $\angle GHD$ (b) $\angle AGH$ and $\angle GHD$ (c) $\angle BGH$ and $\angle GHD$

(d) $\angle AGH$ and $\angle CHF$ (e) $\angle AGE$ and $\angle BGH$ (f) $\angle AGH$ and $\angle CHG$

ShowHide solution

Find each angle by its vertex, the middle letter, and its two arms.

(a) $\angle EGB$ is at $G$ , above $AB$ , on the right of $EF$ . $\angle GHD$ is at $H$ , above $CD$ , on the right of $EF$ . The same position: corresponding angles.

(b) $\angle AGH$ is below $AB$ , on the left of $EF$ . $\angle GHD$ is above $CD$ , on the right. Both are between the lines, on opposite sides of $EF$ : alternate angles.

(c) $\angle BGH$ is below $AB$ , on the right. $\angle GHD$ is above $CD$ , on the right. Both are between the lines, on the same side: interior angles.

(d) $\angle AGH$ is below $AB$ , on the left. $\angle CHF$ is below $CD$ , on the left. The same position: corresponding angles.

(e) Both angles are at $G$ , so they are none of these. They are vertically opposite angles.

(f) $\angle AGH$ is below $AB$ , on the left. $\angle CHG$ is above $CD$ , on the left. Both are between the lines, on the same side: interior angles.

(a) Corresponding (b) Alternate (c) Interior (d) Corresponding (e) None of these: they are vertically opposite angles at $G$ . (f) Interior

Question 2

In the figure, $PQ \parallel RS$ .

(a) Say whether each pair is corresponding, alternate or interior: $a$ and $b$ ; $a$ and $c$ ; $a$ and $d$ .

(b) The angle marked $a$ is $100^{\circ}$ . Write down the size of each of the angles marked $b$ , $c$ and $d$ , with its reason.

ShowHide solution

Here the parallel lines slope, so look at which side of its own line each angle is on, and which side of the transversal.

(a) $a$ is below $PQ$ , so it is between the lines, on the right of the transversal.

$b$ is above $RS$ , between the lines, on the left of the transversal. Both between the lines, on opposite sides: $a$ and $b$ are alternate angles.

$c$ is above $RS$ , between the lines, on the right of the transversal. Both between the lines, on the same side: $a$ and $c$ are interior angles.

$d$ is below $RS$ , on the right of the transversal, just as $a$ is below $PQ$ on the right. The same position: $a$ and $d$ are corresponding angles.

(b) Angle $b$ is $100^{\circ}$ , the same as angle $a$ (alt. ∠s, PQ // RS).

Angle $c$ is $180^{\circ} - 100^{\circ} = 80^{\circ}$ (int. ∠s, PQ // RS).

Angle $d$ is $100^{\circ}$ , the same as angle $a$ (corr. ∠s, PQ // RS).

Check: angles $b$ and $c$ are adjacent angles on the straight line $RS$ , and $100^{\circ} + 80^{\circ} = 180^{\circ}$ .

(a) $a$ and $b$ : alternate; $a$ and $c$ : interior; $a$ and $d$ : corresponding. (b) Angle $b$ is $100^{\circ}$ , angle $c$ is $80^{\circ}$ and angle $d$ is $100^{\circ}$ .

Question 3

In the figure, $AB \parallel CD$ . Find $x$ .

ShowHide solution

$x^{\circ}$ is below $AB$ , on the right of $EF$ . The $110^{\circ}$ angle is below $CD$ , on the right of $EF$ . They are corresponding angles.

$x = 110$ (corr. ∠s, AB // CD)

$x = 110$ , so the angle is $110^{\circ}$ .

Question 4

In the figure, $AB \parallel CD$ . Find $y$ .

ShowHide solution

$y^{\circ}$ is below $AB$ , on the left of $EF$ . The $48^{\circ}$ angle is above $CD$ , on the right of $EF$ . Both are between the lines, on opposite sides of $EF$ : alternate angles.

$y = 48$ (alt. ∠s, AB // CD)

$y = 48$ , so the angle is $48^{\circ}$ .

Question 5

In the figure, $AB \parallel CD$ . Find $z$ .

ShowHide solution

$z^{\circ}$ is below $AB$ and the $125^{\circ}$ angle is above $CD$ , so both are between the lines, and both are on the left of $EF$ . They are interior angles.

$z + 125 = 180$ (int. ∠s, AB // CD)

$z = 180 - 125 = 55$

Check: $55^{\circ} + 125^{\circ} = 180^{\circ}$ .

$z = 55$ , so the angle is $55^{\circ}$ .

Question 6

In the figure, $AB \parallel CD$ , and $EF$ crosses them at $P$ and $Q$ . Find $x$ , giving a reason for each step.

ShowHide solution

The $108^{\circ}$ angle is above $AB$ on the left of $EF$ , and $x^{\circ}$ is below $CD$ on the right. They are not one of the three pairs, so go through the angle $\angle BPQ$ , below $AB$ on the right of $EF$ .

$\angle BPQ = 108^{\circ}$ (vert. opp. ∠s)

$\angle BPQ$ and $x^{\circ}$ are both below their line, on the right of $EF$ : corresponding angles.

$x = 108$ (corr. ∠s, AB // CD)

$x = 108$ , so the angle is $108^{\circ}$ .

Question 7

In the figure, $AB \parallel CD$ . Find $x$ , and the size of each marked angle.

ShowHide solution

Both angles are above their line, on the left of $EF$ : corresponding angles, so they are equal.

$6x - 30 = 4x + 20$ (corr. ∠s, AB // CD)

Subtract $4x$ from both sides: $2x - 30 = 20$

Add $30$ to both sides: $2x = 50$

Divide both sides by $2$ : $x = 25$

The angles: $4x + 20 = 4(25) + 20 = 120$ and $6x - 30 = 6(25) - 30 = 120$ .

Check: both angles are $120^{\circ}$ , so they are equal.

$x = 25$ , and each marked angle is $120^{\circ}$ .

Question 8

In the figure, $AB \parallel CD$ . Find $x$ , and the size of each marked angle.

ShowHide solution

Both angles are between the lines, on the left of $EF$ : interior angles, so they add up to $180^{\circ}$ .

$(2x + 25) + (5x + 15) = 180$ (int. ∠s, AB // CD)

Collect like terms: $7x + 40 = 180$

Subtract $40$ from both sides: $7x = 140$

Divide both sides by $7$ : $x = 20$

The angles: $2x + 25 = 2(20) + 25 = 65$ and $5x + 15 = 5(20) + 15 = 115$ .

Check: $65^{\circ} + 115^{\circ} = 180^{\circ}$ .

$x = 20$ ; the marked angles are $65^{\circ}$ and $115^{\circ}$ .

Question 9

The railing of an HDB staircase has upright bars, which are parallel, and a straight slanting middle rail. In the figure, $AB$ and $CD$ are two of the upright bars and $EF$ is the middle rail, crossing them at $P$ and $Q$ .

(a) Find $y$ .

(b) Find $\angle DQP$ .

ShowHide solution

(a) Here the parallel lines are upright and the rail $EF$ is the transversal. $\angle BPQ$ is on the right of $AB$ , so it is between the bars, and it is below the rail. $y^{\circ}$ is on the left of $CD$ , between the bars, above the rail. Both are between the lines, on opposite sides of the transversal: alternate angles.

$y = 125$ (alt. ∠s, AB // CD)

(b) $\angle DQP$ and $\angle CQP$ , which is $125^{\circ}$ , are adjacent angles on the straight line $CD$ .

$\angle DQP = 180^{\circ} - 125^{\circ} = 55^{\circ}$ (adj. ∠s on a str. line)

(a) $y = 125$ , so $\angle CQP = 125^{\circ}$ . (b) $\angle DQP = 55^{\circ}$

Question 10

In the figure, $AB$ and $CD$ are straight lines cut by the transversal $EF$ .

Ahmad says, " $x = 70$ , because corresponding angles are equal."

(a) Explain why Ahmad is wrong.

(b) What would have to be true of $AB$ and $CD$ for Ahmad to be right?

ShowHide solution

(a) $x^{\circ}$ and the $70^{\circ}$ angle are corresponding angles: both are above their line, on the right of $EF$ . But corresponding angles are equal only when the lines are parallel. $AB$ and $CD$ have no arrow marks, and the question does not say they are parallel (they even look as if they get closer together, but how they look is not a reason). So $x$ need not be $70$ , and it cannot be found from the $70^{\circ}$ angle.

(b) $AB$ would have to be parallel to $CD$ . Then $x = 70$ (corr. ∠s, AB // CD).

(a) $AB$ and $CD$ are not marked or stated to be parallel, so the rule for corresponding angles cannot be used, and $x$ cannot be found from the $70^{\circ}$ angle. (b) $AB \parallel CD$

Question 11

A gate has three straight bars, $AB$ , $CD$ and $EF$ , that are meant to be parallel, and a straight diagonal bar $GH$ that crosses them at $P$ , $Q$ and $R$ . A worker measures the angles shown.

(a) Show that $AB \parallel CD$ .

(b) Is $CD \parallel EF$ ? Give a reason.

ShowHide solution

(a) $\angle BPQ$ is below $AB$ and $\angle PQD$ is above $CD$ , both on the right of $GH$ : interior angles.

$108^{\circ} + 72^{\circ} = 180^{\circ}$

So $AB \parallel CD$ (int. ∠s supp.).

(b) $\angle PQD$ is above $CD$ , on the right of $GH$ . $\angle QRF$ is above $EF$ , on the right of $GH$ . They are corresponding angles.

$72^{\circ} \neq 70^{\circ}$ , so these corresponding angles are not equal.

So $CD$ is not parallel to $EF$ : the bottom bar needs to be fixed.

(a) The interior angles add up to $108^{\circ} + 72^{\circ} = 180^{\circ}$ , so $AB \parallel CD$ . (b) No: the corresponding angles, $72^{\circ}$ and $70^{\circ}$ , are not equal.

Question 12

Mr Lim paints the slanting lines of the lots in a car park. The lines start from a straight kerb $KL$ at $P$ , $Q$ and $R$ , and they should all be parallel. He measures the angles shown.

(a) Show that $PB \parallel QD$ .

(b) Find $\angle DQR$ .

ShowHide solution

(a) Here the kerb $KL$ is the transversal. $\angle BPQ$ and $\angle PQD$ are both between the lines $PB$ and $QD$ , on the same side of $KL$ : interior angles.

$62^{\circ} + 118^{\circ} = 180^{\circ}$ , so $PB \parallel QD$ (int. ∠s supp.).

(b) $\angle PQD$ and $\angle DQR$ are adjacent angles on the straight line $KL$ .

$\angle DQR = 180^{\circ} - 118^{\circ} = 62^{\circ}$ (adj. ∠s on a str. line)

$62^{\circ} + 120^{\circ} = 182^{\circ}$ , which is not $180^{\circ}$ .

So $QD$ is not parallel to $RF$ : the line at $R$ must be painted again.

(a) The interior angles add up to $62^{\circ} + 118^{\circ} = 180^{\circ}$ , so $PB \parallel QD$ . (b) $\angle DQR = 62^{\circ}$ (c) No: the interior angles add up to $182^{\circ}$ , not $180^{\circ}$ .

Key Points

Parallel lines never meet and stay the same distance apart; they are marked with matching arrows and written $AB \parallel CD$ .
A transversal crosses two lines at different points and makes four angles at each crossing.
Corresponding angles: one at each crossing, in the same position (the same side of the transversal, the same side of their own lines).
Alternate angles: one at each crossing, both between the lines, on opposite sides of the transversal.
Interior angles: one at each crossing, both between the lines, on the same side of the transversal.
If $AB \parallel CD$ : corresponding angles are equal (corr. ∠s, AB // CD), alternate angles are equal (alt. ∠s, AB // CD), and interior angles add up to $180^{\circ}$ (int. ∠s, AB // CD).
The rules hold only for parallel lines: check for arrows, or the words of the question, first.
Two lines are parallel if corresponding angles are equal, alternate angles are equal, or interior angles add up to $180^{\circ}$ .
With expressions in $x$ : name the pair, form an equation from its rule, solve it, and check every angle.