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Intersecting lines

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visibility 96update 6 months agobookmarkshare

🎯 In this topic you will

Recognise the properties of angles on perpendicular lines and intersecting lines
Recognise the properties of angles on parallel lines

🧠 Key Words

intersect
opposite angles
perpendicular
parallel
transversal

Show Definitions

intersect: To cross or meet at a point, such that the lines share a common point.
opposite angles: Angles formed opposite each other when two lines intersect, which are always equal in measure.
perpendicular: Lines that meet at a right angle (90°).
parallel: Lines that are always the same distance apart and never meet, no matter how far they are extended.
transversal: A line that cuts across two or more other lines at different points.

➗ Intersecting Lines

These two lines intersect at $A$.

When two lines intersect, opposite angles are equal.

In this diagram, $a$ and $c$ are opposite angles. Angles $a$ and $c$ are equal. Also $b$ and $d$ are opposite angles. Angles $b$ and $d$ are equal.

You can use your algebra skills to find unknown angles, represented by letters.

📘 Worked example

Work out angles $p$, $q$ and $r$.

Answer:

$53^\circ$ and $p$ are angles on a straight line.
The sum of $53^\circ$ and $p$ is $180^\circ$.
So $p = 180^\circ - 53^\circ = 127^\circ$.

$53^\circ$ and $q$ are opposite angles, so $q = 53^\circ$.
$p$ and $r$ are opposite angles, so $r = 127^\circ$.

Angles on a straight line add up to $180^\circ$. Subtract the given angle from $180^\circ$ to find the adjacent angle.

Opposite (vertically opposite) angles are equal, so the angle directly across from a given angle has the same measure.

➗ Perpendicular & Parallel Lines with a Transversal

These two lines intersect, as shown. The angle between the two lines is a right angle. They are perpendicular lines.

$AB$ and $CD$ are two lines that do not intersect. They are parallel.

The arrows show that the lines are parallel. The line $EF$ crosses the parallel lines. This line is called a transversal.

In the previous diagram there are only two different sizes of angle. The four angles at the top are the same as the four angles at the bottom.

$70^\circ 110^\circ = 180^\circ$

$70^\circ 110^\circ 70^\circ 110^\circ = 360^\circ$

📘 Worked example

Work out the unknown angles, $a$, $b$, $c$ and $d$, in this diagram.

Answer:

$82^\circ$ and $a$ are angles on a straight line. The sum is $180^\circ$.
$a = 180^\circ - 82^\circ = 98^\circ$

$82^\circ$ and $d$ are opposite angles. They are equal.
$d = 82^\circ$

$a$ and $b$ are in the same position. They are equal.
$b = 98^\circ$

$c$ and $b$ are opposite angles. They are equal.
$c = 98^\circ$

Angles on a straight line add up to $180^\circ$. Subtract the known angle from $180^\circ$ to find the unknown adjacent angle.

Vertically opposite angles are always equal.

Corresponding angles on parallel lines are equal when a transversal cuts through them.

🧠 PROBLEM-SOLVING Strategy

Angles with Intersecting & Parallel Lines

Use these steps to find unknown angles labelled with letters.

Mark given facts. Circle known angles and write them clearly (e.g., $53^\circ$, $114^\circ$, $77^\circ$).
At a single crossing (two lines intersect): use vertically opposite angles are equal and linear pairs add to $180^\circ$.
$\triangleright$ If one angle is $x^\circ$, the opposite is also $x^\circ$, and each adjacent angle is $180^\circ - x^\circ$.
With parallel lines cut by a transversal: identify pair types—corresponding, alternate, and co-interior—and apply their rules (see table below).
Around a point: if several angles meet, their sum is $360^\circ$. Add the known ones and subtract from $360^\circ$.
Write short reasons next to each calculation (e.g., “$\because$ vertically opposite,” “$\because$ on a straight line,” “$\because$ corresponding”).
Check consistency: opposite angles equal; each pair on a straight line totals $180^\circ$; totals around a point give $360^\circ$.

Angle pair	Rule	Equation
Vertically opposite	Equal	$\angle A = \angle C$
Linear pair (straight line)	Sum to $180^\circ$	$\angle A \angle B = 180^\circ$
Corresponding (‖)	Equal	$\angle A = \angle A'$
Alternate interior (‖)	Equal	$\angle B = \angle B'$
Co-interior (same side, ‖)	Sum to $180^\circ$	$\angle C \angle C' = 180^\circ$
Around a point	Sum to $360^\circ$	$\sum \text{angles} = 360^\circ$

Tip: When letters are repeated in several crossings on parallel lines, first transfer values using corresponding/alternate rules, then finish each crossing with the straight-line rule.

❓ EXERCISES

1. Work out the angles that have a letter.

👀 Show answer

a. On a straight line: $x = 180^\circ - 53^\circ = 127^\circ$. Opposite angles are equal: $y = 53^\circ$.

b. On a straight line: $w = 180^\circ - 114^\circ = 66^\circ$. Opposite angles are equal: $z = 114^\circ$.

2. Two straight lines are shown.
There are four angles. One of the angles is $87^\circ$.
Work out the other three angles.

👀 Show answer

Opposite angles are equal, so the angle opposite $87^\circ$ is also $87^\circ$. Adjacent angles on a straight line sum to $180^\circ$, so the other two angles are each $180^\circ - 87^\circ = 93^\circ$.

❓ EXERCISES

3. Three straight lines meet at a point.
Calculate the values of $a$, $b$, $c$ and $d$. Give reasons for your answers.

👀 Show answer

Angles on a straight line add to $180^\circ$.
$a = 180^\circ - (61^\circ 46^\circ) = 73^\circ$.
$c = 180^\circ - a = 107^\circ$.
Opposite angles are equal: $b = 61^\circ$, $d = 46^\circ$.

4. There are two parallel lines in this diagram. One angle is $42^\circ$.
Copy the diagram and write in the size of all the other angles.

👀 Show answer

Corresponding angles are equal, so the matching angle is $42^\circ$.
Adjacent angles on a straight line add to $180^\circ$, so the supplementary angles are $180^\circ - 42^\circ = 138^\circ$.

5. Work out the unknown angles $a$, $b$ and $c$.

👀 Show answer

$a = 113^\circ$ (vertically opposite angles).
$b = 180^\circ - a = 67^\circ$ (angles on a straight line).
$c = b = 67^\circ$ (vertically opposite angles).

6. Lines $WX$ and $YZ$ are parallel.
One angle is $77^\circ$. Find $a$, $b$ and $c$.

👀 Show answer

$a = 77^\circ$ (corresponding angles).
$b = 180^\circ - a = 103^\circ$ (angles on a straight line).
$c = b = 103^\circ$ (vertically opposite angles).

❓ EXERCISES

7. $AB$ and $CD$ are parallel lines. Calculate $s$ and $t$.

👀 Show answer

Corresponding angles are equal: $t = 75^\circ$.
Angles on a straight line: $s = 180^\circ - 75^\circ = 105^\circ$.

8. Look at the diagram.

a. Explain why these two lines cannot be parallel.

b. Give your answer to part a to a partner to read. Can your answer be improved?

👀 Show answer

If lines were parallel, corresponding angles would be equal, but $56^\circ \neq 126^\circ$.

9. This shape is made from eight identical triangles.

a. Sketch the diagram and label the other angles equal to $a$, $b$ or $c$.

b. Use arrows to mark any parallel lines.

👀 Show answer

Angles equal to $a$, $b$, $c$ can be found by symmetry of the identical triangles. Mark parallel lines using arrow notation.

10. The diagram shows angle $X$ is $45^\circ$.

a. Calculate $a$.

b. Angle $X$ is increased to $90^\circ$. Find the new value of $a$.

c. Angle $X$ is increased to $119^\circ$. Find the new value of $a$.

d. Can angle $X$ be more than $119^\circ$? Give a reason for your answer.

👀 Show answer

a. $a = 180^\circ - (45^\circ 60^\circ) = 75^\circ$.
b. $a = 180^\circ - (90^\circ 60^\circ) = 30^\circ$.
c. $a = 180^\circ - (119^\circ 60^\circ) = 1^\circ$.
d. No, because the sum of angles in a triangle is $180^\circ$, so $a$ would be negative if $X > 119^\circ$.

11. This trapezium has a pair of parallel sides. Use this fact to calculate the missing angles.

👀 Show answer

Angles on the same side of a transversal add to $180^\circ$.
Angle $D = 180^\circ - 45^\circ = 135^\circ$.
Angle $B = 180^\circ - 67^\circ = 113^\circ$.

🧠 Think like a Mathematician

12. These shapes are an equilateral triangle, a rhombus and a square.

All the sides are the same length. Two squares and three triangles can be placed around a point, as shown.

a. How do you know that the shapes fit exactly around a point?
b. Find a different way to fit two squares and three triangles around a point.
c. Show how to fit only triangles around a point.
d. Find all the possible ways of fitting only rhombuses around a point.

👀 Show answer

a. Angles around a point sum to $360^\circ$. Two squares and three equilateral triangles give $2\times90^\circ 3\times60^\circ = 180^\circ 180^\circ = 360^\circ$, so they fit exactly.
b. Any order that still uses two square corners ($90^\circ$ each) and three triangle corners ($60^\circ$ each) works. Example: T–S–T–S–T around the point (T = triangle corner, S = square corner).
c. Use six equilateral triangle corners: $6\times60^\circ = 360^\circ$. Arrange $6$ triangles meeting at the point.
d. A rhombus has angles $60^\circ$ and $120^\circ$. Let $x$ be the number of $60^\circ$ corners at the point and $y$ the number of $120^\circ$ corners. Then $60x 120y=360$ ⇒ $x 2y=6$. Possible fittings (numbers of rhombuses meeting at the point) are:
$\bullet$$6$ rhombuses using $6$ acute corners ($x=6,y=0$)
$\bullet$$5$ rhombuses: $4$ acute $1$ obtuse ($x=4,y=1$)
$\bullet$$4$ rhombuses: $2$ acute $2$ obtuse ($x=2,y=2$)
$\bullet$$3$ rhombuses using $3$ obtuse corners ($x=0,y=3$)

Intersecting lines

🎯 In this topic you will

🧠 Key Words

➗ Intersecting Lines

➗ Perpendicular & Parallel Lines with a Transversal

🧠 PROBLEM-SOLVING Strategy

Angles with Intersecting & Parallel Lines

❓ EXERCISES

❓ EXERCISES

❓ EXERCISES

🧠 Think like a Mathematician

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