In this chapter, you will explore the relationships between pairs of angles that are created when straight lines intersect (meet or cross). You will examine the pairs of angles that are formed by perpendicular lines, by any two intersecting lines, and by a third line that cuts two parallel lines. You will come to understand what is meant by vertically opposite angles, corresponding angles, alternate angles and co-interior angles. You will be able to identify different angle pairs, and then use your knowledge to help you work out unknown angles in geometric figures.

Angles on a straight line

Sum of angles on a straight line

In the figures below, each angle is given a label from 1 to 5.

1. Use a protractor to measure the sizes of all the angles in each figure. Write your answers on each figure.

A

B

1. $\hat{1} + \hat{2} = \text{______}^{\circ}$
2. $\hat{3} + \hat{4} + \hat{5}= \text{______}^{\circ}$

The sum of angles that are formed on a straight line is equal to 180°. (We can shorten this property as: $\angle$s on a straight line.)

Two angles whose sizes add up to 180° are also called supplementary angles, for example $\hat{1} + \hat{2}$.

Angles that share a vertex and a common side are said to be adjacent. So $\hat{1} + \hat{2}$ are therefore also called supplementary adjacent angles.

When two lines are perpendicular, their adjacent supplementary angles are each equal to 90°.

In the drawing below, DC A and DC B are adjacent supplementary angles because they are next to each other (adjacent) and they add up to 180° (supplementary).

Finding unknown angles on straight lines

Work out the sizes of the unknown angles below. Build an equation each time as you solve these geometric problems. Always give a reason for every statement you make.

1. Calculate the size of $a$.

\begin{align} a + 63^{\circ} &= \text{______} [\angle\text{s on a straight line}] \\ a &= \text{______} - 63^{\circ} \\ &= \text{______} \end{align}

2. Calculate the size of $x$.

3. Calculate the size of $y$.

Finding more unknown angles on straight lines

1. Calculate the size of:

1. $x$
2. $\hat{ECB}$
2. Calculate the size of:

1. $m$
2. $\hat{SQR}$
3. Calculate the size of:

1. $x$
2. $\hat{HEF}$
4. Calculate the size of:

1. $k$
2. $\hat{TYP}$
5. Calculate the size of:

1. $p$
2. $\hat{JKR}$

Vertically opposite angles

What are vertically opposite angles?

1. Use a protractor to measure the sizes of all the angles in the figure. Write your answers on the figure.

2. Notice which angles are equal and how these equal angles are formed.

Vertically opposite angles (vert. opp. $\angle$s) are the angles opposite each other when two lines intersect.

Vertically opposite angles are always equal.

Finding unknown angles

Calculate the sizes of the unknown angles in the following figures. Always give a reason for every statement you make.

1. Calculate $x,~ y$ and $z$.

\begin{align} x &= \text{______}^{\circ} &&[\text{vert. opp.}\angle\text{s}] \\ \\ y + 105^{\circ} &= \text{______}^{\circ} &&[\angle\text{s on a straight line}] \\ y &= \text{______} - 105^{\circ} && \\ & = \text{______} \\ \\ z &= \text{______} &&[\text{vert. opp.}\angle\text{s}] \end{align}

2. Calculate $j,~ k$ and $l$.

3. Calculate $a,~ b,~ c$ and $d$.

Equations using vertically opposite angles

Vertically opposite angles are always equal. We can use this property to build an equation. Then we solve the equation to find the value of the unknown variable.

1. Calculate the value of $m$.

\begin{align} m + 20^{\circ} &= 100^{\circ} [\text{vert. opp.}\angle\text{s}] \\ m &= 100^{\circ} - 20^{\circ} \\ &= \text{______} \end{align}
2. Calculate the value of $t$.

3. Calculate the value of $p$.

4. Calculate the value of $z$.

5. Calculate the value of $y$.

6. Calculate the value of $r$.

Lines intersected by a transversal

Pairs of angles formed by a transversal

A transversal is a line that crosses at least two other lines.

When a transversal intersects two lines, we can compare the sets of angles on the two lines by looking at their positions.

The angles that lie on the same side of the transversal and are in matching positions are called corresponding angles (corr.$\angle$s). In the figure, these are corresponding angles:

• $a$ and $e$
• $b$ and $f$
• $d$ and $h$
• $c$ and $g$.
1. In the figure, $a$ and $e$ are both left of the transversal and above a line.

Write down the location of the following corresponding angles. The first one is done for you.

$b$ and $f$: Right of the transversal and above lines

$d$ and $h$:

$c$ and $g$:

Alternate angles (alt.$\angle$s) lie on opposite sides of the transversal, but are not adjacent or vertically opposite. When the alternate angles lie between the two lines, they are called alternate interior angles. In the figure, these are alternate interior angles:

• $d$ and $f$
• $c$ and $e$

When the alternate angles lie outside of the two lines, they are called alternate exterior angles. In the figure, these are alternate exterior angles:

• $a$ and $g$
• $b$ and $h$
1. Write down the location of the following alternate angles:

$d$ and $f$:

$c$ and $e$:

$a$ and $g$:

$b$ and $h$:

Co-interior angles (co-int.$\angle$s) lie on the same side of the transversal and between the two lines. In the figure, these are co-interior angles:

• $c$ and $f$
• $d$ and $e$
1. Write down the location of the following co-interior angles:

$d$ and $e$:

$c$ and $f$:

Identifying types of angles

Two lines are intersected by a transversal as shown below.

Write down the following pairs of angles:

1. two pairs of corresponding angles:
2. two pairs of alternate interior angles:
3. two pairs of alternate exterior angles:
4. two pairs of co-interior angles:
5. two pairs of vertically opposite angles:

Parallel lines intersected by a transversal

Investigating angle sizes

In the figure below left, EF is a transversal to AB and CD. In the figure below right, PQ is a transversal to parallel lines JK and LM.

1. Use a protractor to measure the sizes of all the angles in each figure. Write the measurements on the figures.
2. Use your measurements to complete the following table.
 Angles When two lines are not parallel When two lines are parallel Corr.$\angle$s $\hat{1} = \text{_______};~\hat{5} = \text{_______}$ $\hat{4} = \text{_______};~\hat{8} = \text{_______}$ $\hat{2} = \text{_______};~\hat{4} = \text{_______}$ $\hat{3} = \text{_______};~\hat{7} = \text{_______}$ $\hat{9} = \text{_______};~\hat{13} = \text{_______}$ $\hat{12} = \text{_______};~\hat{16} = \text{_______}$ $\hat{10} = \text{_______};~\hat{14} = \text{_______}$ $\hat{11} = \text{_______};~\hat{15} = \text{_______}$ Alt.int.$\angle$s $\hat{4} = \text{_______};~\hat{6} = \text{_______}$ $\hat{3} = \text{_______};~\hat{5} = \text{_______}$ $\hat{12} = \text{_______};~\hat{14} = \text{_______}$ $\hat{11} = \text{_______};~\hat{13} = \text{_______}$ Alt.ext.$\angle$s $\hat{1} = \text{_______};~\hat{7} = \text{_______}$ $\hat{2} = \text{_______};~\hat{8} = \text{_______}$ $\hat{9} = \text{_______};~\hat{15} = \text{_______}$ $\hat{10} = \text{_______};~\hat{16} = \text{_______}$ Co-int.$\angle$s $\hat{4} + \hat{5} = \text{_______}$ $\hat{3} + \hat{6} = \text{_______}$ $\hat{12} + \hat{13} = \text{_______}$ $\hat{11} + \hat{14} = \text{_______}$
3. Look at your completed table in question 2. What do you notice about the angles formed when a transversal intersects parallel lines?

When lines are parallel:

• corresponding angles are equal
• alternate interior angles are equal
• alternate exterior angles are equal
• co-interior angles add up to 180°

Identifying angles on parallel lines

1. Fill in the corresponding angles to those given.

2. Fill in the alternate exterior angles.

1. Fill in the alternate interior angles.
2. Circle the two pairs of co-interior angles in each figure.

1. Without measuring, fill in all the angles in the following figures that are equal to $x$ and $y$.
2. Explain your reasons for each $x$ and $y$ that you filled in to your partner.
3. Give the value of $x$ and $y$ below.

Finding unknown angles on parallel lines

Working out unknown angles

Work out the sizes of the unknown angles. Give reasons for your answers. (The first one has been done as an example.)

1. Find the sizes of $x,~y$ and $z$.

\begin{align} x &= 74^{\circ} &&[\text{alt.}\angle\text{ with given }74^{\circ}; AB \parallel CD] \\ \\ y &= 74^{\circ} &&[\text{corr.}\angle\text{ with }x; AB \parallel CD] \\ \text{or } y &= 74^{\circ} &&[\text{vert. opp.}\angle\text{ with given }74^{\circ}] \\ \\ z &= 106^{\circ} &&[\text{co-int.}\angle\text{ with }x; AB \parallel CD] \\ \text{or } z &= 106^{\circ} &&[\angle\text{s on a straight line}] \end{align}

2. Work out the sizes of $p,~ q$ and $r$.

3. Find the sizes of $a,~b,~c$ and $d$.

4. Find the sizes of all the angles in this figure.

5. Find the sizes of all the angles. (Can you see two transversals and two sets of parallel lines?)

Extension

Two angles in the following diagram are given as $x$ and $y$. Fill in all the angles that are equal to $x$ and $y$.

Sum of the angles in a quadrilateral

The diagram below is a section of the previous diagram.

2. Look at the top left intersection. Complete the following equation:

Angles around a point$= 360^{\circ}$

$\therefore x + y+ \text{______} + \text{______} = 360^{\circ}$

3. Look at the interior angles of the quadrilateral. Complete the following equations:

Sum of angles in the quadrilateral $= x + y + + \text{______} + \text{______}$

From question 2: $x + y+ \text{______} + \text{______} = 360^{\circ}$

$\therefore$ Sum of angles in a quadrilateral = $\text{______}^{\circ}$

Can you think of another way to use the diagram above to work out the sum of the angles in a quadrilateral?

Solving more geometric problems

Angle relationships on parallel lines

1. Calculate the sizes of $\hat{1}$ to $\hat{7}$.

2. Calculate the sizes of $x,~y$ and $z$.

3. Calculate the sizes of $a, ~b, ~c$ and $d$.

4. Calculate the size of $x$.

5. Calculate the size of $x$.

6. Calculate the size of $x$.

7. Calculate the sizes of $a$ and $\hat{CEP}$.

Including properties of triangles and quadrilaterals

1. Calculate the sizes of $\hat{1}$ to $\hat{6}$.

2. RSTU is a trapezium. Calculate the sizes of $\hat{T}$ and $\hat{R}$.

3. JKLM is a rhombus. Calculate the sizes of $\hat{JML}, \hat{M_2}$ and $\hat{K_1}$.

4. ABCD is a parallelogram. Calculate the sizes of $\hat{ADB}, \hat{ABD}, \hat{C}$ and $\hat{DBC}$

1. Look at the drawing below. Name the items listed alongside.

1. a pair of vertically opposite angles
2. a pair of corresponding angles
3. a pair of alternate interior angles
4. a pair of co-interior angles
2. In the diagram, AB $\parallel$ CD. Calculate the sizes of $\hat{FHG}, \hat{F}, \hat{C}$ and $\hat{D}$. Give reasons for your answers.

3. In the diagram, OK = ON, KN $\parallel$ LM, KL$\parallel$ MN and $\hat{LKO} = 160^{\circ}$.

Calculate the value of $x$. Give reasons for your answers.