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\title{Review of geometry }

\author{}
\date{}


\begin{document}
\maketitle
The word geometry comes from two Greek words meaning 'earth measurement'. Its name explains the origin of the subject. In ancient Egypt, the river Nile regularly overflowed its banks and washed away boundary markers between neighbouring properties. The boundaries had to be found again and this led to the study of the different shapes and sizes of triangles.

Geometry deals with points, lines, planes, angles, circles, triangles, and so on. The ancient Greeks valued its study greatly, and their work established the geometry you will meet in this book.

\section*{Angles at a point}
Clear, logical reasoning has always been an important part of geometry. It is very important in geometry to draw diagrams. This makes it easier to construct convincing arguments.

\section*{Angles at a point - geometric arguments}
\begin{itemize}
  \item The following can be used in arguments:
  \item Adjacent angles can be added.
  \item Angles in a revolution add to \(360^{\circ}\).
  \item Angles in a straight angle add to \(180^{\circ}\).
  \item Vertically opposite angles are equal.
  \item Two lines are called perpendicular if they meet at right angles.
  \item The Greek letters \(\alpha\) (alpha), \(\beta\) (beta), \(\gamma\) (gamma) and \(\theta\) (theta) are often used as pronumerals to represent angle size in geometry.
\end{itemize}

\section*{Example 1}
Find \(\angle A O C\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-02}
\end{center}

\section*{Solution}
\(\angle A O C+27^{\circ}=90^{\circ}(\) adjacent angles at \(O)\)

so \(\angle A O C=63^{\circ}\)

\section*{Example 2}
Find \(\angle A O D\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-02(1)}
\end{center}

\section*{Solution}
\(\angle A O D+90^{\circ}+160^{\circ}+90^{\circ}=360^{\circ} \quad(\) revolution at \(O)\)

so \(\angle A O D=20^{\circ}\)

\section*{Example 3}
Find \(\angle B O C\) in the diagram below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-03}
\end{center}

\section*{Solution}
\(\angle B O C+135^{\circ}=180^{\circ}\) (straight angle \(\angle A O C\) )

so \(\angle B O C=45^{\circ}\)

\section*{Example 4}
Find \(\alpha\) and \(\beta\) in the diagram below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-03(1)}
\end{center}

\section*{Solution}
\(\alpha=40^{\circ}\) (vertically opposite angles at \(A\) )

Also, \(\beta+40^{\circ}=180^{\circ}\) (straight angle \(\angle P A Q\) ),

so \(\beta=140^{\circ}\)

\section*{Exercise 5A}
1 Find the value of \(\angle A O B\) in each diagram below, giving reasons for your answer. In parts \(\mathbf{g}\) and \(\mathbf{j}\), find the reflex angle \(\angle A O B\).

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(8)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(5)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(4)}
\end{center}

j

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04}
\end{center}

b \(B\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(6)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(2)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(14)}
\end{center}

\(\mathbf{k}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(7)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(3)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(12)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(13)}
\end{center}

I

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(9)}
\end{center}

2 Name the angle that is vertically opposite to \(\angle P Q R\) in each diagram below.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(10)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-04(11)}
\end{center}

3 Find the sizes of the angles denoted by \(\alpha, \beta, \gamma\) and \(\theta\) in each diagram below, giving reasons.

a \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(1)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(13)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(12)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(10)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(14)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(9)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(5)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(6)}
\end{center}

4 In each diagram below, angles marked with the same Greek letter are equal in size. Find the value of each, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(15)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(4)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(7)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(8)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05(11)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-05}
\end{center}

\section*{Angles associated with transversals}
This section and the next one involve the relationships between angles and parallel lines.

As always in geometry, reasons should be as specific as possible. Make sure that you name any parallel lines that you are using in an argument.

\section*{Transversal}
A transversal is a line that crosses two other lines. In both diagrams below, the line \(P Q\) is a transversal to the lines \(A B\) and \(C D\).
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-06(3)}

Notice that \(P Q\) is a transversal whether or not the two lines that it crosses are parallel.

\section*{Corresponding angles}
The two marked angles in the diagram are called corresponding angles, because they are in corresponding positions around the two vertices \(F\) and \(G\).

There are actually four pairs of corresponding angles in the diagram. Can you name the other three pairs?

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-06(2)}
\end{center}

\section*{Alternate angles}
In the diagram opposite, the two marked angles are called alternate angles, because they are on alternate sides of the transversal \(P Q\). The two angles must also be inside the two parallel lines.

There is a second pair of alternate angles in the diagram. Can you name the angles of this pair?

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-06(1)}
\end{center}

\section*{Co-interior angles}
In the diagram opposite, the two marked angles are called co-interior angles, because they are between the two lines and on the same side of the transversal \(P Q\).

There is a second pair of co-interior angles in the diagram.

Name the angles of this pair.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-06}
\end{center}

\section*{A transversal lying across parallel lines}
We learned in Chapter 6 of ICE-EM Mathematics Year 7 that when a transversal lies across two parallel lines, the alternate angles, the corresponding angles and the co-interior angles are closely related.

\section*{Angles associated with transversals}
A transversal is a line that crosses two other lines. If the lines crossed by the transversal are parallel, then:

\begin{itemize}
  \item corresponding angles are equal
  \item alternate angles are equal
  \item co-interior angles are supplementary.
\end{itemize}

Here are three examples that show how to set out your work with corresponding angles, alternate angles and co-interior angles. Your reason should first name the type of angles involved, and then name the relevant pair of parallel lines.

\section*{Example 5}
Find \(\theta\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-07}
\end{center}

\section*{Solution}
\(\theta=110^{\circ}\) (corresponding angles, \(A B \| C D\) )

\section*{Example 6}
Find \(\angle A S T\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-07(1)}
\end{center}

\section*{Solution}
\(\angle A S T=38^{\circ}\) (alternate angles, \(A B \| C D\) )

\section*{Example 7}
Find \(\alpha\) in the diagram below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-08}
\end{center}

\section*{Solution}
\(\alpha+35^{\circ}=180^{\circ}\) (co-interior angles, \(K L \| M N\) )

\(\alpha=145^{\circ}\)

\section*{Two-step solutions}
The solution of the problem below needs two steps, with a reason for each step. Notice that the pairs of parallel lines used are different in the two steps.

\section*{Example 8}
Find \(\angle B A C\) in the diagram below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-08(1)}
\end{center}

\section*{Solution}
First, \(\angle D C A=102^{\circ}\) (alternate angles, \(A C \| B D\) ).

Hence, \(\angle B A C=78^{\circ}\) (co-interior angles, \(A B \| C D\) ).

\section*{Exercise 5B}
1 In each diagram below, identify each pair of angles marked with \(\alpha\) and \(\beta\) as corresponding angles, alternate angles or co-interior angles.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(12)}
\end{center}

j

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(1)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(5)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(14)}
\end{center}

\(\mathbf{k}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(10)}
\end{center}

\(\mathbf{c}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(6)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(13)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(9)}
\end{center}

I

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(8)}
\end{center}

2 Find the values of the pronumerals \(\alpha, \beta, \gamma\) and \(\theta\) in the diagrams below. Give careful reasons for all your statements, mentioning the relevant parallel lines.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-09(7)}
\end{center}

\(\mathbf{c}\)

\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-09(11)}
d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(15)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(9)}
\end{center}

j

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(4)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(13)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(5)}
\end{center}

k

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(1)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(2)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(6)}
\end{center}

I

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(12)}
\end{center}

3

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(7)}
\end{center}

d

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(14)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(3)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(10)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(8)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(16)}
\end{center}

f \(\mathrm{C}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-10(11)}
\end{center}

4 Find the size of the marked angle \(\angle A V B\) in each diagram below. The solution to each part will require two steps, each with its own reason.

a \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-11(3)}
\end{center}

\(\mathbf{c}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-11(1)}
\end{center}

b \(W\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-11(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-11(2)}
\end{center}

\section*{Further problems involving parallel lines}
This section deals with more complicated problems involving parallel lines:

\begin{itemize}
  \item problems where construction lines need to be added
  \item problems involving algebra.
\end{itemize}

\section*{Adding construction lines to solve a problem}
Some problems cannot be solved until one or more extra lines, called construction lines, have been added to the diagram, as in the example below.

\section*{Example 9}
Find \(\angle B C D\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-11}
\end{center}

(continued over page)

\section*{Solution}
Construct the line \(C M\) through \(C\) parallel to \(A B\) and \(E D\), as shown in the diagram opposite.

Then \(\angle B C M=50^{\circ}\) (alternate angles, \(A B \| C M\) ), and \(\angle D C M=45^{\circ}\) (alternate angles, \(E D \| C M\) ).

Hence, \(\angle B C D=95^{\circ}\) (adjacent angles at \(C\) ).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-12}
\end{center}

\section*{Problems involving algebra}
In the example below, the value of \(\theta\) is found using geometric arguments and algebra.

\section*{Example 10}
Find \(\theta\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-12(1)}
\end{center}

\section*{Solution}
\[
\begin{aligned}
\theta+3 \theta & =180^{\circ}(\text { co-interior angles, } A D \| B C) \\
4 \theta & =180^{\circ} \\
\theta & =45^{\circ}
\end{aligned}
\]

\section*{Further problems involving parallel lines}
\begin{itemize}
  \item Extra construction lines may need to be added to the diagram.
  \item Algebra may be required.
\end{itemize}

\section*{Exercise 5C}
1 Find the values of \(\alpha, \beta, \gamma\) and \(\theta\) in the diagrams below, giving reasons.
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-12(2)}
d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(2)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(9)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(7)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(6)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(1)}
\end{center}

2 Find the size of the marked angle \(\angle P O Q\) in each diagram below. Several steps may be required.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(3)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(4)}
\end{center}

b
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-13(10)}

3 Copy each diagram, then add a suitable construction line in order to find the size of the marked angle \(\angle V O W\) - the first has been done for you. Give reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(8)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13}
\end{center}

c \(P\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-13(5)}
\end{center}

ICE-EM MATHEMATICS YEAR 8

ICE-EM Mathematics 8 3ed ISBN 978-1-108-40429-7 (c) Brown et al. 2017 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party.
d \(F\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(1)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(2)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(6)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(5)}
\end{center}

c

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(4)}
\end{center}

e
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-14(7)}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(8)}
\end{center}

S
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-14(9)}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-14(3)}
\end{center}

\section*{Proving that two lines are parallel}
Corresponding, alternate and co-interior angles can be used to prove that two lines are parallel.

\section*{Proving that two lines are parallel}
Suppose that a transversal crosses two other lines.

\begin{itemize}
  \item If the corresponding angles are equal, then the lines are parallel.
  \item If the alternate angles are equal, then the lines are parallel.
  \item If the co-interior angles are supplementary, then the lines are parallel.
\end{itemize}

These three results are called the converses of the previous three results, because the logic works in reverse. Here are three examples that show how to set out arguments using them.

\section*{Example 11}
Find any parallel lines in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-15}
\end{center}

\section*{Solution}
\(J K \| L M\) (corresponding angles are equal)

\section*{Example 12}
Find any parallel lines in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-15(1)}
\end{center}

\section*{Solution}
\(S T \| Y Z\) (alternate angles are equal)

\section*{Example 13}
Find any parallel lines in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16}
\end{center}

\section*{Solution}
\(135^{\circ}+45^{\circ}=180^{\circ}\) (co-interior angles are supplementary)

Hence, \(A B \| C D\)

\section*{Exercise 5D}
1 In each diagram below, name all pairs of parallel lines, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(6)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(7)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(2)}
\end{center}

e \(R\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(3)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(8)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(4)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-16(1)}
\end{center}

2 In each diagram below, give a reason why \(A B \| C D\). Hence, find the values of \(\alpha, \beta, \gamma\) and \(\theta\).

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(3)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(6)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(1)}
\end{center}

d \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(7)}
\end{center}

3 In each diagram below, write down the values of \(\alpha, \beta, \gamma\) and \(\theta\) that will make \(A B\) parallel to \(C D\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(8)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(2)}
\end{center}

b \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17}
\end{center}

\section*{Angles in triangles}
This section proves and applies two theorems about the angles of any triangle. In mathematics, a true statement and its proof are called a theorem - a Greek word meaning 'a thing to be gazed upon' or 'a thing contemplated by the mind'. You may have seen these theorems already, but proving them will probably be new to you.

\section*{Triangles}
A triangle is formed by taking three non-collinear points \(A, B\) and \(C\) and joining the three intervals \(A B, B C\) and \(C A\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-17(4)}
\end{center}

These intervals are called the sides of the triangle, and the three points are called its vertices. (The singular of vertices is vertex.)

\section*{Proving the theorem about the angle sum of a triangle}
We proved last year that the sum of the interior angles of a triangle is \(180^{\circ}\). As we discussed then, checking a theorem many times doesn't prove it - however many triangles you check, there are always more to check. Here is a totally convincing proof that applies to every triangle, whatever its shape.

The theorem and its proof are set out very formally in the manner traditional for geometry. The proof uses only the methods we have already developed about angles at a point and across transversals.

Theorem: The sum of the interior angles of a triangle is \(180^{\circ}\).

Proof: \(\quad\) Let \(A B C\) be a triangle.

Let \(\angle B A C=\alpha, \angle A B C=\beta\) and \(\angle A C B=\gamma\).

We must prove that \(\alpha+\beta+\gamma=180^{\circ}\).

Draw the line \(X Y\) parallel to \(B C\) and passing through the vertex \(A\).

Then \(\angle X A B=\beta\) (alternate angles, \(X Y \| B C\) ),

\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-18}
and \(\angle Y A C=\gamma\) (alternate angles, \(X Y \| B C)\).

Hence, \(\quad \alpha+\beta+\gamma=180^{\circ}\) (straight angle \(\angle X A Y\) ).

\section*{The exterior angles of a triangle}
Let \(A B C\) be a triangle, with the side \(B C\) produced to \(D\). (The word 'produced' means 'extended'.) Then the angle \(\angle A C D\) between the side \(A C\) and the extension \(C D\) is called an exterior angle of the triangle.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-18(1)}
\end{center}

The angles \(\angle C A B\) and \(\angle A B C\) are called the opposite interior angles, because they are opposite the exterior angle at the vertex \(C\).

There are two exterior angles at each vertex, as shown in the lower diagram opposite, and because they are vertically opposite, they are equal in size:

\(\angle A C D=\angle B C E\) (vertically opposite at \(C\) ).

\section*{Proving the exterior angle theorem}
\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-18(2)}
\end{center}

We proved last year that an exterior angle of a triangle is the sum of the opposite interior angles. The following is valid in every situation, whatever shape the triangle may have.

Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles.

Proof: \(\quad\) Let \(A B C\) be a triangle, with \(B C\) produced to \(D\).

Let \(\angle B A C=\alpha\) and \(\angle C B A=\beta\).

We must prove that \(\angle A C D=\alpha+\beta\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-18(3)}
\end{center}

Draw the line \(C Z\) through \(C\) parallel to \(B A\).

\(\angle Z C D=\beta\) (corresponding angles, \(B A \| C Z\) ), and \(\angle A C Z=\alpha\)

(alternate angles, \(B A \| C Z\) ).

Hence, \(\angle A C D=\alpha+\beta(\) adjacent angles at \(C)\).

\section*{Angles in triangles}
\begin{itemize}
  \item The sum of the interior angles of a triangle is \(180^{\circ}\).
  \item An exterior angle of a triangle equals the sum of the opposite interior angles.
\end{itemize}

\section*{Example 14}
Find \(\angle A\) in the triangle below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-19(1)}
\end{center}

\section*{Solution}
\(\angle A+55^{\circ}+35^{\circ}=180^{\circ}(\) angle sum of \(\triangle A B C)\)

so \(\angle A=90^{\circ}\)

\section*{Example 15}
Find \(\theta\) in the triangle below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-19}
\end{center}

\section*{Solution}
\(\theta=60^{\circ}+50^{\circ}\) (exterior angle of \(\left.\triangle L M N\right)\)

so \(\theta=110^{\circ}\)

\section*{Exercise 5E}
1 Find the third angle of a triangle, two of whose angles are given.
a \(60^{\circ}, 30^{\circ}\)
b \(120^{\circ}, 30^{\circ}\)
c \(111^{\circ}, 11^{\circ}\)
d \(62^{\circ}, 90^{\circ}\)
e \(28^{\circ}, 58^{\circ}\)

2 Find the values of \(\alpha, \beta, \gamma\) and \(\theta\) in the diagrams below, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(7)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(10)}
\end{center}

e \(B\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(3)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(13)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(1)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(12)}
\end{center}

3 Use exterior angles - not interior angles - to find \(\alpha, \beta, \gamma\) and \(\theta\) in each diagram below. Give reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(11)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(5)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(6)}
\end{center}

b \(N\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(9)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(2)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-20(8)}
\end{center}

c
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-20(14)}

4 Find the values of the pronumerals in each diagram, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(11)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(6)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(7)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(13)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(3)}
\end{center}

5 Find \(\alpha, \beta, \gamma\) and \(\theta\) in these diagrams, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(8)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(14)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(12)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(10)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(9)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(15)}
\end{center}

c \(P\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(4)}
\end{center}

f \(L\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-21(5)}
\end{center}

6 Find \(\alpha, \beta, \gamma\) and \(\theta\) in these diagrams, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(6)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(7)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(4)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(12)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(5)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(11)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(15)}
\end{center}

7 Find the size of the marked angle \(\angle A V B\) in each diagram, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(2)}
\end{center}

d \(R\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(8)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(3)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(10)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(14)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(13)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-22(9)}
\end{center}

\section*{Isosceles triangles}
An isosceles triangle is a triangle in which two (or more) sides have equal length.

\begin{itemize}
  \item The equal sides \(A B\) and \(A C\) in the isosceles triangle \(A B C\) to the right are called the legs. They have been marked with double dashes to indicate that they are equal in length.
  \item The vertex \(A\) where the legs meet is called the apex.
  \item The third side \(B C\) is called the base.
  \item The angles \(\angle A B C\) and \(\angle A C B\) at the base are called base angles.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-23(1)}
\end{center}

The word isosceles is a Greek word meaning 'equal legs' - iso means 'equal', and sceles means 'legs'.

\section*{Constructing an isosceles triangle using a circle}
In the diagram below, the equal radii \(A O\) and \(B O\) of the circle form the equal legs of the isosceles triangle \(O A B\). The interval \(A B\) is the base and the centre \(O\) is the apex.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-23}
\end{center}

Using two radii of a circle is usually the simplest way to construct an isosceles triangle.

\section*{Isosceles triangles and their base angles}
Last year, in Chapter 13 of ICE-EM Mathematics Year 7 we developed in an informal way two results about isosceles triangles. The first is a property that all isosceles triangles have.

\begin{itemize}
  \item The base angles of an isosceles triangle are equal.
\end{itemize}

(This result can also be stated as 'If two sides of a triangle are equal, then the angles opposite those sides are equal.')

The second is a test for a triangle to be isosceles, and is the converse of the first result.

\begin{itemize}
  \item If two angles of a triangle are equal, then the sides opposite those angles are equal.
\end{itemize}

In Chapter 12 we will introduce congruence, which will allow us to prove these theorems properly.

\section*{Isosceles triangles}
\begin{itemize}
  \item An isosceles triangle is a triangle with two (or more) sides equal.
  \item The equal sides are called the legs; the legs meet at the apex; the third side is the base; and the angles opposite the legs are called base angles.
  \item Two radii of a circle and the interval joining them form an isosceles triangle.
  \item The base angles of an isosceles triangle are equal.
  \item Conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal.
\end{itemize}

\section*{Equilateral triangles}
An equilateral triangle is a triangle in which all three sides have equal length. (The name comes from Latin - equi means 'equal', and latus means 'side'.)

The diagram below shows an equilateral triangle \(A B C\). Notice how it is an isosceles triangle in three different ways, because the base could be taken as \(A B, B C\) or \(C A\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-24(1)}
\end{center}

\section*{The interior angles of an equilateral triangle are all \(60^{\circ}\)}
It is not hard to prove that all the angles of an equilateral triangle are \(60^{\circ}\) provided that we use the earlier theorem that the base angles of an isosceles triangle are equal.

Theorem: The interior angles of an equilateral triangle are all \(60^{\circ}\).

Proof: \(\quad\) Let \(A B C\) be an equilateral triangle.

Let \(\angle B A C=\alpha, \angle A B C=\beta\) and \(\angle A C B=\gamma\).

We must prove that \(\alpha=\beta=\gamma=60^{\circ}\).

\(\beta=\gamma\) (opposite sides \(A C\) and \(A B\) are equal), and \(\beta=\alpha\)

(opposite sides \(A C\) and \(B C\) are equal), so \(\alpha=\beta=\gamma\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-24}
\end{center}

But \(\alpha+\beta+\gamma=180^{\circ}\) (angle sum of \(\angle A B C\) ), so \(\alpha=\beta=\gamma=60^{\circ}\).

\section*{Equilateral triangles}
\begin{itemize}
  \item An equilateral triangle is a triangle with all three sides equal.
  \item The angles in an equilateral triangle are all \(60^{\circ}\).
\end{itemize}

Here are examples of arguments using each of the three theorems in this section, and the fact that all the radii of any circle are equal.

\section*{Example 16}
Find \(\angle O B A\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-25(2)}
\end{center}

\section*{Solution}
\(A O=B O\) (radii)

Hence, \(\angle O B A=35^{\circ}\) (base angles of isosceles \(\triangle A B O\) )

\section*{Example 17}
Find the length of \(P Q\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-25}
\end{center}

\section*{Solution}
\(P Q=6 \mathrm{~cm}\) (opposite angles \(\angle P Q R\) and \(\angle P R Q\) are equal)

\section*{Example 18}
Find \(\theta\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-25(1)}
\end{center}

\section*{Solution}
\(\angle A B C=60^{\circ}\) (equilateral \(\triangle A B C\) )

Hence, \(\theta+60^{\circ}=180^{\circ}\) (straight angle \(\angle B C D\) )

so \(\quad \theta=120^{\circ}\)

\section*{Exercise 5F}
1 Find the value of each pronumeral, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(8)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(2)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(11)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(3)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(6)}
\end{center}

2 Determine whether \(\triangle A B C\) in each part is an isosceles triangle, naming the equal sides. Give reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(13)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(10)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(12)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(1)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(14)}
\end{center}

3 a Let \(\triangle A B C\) be isosceles, with \(A B=A C\). Produce \(B C\) to \(D\), and let \(\angle A C D=124^{\circ}\). Draw a diagram and find \(\angle A B C\), giving reasons.

b \(P\) and \(Q\) are points on a circle with centre \(O\) such that \(\angle P O Q=48^{\circ}\). Draw a diagram and calculate the size of \(\angle O P Q\).

4 Find the values of the pronumerals in these diagrams, giving reasons.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(9)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-26(7)}
\end{center}

c

\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-26(5)}
d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(4)}
\end{center}

e \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(1)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(3)}
\end{center}

5 Find the values of the pronumerals in these diagrams, giving reasons. In part d, \(O\) is the centre of the circle.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(2)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(8)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(9)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(7)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(10)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(6)}
\end{center}

\section*{Quadrilaterals}
Take any four points \(A, B, C\) and \(D\) in the plane, no three of which are collinear, and join the intervals \(A B, B C, C D\) and \(D A\). Provided that no two of these intervals cross over each other, the figure \(A B C D\) is called a quadrilateral (from Latin - quadri means 'four', and latus means 'side'). The four points are its vertices and the four intervals are its sides.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27(5)}
\end{center}

The quadrilateral above is called a convex quadrilateral because none of its four interior angles is a reflex angle.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-27}
\end{center}

The quadrilateral above is called a nonconvex quadrilateral because one of its interior angles \((\angle A B C)\) is a reflex angle.

(Note: No interior angle can be exactly \(180^{\circ}\) because the definition requires that no three of the vertices be collinear.)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-28}
\end{center}

The intervals \(A C\) and \(B D\) joining non-adjacent vertices are called the diagonals of the quadrilateral. The left-hand quadrilateral above is convex and so both diagonals are inside the quadrilateral. The right-hand quadrilateral above is non-convex and one diagonal is inside the quadrilateral and the other is outside.

\section*{The interior angles of a quadrilateral add to \(360^{\circ}\)}
The interior angles of a quadrilateral always add to \(360^{\circ}\).

The general proof divides the quadrilateral into two triangles and then applies the earlier theorem about the angle sum of a triangle.

Theorem: The sum of the interior angles of a quadrilateral is \(360^{\circ}\).

Proof: \(\quad\) Let \(A B C D\) be a quadrilateral, labelled so that the diagonal \(A C\) is inside the quadrilateral.

(At least one diagonal lies inside the quadrilateral.)
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-28(1)}

We must prove that \(\angle A B C+\angle B C D+\angle C D A+\angle D A B=360^{\circ}\).

Join the diagonal \(A C\).

The interior angles of \(\triangle A B C\) add to \(180^{\circ}\).

Similarly, the interior angles of \(\triangle A D C\) add to \(180^{\circ}\).

But the sum of the interior angles of \(A B C D\) is the sum of the angles of \(\triangle A B C\) and \(\triangle A D C\).

Hence, the interior angles of \(A B C D\) add to \(360^{\circ}\).

\section*{Example 19}
Find \(\theta\) in the diagram opposite.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-29(2)}
\end{center}

\section*{Solution}
\[
\begin{aligned}
\theta+60^{\circ}+110^{\circ}+140^{\circ} & =360^{\circ}(\text { angle sum of quadrilateral } A B C D) \\
\theta+310^{\circ} & =360^{\circ} \\
\theta & =50^{\circ}
\end{aligned}
\]

\section*{Quadrilaterals}
\begin{itemize}
  \item A quadrilateral \(A B C D\) has four vertices, no three of which are collinear, and four sides, no two crossing each other.
  \item A convex quadrilateral has no reflex interior angle. Both diagonals of a convex quadrilateral lie inside the figure.
  \item A non-convex quadrilateral has one reflex interior angle. One diagonal of a non-convex quadrilateral lies outside the figure, the other inside.
  \item The sum of the interior angles of a quadrilateral is \(360^{\circ}\).
\end{itemize}

\section*{Exercise 5G}
1 Find the values of the pronumerals in these diagrams, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-29(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-29(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-29(5)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-29(4)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-29}
\end{center}

2 In each part, three angles of a quadrilateral are given. Calculate the size of the fourth angle of the quadrilateral. Sketch a quadrilateral with these angle sizes.
a \(90^{\circ}, 90^{\circ}, 150^{\circ}\)
b \(45^{\circ}, 90^{\circ}, 135^{\circ}\)
c \(70^{\circ}, 160^{\circ}, 20^{\circ}\)
d \(40^{\circ}, 40^{\circ}, 40^{\circ}\)

3 Find the values of the pronumerals, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(15)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(14)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(13)}
\end{center}

4 Find the values of the pronumerals, giving reasons. In part \(\mathbf{f}, O\) is the centre of the circle.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(9)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(11)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(5)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(12)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(10)}
\end{center}

5 Find the values of the pronumerals, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30}
\end{center}

\[
Q P=Q R
\]

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(3)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(7)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(2)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(6)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-30(8)}
\end{center}

6 The following diagram is a map of the first floor of Tony's house.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31(6)}
\end{center}

a Tony would like to build some wooden shelves at the end of the recreation room. Each shelf would be an isosceles triangle. Evaluate the internal angles of these shelves.

b Tony would also like to construct a TV table in the shape of an isosceles triangle. Evaluate the internal angles of the TV table.

\section*{Review exercise}
1 Find the value of the pronumeral in each diagram. Give reasons in each case.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31(1)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31(5)}
\end{center}

e \(L\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31(3)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-31(4)}
\end{center}

2 State the complement of each angle.
a \(30^{\circ}\)
b \(63^{\circ}\)
c \(74^{\circ}\)
d \(84^{\circ}\)

3 State the supplement of each angle.
a \(127^{\circ}\)
b \(76^{\circ}\)
c \(134^{\circ}\)
d \(15^{\circ}\)

4 Find the value of each pronumeral.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(10)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(8)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(11)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(13)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(2)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(6)}
\end{center}

f
\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-32(12)}

5 Find the values of the pronumerals in these diagrams.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(7)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(1)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(5)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-32(9)}
\end{center}

6 Find the value of \(\alpha\) in each of these diagrams.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33(3)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33(2)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33(4)}
\end{center}

7 Explain why a triangle cannot have a reflex angle as one of its internal angles.

8 Find the values of \(\alpha, \beta\) and \(\gamma\) in the diagrams below, and explain your reasoning.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33(5)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33(1)}
\end{center}

\section*{Challenge exercise}
1 Calculate the unknown angle.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-33(6)}
\end{center}

2 a Find the value of \(\alpha\).

\includegraphics[max width=\textwidth, center]{2024_02_22_7550d257909da2a4b5a7g-33(7)}
b Find \(\alpha\) in terms of \(\beta, \gamma\) and \(\theta\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(8)}
\end{center}

c Find \(\alpha\) in terms of \(\beta, \gamma\) and \(\theta\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(6)}
\end{center}

3 The point \(O\) is the centre of the circle. Find \(\alpha\) in terms of \(\beta\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(7)}
\end{center}

4 The sum of the internal angles of a triangle is \(180^{\circ}\). Use this property of a triangle to find the sum of the internal angles in each of the diagrams below. What relationship do you notice?

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(4)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(3)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-34(2)}
\end{center}

5 Name all the isosceles triangles in the diagram below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-35}
\end{center}

6 Consider the isosceles triangle \(A B C\), with \(A B=A C\). Suppose \(D\) is on \(A B\), and \(E\) is on \(A C\), so that \(A D=D E\). If \(\angle \mathrm{CED}=142^{\circ}\), calculate \(\angle A B C\).

\(7 A B C D\) is a square and \(A E B\) is an equilateral triangle. Calculate the two possible values of the size of \(\angle D E C\).

8 Find the value of \(\alpha+\beta+\gamma\) in the diagram below.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_7550d257909da2a4b5a7g-35(1)}
\end{center}


\end{document}