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\title{Congruence and special quadrilaterals }

\author{}
\date{}


\begin{document}
\maketitle
This chapter reviews work on angles, congruence tests and their applications. Congruence is an extremely useful tool in geometrical arguments. It is used here to prove properties of various types of quadrilaterals, and to develop tests for special quadrilaterals.

This chapter is a revision of all the geometry presented in Years 7 and 8.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-01(1)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-01}
\end{center}

\section*{Review of angles}
Most geometrical arguments rely on the relationship between different angles in the one figure. This section reviews the methods introduced so far, and the next section uses these methods to prove some interesting general results.

\section*{Describing angles by their sizes}
Angles of three particular sizes have special names:

\begin{itemize}
  \item An angle of \(90^{\circ}\) is a right angle.
  \item An angle of \(180^{\circ}\) is a straight angle.
  \item An angle of \(360^{\circ}\) is a revolution.
\end{itemize}

Other angles are described by being within a range of angle sizes:

\begin{itemize}
  \item An acute angle is an angle between \(0^{\circ}\) and \(90^{\circ}\).
  \item An obtuse angle is an angle between \(90^{\circ}\) and \(180^{\circ}\).
  \item A reflex angle is an angle between \(180^{\circ}\) and \(360^{\circ}\).
\end{itemize}

For example, \(30^{\circ}\) is an acute angle, \(140^{\circ}\) is obtuse, and \(220^{\circ}\) is reflex.

Particular pairs of angles also have special names:

\begin{itemize}
  \item Two angles whose sum is \(90^{\circ}\) are called complementary angles.
  \item Two angles whose sum is \(180^{\circ}\) are called supplementary angles.
\end{itemize}

For example, \(35^{\circ}\) and \(55^{\circ}\) are complementary angles because \(35^{\circ}+55^{\circ}=90^{\circ}\), and \(70^{\circ}\) and \(110^{\circ}\) are supplementary angles because \(70^{\circ}+110^{\circ}=180^{\circ}\).

\section*{Angles at a point}
The first group of results concern angles at a point.

\begin{itemize}
  \item Adjacent angles can be added and subtracted.
  \item Adjacent angles on a straight line are supplementary.
  \item Angles in a revolution add to \(360^{\circ}\).
  \item Vertically opposite angles are equal.
\end{itemize}

\section*{Example 1}
Find the value of the pronumeral in each figure.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-02}
\end{center}

b

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-02(1)}
a \(\quad \beta=60^{\circ} \quad\) (complementary angles)

\(2 \alpha=90^{\circ} \quad\) (supplementary angles)

\(\alpha=45^{\circ}\)

b \(3 \alpha=75^{\circ} \quad\) (vertically opposite angles at \(M\) )

\(\alpha=25^{\circ}\)

\section*{Angles across transversals}
Suppose that a transversal crosses two lines.

\begin{itemize}
  \item If the lines are parallel, then the alternate angles (for example, \(\delta\) and \(\gamma\) ) are equal.
  \item If the lines are parallel, then the corresponding angles (for example, \(\alpha\) and \(\beta\) ) are equal.
  \item If the lines are parallel, then the co-interior angles (for example, \(\delta\) and \(\beta\) )
\end{itemize}

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-03}
are supplementary.

\section*{Example 2}
Find \(\alpha, \beta\) and \(\gamma\) in the figure to the right.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-03(1)}
\end{center}

\section*{Solution}
\begin{center}
\begin{tabular}{ll}
\(\alpha=130^{\circ}\) & (corresponding angles, \(P Q \| R S\) ) \\
\(\beta=50^{\circ}\) & (co-interior angles, \(A B \| C D\) ) \\
\(\gamma=50^{\circ}\) & (alternate angles, \(P Q \| R S\) ) \\
\end{tabular}
\end{center}

\section*{Proving that two lines are parallel}
The converses of the previous three results are tests to prove that two lines are parallel. Suppose that a transversal crosses two lines.

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

\section*{Example 3}
Identify every pair of parallel lines in the figure to the right.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-04(3)}
\end{center}

\section*{Solution}
First, \(A B \| D C\) (alternate angles \(\angle B A C\) and \(\angle A C D\) are equal).

Secondly, \(\angle B A D=115^{\circ}(\) adjacent angles at \(A)\).

So \(A D \| B C\) (co-interior angles \(\angle B A D\) and \(\angle A B C\) are supplementary).

\section*{Angle sums of triangles and quadrilaterals}
\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.
  \item The sum of the interior angles of a quadrilateral is \(360^{\circ}\).
\end{itemize}

Proofs of these results are reviewed in Exercise 7B.

\section*{Example 4}
Find \(\theta\) in each diagram below.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-04}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-04(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-04(1)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-04(2)}
\end{center}

Solution

a \(3 \theta+2 \theta+60^{\circ}=180^{\circ}\)

(angle sum of \(\Delta F G H\) )

\[
\begin{aligned}
5 \theta & =120^{\circ} \\
\theta & =24^{\circ}
\end{aligned}
\]

(continued over page)
b \(\theta+130^{\circ}=150^{\circ}\)

\[
\theta=20^{\circ}
\]

c \(3 \theta+3 \theta+4 \theta+90^{\circ}=360^{\circ} \quad\) (angle sum of quadrilateral \(A B C D\) )

\[
\begin{aligned}
10 \theta & =270^{\circ} \\
\theta & =27^{\circ}
\end{aligned}
\]

d \(\theta+\frac{\theta}{2}+90^{\circ}=180^{\circ} \quad\) (angle sum of \(\triangle A B C\) )

\[
\begin{aligned}
\frac{3 \theta}{2} & =90^{\circ} \\
\theta & =60^{\circ}
\end{aligned}
\]

\section*{Isosceles and equilateral triangles}
A triangle is called isosceles if it has two equal sides; it is called equilateral if all three sides are equal. Thus all equilateral triangles are also isosceles triangles.

\begin{itemize}
  \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.
  \item Each interior angle of an equilateral triangle is \(60^{\circ}\).
  \item Conversely, if the three angles of a triangle are equal, then the triangle is equilateral.
\end{itemize}

Proofs of these results are reviewed in Exercise 7C.

\section*{Example 5}
Find \(\alpha, \beta, \gamma\) and \(x\) in the diagrams below.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-05(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-05(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-05}
\end{center}

c

\section*{Solution}
a First, \(\alpha=60^{\circ}\)

Secondly, \(\angle B C D=\beta\)

so \(2 \beta+30^{\circ}=180^{\circ}\)

\[
\beta=75^{\circ}
\]

b \(2 x+3=5 x+6\)

\[
-3=3 x
\]

\[
x=-1
\]

Thus \(P Q\) and \(P R\) each have length of 1 .

c

\[
\alpha=65^{\circ}
\]

(base angles of isosceles \(\triangle A B C\) )

\(\angle B C X=40^{\circ}\)

(base angles of isosceles \(\triangle B X C\) )

\(\gamma=100^{\circ}\)
( \(\triangle A B C\) is equilateral)

(base angles of isosceles \(\triangle B C D\) )

(angle sum of \(\triangle B C D\) )

(opposite angles of \(\triangle P Q R\) are equal)

\[
\text { (angle sum of } \triangle X B C \text { ) }
\]

\section*{Exercise 7A}
1 a Classify these angles using the standard terms.
i \(140^{\circ}\)
ii \(360^{\circ}\)
iii \(33^{\circ}\)
iv \(180^{\circ}\)
v \(90^{\circ}\)
vi \(350^{\circ}\)

b Write down the complements of:
i \(20^{\circ}\)
ii \(72^{\circ}\)
iii \(45^{\circ}\)

c Write down the supplements of:
i \(20^{\circ}\)
ii \(172^{\circ}\)
iii \(90^{\circ}\)

2 This question reviews angles at a point. Find the angles marked with pronumerals, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-06(2)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-06}
\end{center}

b
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-06(1)}

c
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-06(3)}

3 This question reviews angles across transversals. Find the angles marked with pronumerals, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(10)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(14)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(2)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(6)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(8)}
\end{center}

Example 3 4 This question reviews tests for two lines to be parallel. In each part, name all pairs of parallel lines, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(13)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(11)}
\end{center}

e \(K\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(5)}
\end{center}

c \(Q\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(7)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(12)}
\end{center}

5 This question reviews angle sums of triangles and quadrilaterals. Find the angles marked with pronumerals, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(4)}
\end{center}

b \(K\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-07(3)}
\end{center}

c \(P\)

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-07(9)}
d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(1)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(2)}
\end{center}

f \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(7)}
\end{center}

6 This question reviews isosceles and equilateral triangles. Find the angles and sides marked with pronumerals, giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(8)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(6)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(3)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(9)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(5)}
\end{center}

7 Find the values of the pronumerals \(\alpha, \beta, \gamma, \theta\) and \(x\). Different methods are combined in each part of this question.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(10)}
\end{center}

b A

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-08(4)}
\end{center}

c

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-08(11)}
d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09(2)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09(4)}
\end{center}

\(\mathbf{h}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09(5)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09(3)}
\end{center}

\section*{Reasoning with angles}
This section uses the methods of Section 7A in more general situations. Construction lines are often needed.

\section*{Example 6}
Find the angle \(\angle A O B\) in terms of \(\theta\), giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09(6)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-09(1)}
\end{center}

\section*{Solution}
\[
\begin{aligned}
& \text { a } \angle F O A=\theta \\
& \text { (corresponding angles, } E D \| F B \text { ) } \\
& \angle A O B=180^{\circ}-\theta \\
& \text { (supplementary angle) } \\
& \text { b } \angle A O C=\frac{180^{\circ}-\theta}{2} \\
& \text { ( } \triangle A C O \text { is isosceles) } \\
& =90^{\circ}-\frac{\theta}{2} \\
& \angle A O B=180^{\circ}-\left(90^{\circ}-\frac{\theta}{2}\right) \quad(\text { supplementary angles }) \\
& =90^{\circ}+\frac{\theta}{2}
\end{aligned}
\]

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Then & \(\angle A O X=\alpha\) & (alternate angles, \(O X \| F A\) ) \\
\hline
and & \(\angle B O X=\beta\) & (alternate angles, \(O X \| G B\) ) \\
\hline
SO & \(\alpha+\beta=90^{\circ}\) & (adjacent angles) \\
\hline
\end{tabular}
\end{center}

\section*{Example 7}
Show that \(\alpha+\beta=90^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-10(1)}
\end{center}

\section*{Solution}
Draw \(X O\) parallel to \(A F\).

\section*{Example 8}
Prove that \(\alpha+\beta-\gamma=180^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-10}
\end{center}

\section*{Solution}
Draw \(C X\) parallel to \(B A\).

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Then & \(\angle B C X=180^{\circ}-\alpha\) & (co-interior angles, \(B A \| C X\) ) \\
\hline
and & \(\angle D C X=\gamma\) & (alternate angles, \(C X \| E D\) ) \\
\hline
Hence & \(\beta=180^{\circ}-\alpha+\gamma\) & \((\angle B C D=\angle B C X+\angle D C X)\) \\
\hline
 & \(-\beta-\gamma=180^{\circ}\) &  \\
\hline
\end{tabular}
\end{center}

\section*{Proving theorems}
The results of the previous section can be applied to prove many interesting general results, called theorems. For the rest of this chapter, each theorem to be proven is stated in italics. Usually the question includes a diagram, introduces the necessary pronumerals and breaks the proof down into a number of steps. Sometimes a construction is required.

\section*{Example 9}
Prove that: A quadrilateral with two pairs of equal adjacent angles is a trapezium.

In the quadrilateral \(A B C D\) to the right, \(\angle A=\angle D=\alpha\) and \(\angle B=\angle C=\beta\).

a Prove that \(\alpha+\beta=180^{\circ}\).

b Hence prove that \(A B C D\) is a trapezium.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-11}
\end{center}

\section*{Solution}
a \(\alpha+\alpha+\beta+\beta=360^{\circ} \quad\) (angle sum of quadrilateral \(A B C D\) )

\[
\text { so } \alpha+\beta=180^{\circ}
\]

b Hence \(A D \| B C \quad\) (co-interior angles are supplementary)

Therefore \(A B C D\) is a trapezium.

\section*{Angle sums of polygons}
The last four questions of the following exercise deal with interior and exterior angle sums of polygons. These are interesting general results that you may want to remember as part of your known geometric facts.

\section*{Exercise 7B}
1 Find the marked angle \(\angle A O B\) in terms of \(\theta\), giving reasons.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(2)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(8)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(12)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(7)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(9)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(10)}
\end{center}

f \(M\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(14)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(6)}
\end{center}

2 a Prove that \(A B \perp P Q\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12}
\end{center}

c Prove that \(P O \perp Q O\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(1)}
\end{center}

b Prove that \(\angle A B P=2 \times \angle A B Q\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(3)}
\end{center}

d Prove that \(A B \| D C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(4)}
\end{center}

Example 83 These questions may require construction lines to be drawn.

a Prove that \(\alpha+\beta=120^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-12(11)}
\end{center}

b Prove that \(\alpha-\beta=40^{\circ}\).

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-12(13)}
c Prove that \(\alpha+\beta=130^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-13}
\end{center}

Example 9 4 This question reviews proofs of theorems about angle sums of triangles.

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

In \(\triangle A B C\) to the right, \(\angle B A C=\alpha, \angle B=\beta\) and \(\angle C=\gamma\).

The line \(X A Y\) has been constructed parallel to \(B C\).

i Explain why \(\angle X A B=\beta\) and \(\angle Y A C=\gamma\).

ii Hence explain why \(\alpha+\beta+\gamma=180^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-13(3)}
\end{center}

b Prove that: An exterior angle of a triangle equals the sum of the opposite interior angles.

In \(\triangle A B C\) to the right, \(\angle A=\alpha\) and \(\angle B=\beta\).

The line \(C G\) has been constructed parallel to \(B A\), and the side \(B C\) has been produced to \(P\).

i Explain why \(\angle A C G=\alpha\) and \(\angle P C G=\beta\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-13(5)}
\end{center}

ii Hence explain why \(\angle A C P=\alpha+\beta\).

5 Prove that: An angle in a semicircle is a right angle.

In the circle to the right, \(A O B\) is a diameter, and \(P\) is any other point on the circle.

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

a Prove that \(\angle A P B=\alpha+\beta . \quad\) b Hence prove that \(\angle A P B=90^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-13(2)}
\end{center}

6 a Prove that: The sum of the exterior angles of a triangle is \(360^{\circ}\).

In the diagram to the right, \(\alpha, \beta\) and \(\gamma\) are the sizes of the three exterior angles of \(\triangle A B C\).

i Find the sizes of the three interior angles of \(\triangle A B C\).

ii Hence show that \(\alpha+\beta+\gamma=360^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-13(1)}
\end{center}

b Draw a quadrilateral \(A B C D\). Use a construction line to help explain why the sum of the interior angles of \(A B C D\) is \(360^{\circ}\).

c Prove that: The sum of the four exterior angles of a convex quadrilateral is \(360^{\circ}\).

In the diagram to the right, \(\alpha, \beta, \gamma\) and \(\theta\) are the sizes of the four exterior angles of the convex quadrilateral \(A B C D\).

(All the interior angles of a convex quadrilateral are less than \(180^{\circ}\).)

Show that \(\alpha+\beta+\gamma+\theta=360^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-13(4)}
\end{center}

\section*{Angle sums of polygons}
7 a The diagrams below show a pentagon, an octagon and a dodecagon. By dividing each figure into triangles, find the sum of its interior angles.

i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-14(2)}
\end{center}

ii

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-14}
\end{center}

iii

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-14(1)}
\end{center}

b Using a similar technique, find the sum of the interior angles of:
i a hexagon
ii a heptagon
iii a nonagon
iv a decagon

c Using these examples as a guide, find a formula for the angle sum of an \(n\)-sided polygon.

8 In a convex polygon, all the interior angles are less than \(180^{\circ}\).

a The diagram to the right shows a convex decagon, and any point \(O\) inside the decagon. Intervals are drawn from \(O\) to each of the 10 vertices, dissecting the decagon into 10 triangles.

i What is the sum of the interior angles of all 10 triangles so formed?

ii Hence find the sum of the interior angles of the decagon.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-14(4)}
\end{center}

b Draw a convex hexagon and repeat the process.

c Using the same technique, find a formula for the sum of the interior angles of a convex polygon with \(n\) sides. Confirm that this matches your answer to Question 7c.

9 a In the convex decagon to the right, each side has been produced to form an exterior angle.

i Explain why the sum of the interior angles plus the sum of the exterior angles is \(1800^{\circ}\).

ii Hence find the sum of the 10 exterior angles.

b Draw a convex hexagon and repeat the process.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-14(3)}
\end{center}

c Using the same technique, find the sum of the exterior angles of a convex polygon with \(n\) sides.

10 Find a formula for the size of each interior angle in a regular \(n\)-sized polygon.

11 The figure shown is a regular pentagram. It is formed by producing the sides of a regular pentagon.

a Find the size of each interior angle of the pentagon \(A B C D E\).

b Find the values of \(\alpha, \beta\) and \(\gamma\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-14(5)}
\end{center}

\section*{Using congruence}
In ICE-EM Mathematics Year 8 we introduced the idea of congruent figures.

\section*{Congruent figures}
\begin{itemize}
  \item Two plane figures are called congruent if one figure can be moved on top of the other figure, by a sequence of translations, rotations and reflections, so that they coincide exactly.
  \item Congruent figures have exactly the same shape and size.
  \item When two figures are congruent, we can match every part of one figure with the corresponding part of the other, so that:
  \item matching angles have the same size
  \item matching paired intervals have the same length
  \item matching paired regions have the same area.
\end{itemize}

The congruence arguments used in this chapter involve only congruent triangles. In Year 8 we developed four tests for two triangles to be congruent.

\section*{The four standard congruence tests for triangles}
Two triangles are congruent if:

\begin{itemize}
  \item SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or
  \item AAS: two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle, or
  \item SAS: two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle, or
  \item RHS: the hypotenuse and one side of one right-angled triangle are respectively equal to the hypotenuse and one side of the other right-angled triangle.
\end{itemize}

\section*{Example 10}
In each part, write a congruence statement, giving a test as the reason. Make sure that you write the vertices of the two triangles in matching order.

a \(P\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-15}
\end{center}

z

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-15(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-15(3)}
\end{center}

D

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-15(2)}
c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-16}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-16(1)}
\end{center}

\section*{Solution}
a \(\triangle P Q R \equiv \triangle X Y Z(\mathrm{SSS})\)
b \(\triangle B A C \equiv \triangle E D F(\mathrm{SAS})\)
c \(\triangle A B P \equiv \triangle A B Q(\mathrm{AAS})\)
d \(\triangle A B Q \equiv \triangle B A P(\mathrm{RHS})\)

Notice that in parts \(\mathbf{c}\) and \(\mathbf{d}\), the side \(A B\) is common to both triangles, and thus provides one of the pairs of equal sides.

\section*{Using congruence to prove that two matching sides or angles are equal}
Once we have established that two triangles are congruent, we know that the remaining matching sides and angles are equal, as in the following example.

\section*{Example 11}
In the diagram to the right, \(A B\) and \(B C\) are equal chords of a circle with centre \(O\).

a Prove that \(\triangle A O B \equiv \triangle C O B\).

b Hence prove that \(\angle A O B=\angle C O B\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-16(2)}
\end{center}

\section*{Solution}
a In the triangles \(A O B\) and \(C O B\) :

\[
\begin{aligned}
O A & =O C & & \text { (radii) } \\
O B & =O B & & \text { (common) } \\
A B & =C B & & \text { (given) } \\
\text { so } \triangle A O B & & & \text { (SSS) }
\end{aligned}
\]

b Hence \(\angle A O B=\angle C O B \quad\) (matching angles of congruent triangles)

\section*{Using congruence to prove that two lines are parallel}
In many situations, congruence is used to prove that two alternate angles, or two corresponding angles, are equal. This allows us to prove that two lines are parallel.

\section*{Example 12}
In the diagram to the right,

\(A M=B M\) and \(P M=Q M\)

a Prove that \(\triangle A M P \equiv \triangle B M Q\).

b Prove that \(A P \| B Q\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-17(3)}
\end{center}

\section*{Solution}
a In the triangles \(A M P\) and \(B M Q\) :

\[
\begin{array}{ll}
A M=B M & \text { (given) } \\
P M=Q M & \text { (given) }
\end{array}
\]

\[
\begin{aligned}
\angle A M P & =\angle B M Q & & (\text { vertically opposite angles at } M) \\
\text { so } \triangle A M P & \equiv \triangle B M Q & & (\mathrm{SAS})
\end{aligned}
\]

b Hence \(\angle A P M=\angle B Q M \quad\) (matching angles of congruent triangles)

so \(A P \| B Q \quad\) (alternate angles are equal)

\section*{Exercise 7C}
1 In each part, write a congruence statement, giving a congruence test as the reason. Make sure that you name the vertices in matching order.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-17(6)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-17(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-17(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-17(5)}
\end{center}

2 In each part, write a congruence statement, giving a congruence test as the reason.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-17}
\end{center}

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-17(1)}
b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-18(9)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-18(4)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-18(3)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-18(1)}
\end{center}

3 From the six triangles drawn, name three pairs of congruent triangles. Give reasons.
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-18(8)}

4 In each of the following, state which triangle is congruent to \(\triangle A B C\), giving the appropriate congruence test as a reason.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-18(2)}
\end{center}

b
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-18(5)}
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-18(6)}
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-18(7)}

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-18}
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-19(1)}

\(5 \triangle A B C \equiv \triangle P Q R\) (where the vertices are in matching order).

a Which angle in \(\triangle P Q R\) is equal in size to \(\angle A B C\) ?

b Which angle in \(\triangle P Q R\) is equal in size to \(\angle C A B\) ?

c Which side in \(\triangle P Q R\) is equal in length to \(A C\) ?

d Which side in \(\triangle A B C\) is equal in length to \(Q R\) ?

6 Find the values of the pronumerals.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-19}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-19(5)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-19(2)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-19(3)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-19(6)}
\end{center}

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-19(4)}
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-19(7)}
\(7 \mathbf{a}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(7)}
\end{center}

i Prove that \(\triangle A B M \equiv \triangle A C M\).

ii Hence prove that \(\alpha=\beta\).

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(2)}
\end{center}

Use congruence to prove that \(F G=G H\), given that \(O\) is the centre of the circle. b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(5)}
\end{center}

i Prove that \(\triangle O A B \equiv \triangle O P Q\).

ii Hence prove that \(x=y\).

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(1)}
\end{center}

Use congruence to prove that \(A D=D C\) and that \(A B=B C\).

\(8 A B\) is a chord of a circle centre \(O\) and \(M\) is the midpoint of \(A B\). Prove that \(O M\) is perpendicular to \(A B\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20}
\end{center}

i Prove that \(\triangle N A B \equiv \triangle N D C\).

ii Hence prove that \(\angle A=\angle D\).

iii Hence prove that \(A B \| C D\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(4)}
\end{center}

Use congruence to prove that \(A B \| C D\), given that \(O\) is the centre of the circle. b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(6)}
\end{center}

i Give a reason why \(P Q \| S R\).

ii Prove that \(\triangle P R S \equiv \triangle R P Q\).

iii Hence prove that \(P S \| Q R\).

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-20(3)}
\end{center}

Prove that \(\angle A B C=\angle A B D\) using congruence, and hence prove that \(A B \perp C D\).

10 a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-21(6)}
\end{center}

Use congruence to prove that \(\triangle A B C\) is isosceles. b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-21}
\end{center}

Use congruence to prove that \(\triangle A B C\) is isosceles, given that \(B P=C Q\).

11 In the triangle \(\triangle A B C, A B=A C\). The bisectors of angles \(A B C\) and \(A C B\) meet at \(X\). Prove that \(X B=X C\).

12 In each part, \(O\) is the centre of the circle or circles. You will need to construct radii and use congruence in these questions.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-21(1)}
\end{center}

Prove that \(P Q\) bisects \(A B\). b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-21(5)}
\end{center}

Prove that \(A B=C D\).

13 This question reviews proofs of several theorems about isosceles and equilateral triangles.

a Prove that: The base angles of an isosceles triangle are equal.

The triangle \(A B C\) to the right is isosceles, with \(A B=A C\).

The bisector of \(\angle B A C\) meets the base \(B C\) at \(M\).

i Prove that \(\triangle A B M \equiv \triangle A C M\).

ii Hence prove that \(\angle B=\angle C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-21(3)}
\end{center}

b Conversely, prove that: If two angles of a triangle are equal, then the sides opposite those angles are equal.

In the triangle \(A B C\) to the right, \(\angle B=\angle C\).

The bisector of \(\angle B A C\) meets the base \(B C\) at \(M\).

i Prove that \(\triangle A B M \equiv \triangle A C M\).

ii Hence prove that \(\triangle A B C\) is isosceles.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-21(2)}
\end{center}

c Prove that: Each interior angle of an equilateral triangle is \(60^{\circ}\).

The triangle \(A B C\) to the right is equilateral. Let \(\angle A=\theta\).

i Give a reason why \(\angle B=\theta\).

ii Give a reason why \(\angle C=\theta\).

iii Explain why \(\theta=60^{\circ}\).

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-21(4)}
d Conversely, prove that: If all angles of a triangle are equal, then the triangle is equilateral.

In the triangle \(A B C\) to the right, all the interior angles are equal.

Let \(\angle A=\angle B=\angle C=\theta\).

i Give a reason why \(A B=A C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-22(4)}
\end{center}

ii Give a reason why \(A C=B C\).

14 This question reviews proofs by congruence of three well-known properties of parallelograms.

Recall that a parallelogram is a quadrilateral whose opposite sides are parallel.

a Prove that: Opposite angles of a parallelogram are equal.

A parallelogram \(A B C D\) is drawn to the right. Let \(\angle A=\alpha\).

i Find the sizes of \(\angle B\) and \(\angle D\) in terms of \(\alpha\).

ii Hence find the size of \(\angle C\) in terms of \(\alpha\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-22(3)}
\end{center}

b Prove that: Opposite sides of a parallelogram are equal.

Let \(A B C D\) be a parallelogram with the diagonal \(A C\) joined.

i Prove that \(\triangle A B C \equiv \triangle C D A\).

ii Hence prove that \(A B=D C\) and \(A D=B C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-22}
\end{center}

c Prove that: The diagonals of a parallelogram bisect each other.

Let the diagonals of the parallelogram \(A B C D\) meet at \(M\).

i Prove that \(\triangle A B M \equiv \triangle C D M\).

ii Hence prove that \(A M=C M\) and \(B M=D M\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-22(2)}
\end{center}

15 This question reviews proofs by congruence of some diagonal properties of rhombuses and rectangles.

Recall that a rhombus is a quadrilateral with all sides equal.

a Prove that: A rhombus is a parallelogram.

b Prove that: The diagonals of a rhombus are perpendicular, and bisect the vertex angles through which they pass.

Let the diagonals of the rhombus \(A P B Q\) meet at \(M\).

i Prove that \(\triangle A P B \equiv \triangle A Q B\).

ii Hence prove that \(\angle P A M=\angle Q A M\).

iii Prove also that \(A M \perp P Q\).
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-22(1)}
c Prove that: The diagonals of a rectangle are equal.

Let \(A B C D\) be a rectangle with the diagonals joined.

i Prove that \(\triangle A D C \equiv \triangle B C D\).

ii Hence prove that \(A C=B D\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-23}
\end{center}

16 Prove that: When two circles intersect, the line joining their centres is the perpendicular bisector of their common chord.

In the diagram to the right, \(A B\) is the common chord of two intersecting circles with centres \(O\) and \(P\). The chord \(A B\) meets the line \(O P\) at \(M\).

a Prove that \(\triangle A O P \equiv \triangle B O P\).

b Hence prove that \(O P\) bisects \(\angle A O B\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-23(3)}
\end{center}

c Prove that \(A O M \equiv B O M\).

d Hence prove that \(A M=B M\) and \(A B \perp O P\).

\section*{Parallelograms}
In Year 8 we defined a parallelogram and discovered some of its important properties.

\section*{Parallelograms}
\section*{Definition of a parallelogram}
\begin{itemize}
  \item A parallelogram is a quadrilateral whose opposite sides are parallel.
\end{itemize}

\section*{Properties of a parallelogram}
\begin{itemize}
  \item The opposite angles of a parallelogram are equal.
  \item The opposite sides of a parallelogram are equal.
  \item The diagonals of a parallelogram bisect each other.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-23(1)}
\end{center}

The diagrams below illustrate these properties. Their proofs were reviewed in Exercise 7C, Question 13.
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-23(2)}

The purpose of this section is to establish, and subsequently apply, four well-known tests for a quadrilateral to be a parallelogram.

\section*{Four tests for a parallelogram}
\begin{itemize}
  \item If the opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram.
  \item If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.
  \item If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.
  \item If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
\end{itemize}

The first question in the following exercise reviews proofs of these tests.

\section*{Exercise 7D}
1 This question reviews the proofs of the four tests for a quadrilateral to be a parallelogram.

a Prove that: If the opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram.

Let \(A B C D\) be a quadrilateral with \(\angle A=\angle C=\alpha\) and \(\angle B=\angle D=\beta\).

i Prove that \(\alpha+\beta=180^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-24}
\end{center}

ii Hence prove that \(A B \| D C\) and \(A D \| B C\).

\(D\)

b Prove that: If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

Let \(R S T U\) be a quadrilateral with \(R S=U T\) and \(R U=S T\).

i Prove that \(\triangle R S U \equiv \triangle T U S\).

ii Hence prove that \(R S \| U T\) and \(R U \| S T\).

c Prove that: If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

Let \(K L M N\) be a quadrilateral with \(K L=N M\) and \(K L \| N M\).

i Prove that \(\triangle K L M \equiv \triangle M N K\).

ii Hence prove that \(K N \| L M\).

d Prove that: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Let \(E F G H\) be a quadrilateral whose diagonals meet at \(O\) and bisect each other.

i Prove that \(\triangle E H O \equiv \triangle G F O\).

ii Hence prove that \(E H=F G\) and \(E H \| F G\).

iii Explain why this proves that \(E F G H\) is a parallelogram.
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-24(1)}

2 a In the diagram to the right, both circles have centre \(O\). The interval \(A O B\) is a diameter of the smaller circle, and \(P O Q\) is a diameter of the larger circle. What sort of figure is \(A P B Q\), and why?

b The opposite edges of my ruler are parallel. I draw

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-25(5)}
intervals \(L M\) and \(S T\) of length \(4 \mathrm{~cm}\) on the opposite edges of the ruler, as shown in the diagram to the right. What sort of figure is LMST, and why?

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(8)}
\end{center}

c In the diagram to the right, \(\angle A O B\) is any given angle. First I draw an arc with centre \(A\) and radius \(O B\). Then I draw an arc with centre \(B\) and radius \(O A\). The two arcs meet at \(M\), and I claim that \(A M B O\) is a parallelogram. Am I correct? Explain why or why not.

\(3 \mathbf{a}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(2)}
\end{center}

\(R\)
Given that \(A B C D\) is a parallelogram, prove that \(A P=Q C\).

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(4)}
\end{center}

\(A B C D\) is a parallelogram with \(X\) and \(Y\) points on the diagonal \(B D\) such that \(B Y=D X\). Prove that \(A Y C X\) is a parallelogram.

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(7)}
\end{center}

\(A B C D\) is a parallelogram and \(B X\) and \(D Y\) are perpendicular to \(A C\). Prove that \(B X=D Y\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(3)}
\end{center}

Given that \(R S T U\) is a parallelogram, prove that \(U X \| Y S\).

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25}
\end{center}

\(A B C D\) is a parallelogram and \(P B=D Q\). Prove that \(A Q=C P\).

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-25(6)}
\end{center}

\(A B C D\) is a parallelogram with \(B P=D Q\). Prove that \(A P C Q\) is a parallelogram.

\section*{Tests for rhombuses, rectangles and squares}
Rhombuses, rectangles and squares are special types of parallelograms. Last year we defined these figures and proved some of their properties. This section establishes some important tests for these figures.

\section*{Rhombuses}
\section*{Definition of a rhombus}
\begin{itemize}
  \item A rhombus is a quadrilateral with four equal sides.
\end{itemize}

\section*{Diagonal properties of a rhombus}
\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-26(1)}
\end{center}

\begin{itemize}
  \item The diagonals of a rhombus bisect each other at right angles.
  \item The diagonals of a rhombus bisect the vertex angles through which they pass.
\end{itemize}

These diagonal properties of a rhombus are far clearer when it is drawn in a diamond orientation, with its diagonals vertical and horizontal as illustrated below.
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-26}

\section*{Tests for a rhombus}
\begin{itemize}
  \item If a quadrilateral is a parallelogram with two adjacent sides equal, then the parallelogram is a rhombus.
  \item If the diagonals of a quadrilateral bisect each other at right angles, then the quadrilateral is a rhombus.
\end{itemize}

These tests are proved in Question 1 of Exercise 7E.

Note: A rhombus is often defined to be a parallelogram with two adjacent sides equal.

\section*{Rectangles}
\section*{Definition of a rectangle}
\begin{itemize}
  \item A rectangle is a parallelogram whose interior angles are right angles.
\end{itemize}

Diagonal properties of a rectangle

\begin{itemize}
  \item The diagonals of a rectangle are equal and bisect each other.
\end{itemize}

\section*{Tests for a rectangle}
\begin{itemize}
  \item A parallelogram with one right angle is a rectangle.
  \item If all angles of a quadrilateral are equal, then the quadrilateral is a rectangle.
  \item If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a rectangle.
\end{itemize}

The three tests for a rectangle mentioned above are proved in Questions \(\mathbf{2 a}, \mathbf{2 b}\), and \(\mathbf{1} \mathbf{c}\), respectively, on the next page as part of Exercise 7E.

Note: A rectangle is often defined to be a parallelogram with one right angle.

\section*{Squares}
\section*{Definition of a square}
\begin{itemize}
  \item A square is a quadrilateral that is both a rectangle and a rhombus. Thus a square has all the properties of a rectangle and a rhombus.
\end{itemize}

\section*{Properties of a square}
\begin{itemize}
  \item All angles of a square are right angles.
  \item All sides of a square are equal.
  \item The diagonals of a square are equal and bisect each other at right angles.
  \item Each diagonal meets each side at \(45^{\circ}\).
\end{itemize}

A square is often defined as a regular polygon with four vertices; that is, as a quadrilateral with four equal sides and four equal interior angles. Note that such a figure is a parallelogram, because its opposite sides are equal, so the two definitions agree.

\section*{Exercise 7E}
1 a Prove that: If a parallelogram has two adjacent sides equal, then the parallelogram is a rhombus.

b Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then the quadrilateral is a rhombus.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-28(1)}
\end{center}

The diagonals of the quadrilateral \(A S B T\) meet at right angles at \(M\) and bisect each other.

i Why is \(A S B T\) a parallelogram?

ii Prove that \(\triangle A M S \equiv \triangle A M T\).

iii Hence prove that \(A S=A T\); that is, \(A S B T\) is a rhombus.

c Prove that: If the diagonals of a quadrilateral have equal length and bisect each other, then the quadrilateral is a rectangle.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-28}
\end{center}

The diagonals of the quadrilateral \(R S T U\) have equal length and bisect each other.

i Why is RSTU a parallelogram?

ii Prove that \(\triangle U R S \equiv \triangle T S R\).

iii Hence prove that \(\angle U R S=\angle T S R\).

iv Hence prove that \(\angle U R S=90^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-28(3)}
\end{center}

2 a Prove that: A parallelogram with one right angle is a rectangle.

b Prove that: If all angles of a quadrilateral are equal, then the quadrilateral is a rectangle.

3 Prove that: If one diagonal of a parallelogram bisects one vertex angle, then the parallelogram is a rhombus.

Let \(A F B G\) be a parallelogram in which the diagonal \(A B\) bisects the vertex angle \(\angle G A F\).

Let \(\angle F A B=\angle G A B=\alpha\).

a Prove that \(\angle A B F=\alpha\).

b Hence prove that \(A F B G\) is a rhombus.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-28(5)}
\end{center}

4 a In the diagram to the right, \(O\) is the centre of the circle, and \(A B\) and \(F G\) are diameters of the circle. What sort of figure is \(A F B G\), and why?

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-28(4)}
\end{center}

b In the diagram to the right, \(O\) is the centre of the circle, and \(A B\) and \(F G\) are perpendicular diameters of the circle. What sort of figure is \(A F B G\), and why?

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-28(2)}
\end{center}

5 Prove that: The circle with centre the midpoint of the hypotenuse of a right-angled triangle, and passing through the endpoints of the hypotenuse, also passes through the third vertex. Let \(\triangle A B F\) be right-angled at \(F\), and let \(M\) be the midpoint of the hypotenuse \(A B\). Produce \(F M\) to \(G\) so that \(F M=M G\).

a Explain why \(A F B G\) is a rectangle.

b Explain why \(A M=B M=F M=G M\).

c Hence explain why the circle with centre \(M\) and radius \(A M\)

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-29(3)}
passes through \(F\).

6 This question justifies some of the constructions that you learnt in earlier years.

a The diagram shows the standard construction of the perpendicular bisector of a given interval \(A B\). We draw two arcs with centres \(A\) and \(B\) respectively, equal radius, intersecting at \(P\) and \(Q\).

i Explain why \(A P B Q\) is a rhombus.

ii Hence explain why \(P Q\) is the perpendicular bisector of \(A B\).

b The diagram shows the standard construction of the bisector of a given angle \(\angle A O B\). We draw three arcs with centres \(O, P\) and \(Q\) respectively, and equal radius.

i Explain why \(O P M Q\) is a rhombus.

ii Hence explain why \(O M\) is the bisector of \(\angle A O B\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-29(2)}
\end{center}

c The diagram shows the standard construction of the perpendicular from a given point \(P\) to a given line \(\ell\). First draw an arc with centre \(P\) cutting \(\ell\) at \(A\) and \(B\). Then with centres \(A\) and \(B\) and the same radius, draw arcs meeting at \(M\).

i Explain why \(A P B M\) is a rhombus.

ii Hence explain why \(P M\) is perpendicular to \(\ell\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-29(1)}
\end{center}

7 The points \(P\) and \(Q\) are chosen on the diagonal \(B D\) of the square \(A B C D\) so that \(B P=D Q\).

a Prove that the triangles \(A B P, C B P, A D Q\) and \(C D Q\) are all congruent.

b Hence prove that \(A P C Q\) is a rhombus.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-29(4)}
\end{center}

A kite is a quadrilateral with two pairs of equal adjacent sides. As with a rhombus, the properties of a kite are best seen when it is drawn in a diamond orientation, as in the diagram.

a Let \(C F B G\) be a kite with \(C F=C G\) and \(B F=B G\).

Let the diagonals \(C B\) and \(F G\) meet at \(M\).

i Show that \(\triangle C B F \equiv \triangle C B G\).

ii Hence show that \(C B\) bisects \(\angle F C G\) and \(\angle F B G\).
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-29}
iii Show that \(\triangle C M F \equiv \triangle C M G\).

iv Hence show that the diagonals are perpendicular.

b Conversely, let \(A F B G\) be a quadrilateral in which the diagonal \(A B\) bisects the diagonal \(F G\) and is perpendicular to it. Prove that \(A F B G\) is a kite.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-30(1)}
\end{center}

c Let \(\ell\) be a line and \(P\) a point not on the line. Choose two points \(A\) and \(B\) on \(\ell\). Draw an arc with centre \(A\) and radius \(A P\). Then draw a second arc with centre \(B\) and radius \(B P\). Let the arcs meet at \(M\). Explain why \(P M\) is perpendicular to \(\ell\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-30(3)}
\end{center}

\(9 A B C D\) is a straight line such that \(A B=B C=C D\) and \(B C P Q\) is a rhombus. The lines produced through \(A Q\) and \(D P\) meet at \(R\). Prove that \(\angle A R D\) is a right angle.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-30}
\end{center}

\(10 P Q R S\) is a parallelogram. \(X\) is a point on \(P Q\) such that \(P X=S P\). Prove that \(S X\) bisects \(\angle R S P\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-30(2)}
\end{center}

11 A trapezium is a quadrilateral with one pair of parallel lines.

a The trapezium \(A B C D\) has \(\angle C=70^{\circ}\) and \(\angle D=50^{\circ}\). Find the sizes of \(\angle A\) and \(\angle B\).

If the non-parallel sides of a trapezium are equal, then the figure is called an isosceles trapezium.

b Prove that the base angles of an isosceles trapezium are equal.

c Prove that the diagonals of an isosceles trapezium are equal in length.

d Prove that if a trapezium has equal diagonals then it is isosceles.

\section*{Review exercise}
1 Classify these angles according to their size.
a \(78^{\circ}\)
b \(180^{\circ}\)
c \(90^{\circ}\)
d \(340^{\circ}\)
e \(360^{\circ}\)
f \(162^{\circ}\)

2 Write down the complements of:
a \(56^{\circ}\)
b \(67^{\circ}\)
c \(47^{\circ}\)
d \(15^{\circ}\)
e \(26^{\circ}\)
f \(46^{\circ}\)

3 Write down the supplements of:
a \(154^{\circ}\)
b \(89^{\circ}\)
c \(34^{\circ}\)
d \(90^{\circ}\)
e \(113^{\circ}\)
f \(116^{\circ}\)

4 Find the angles marked with pronumerals. Give reasons for your answers.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31(4)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31(6)}
\end{center}

b \(R\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31(7)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-31(3)}
\end{center}

5 In the diagram to the right, \(M T=O T\) and \(\angle M N T=\angle O U T\).

Prove that \(N T=U T\).

6 In the diagram to the right, \(\triangle A B C\) is isosceles with \(A B=A C, B D \perp A C\) and \(C E \perp A B\).

a Prove that \(\triangle B D C\) is congruent to \(\triangle B E C\).

b Prove that \(\triangle A E D\) is isosceles.
\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-31(2)}
\(7 \quad A B C D E F\) is a hexagon in which the interval \(A B\) is parallel and equal in length to the interval \(E D\), and the interval \(B C\) is parallel and equal in length to the interval \(F E\).

a Join \(B\) to \(E\) and prove that \(\angle A B C=\angle F E D\).

b Prove that \(A C D F\) is a parallelogram.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-32(1)}
\end{center}

8 In a parallelogram \(A B C D\), the point \(E\) lies on the side \(D C\) such that \(E A\) bisects \(\angle B A D\) and \(E B\) bisects \(\angle A B C\). Prove that:

a \(D E=E C\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-32(2)}
\end{center}

b \(D C=2 C B\)

c \(\angle A E B\) is a right angle

9 In the parallelogram \(A B C D\), the points \(X\) and \(Y\) are on the diagonal \(D B\) such that \(D X=D A\) and \(B Y=B C\). Prove that:

a \(A X=C Y\)

b \(A Y=C X\)

c \(A Y C X\) is a parallelogram

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-32}
\end{center}

\section*{Challenge exercise}
1 a Prove that: If the diagonals of a quadrilateral are perpendicular, then the sums of the squares of opposite sides are equal.

The diagonals of the convex quadrilateral \(A B C D\), to the right, meet at right angles at \(M\).

Prove that \(A B^{2}+C D^{2}=A D^{2}+B C^{2}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-32(4)}
\end{center}

b Conversely, prove that: If the sums of the squares of opposite sides of a quadrilateral are equal, then the diagonals are perpendicular.

In the convex quadrilateral \(A B C D\) to the right, \(A B^{2}+C D^{2}=A D^{2}+B C^{2}\). The perpendiculars to the diagonal \(B D\) from \(A\) and \(C\) meet \(B D\) at \(P\) and \(Q\) respectively. Prove that the points \(P\) and \(Q\) coincide.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-32(3)}
\end{center}

c What happens if the quadrilateral is non-convex?

2 The circle passing through all three vertices of a triangle is called a circumcircle of the triangle.

The following theorem not only shows that every triangle has a unique circumcircle, but also shows how to construct the circumcircle.

Prove that: The three perpendicular bisectors of a triangle are concurrent. Their point of intersection (called the

\includegraphics[max width=\textwidth, center]{2023_12_09_aeea47671fc5a3105b5bg-33(3)}
circumcentre) is the centre of a circle (called a circumcircle).

Prove this theorem using congruence. In the diagram above, the midpoints of \(B C, C A\) and \(A B\) are \(P, Q\) and \(R\) respectively.

3 In Exercise 7B, you proved that the sum of the exterior angles of a convex polygon is \(360^{\circ}\). This question introduces another way of looking at this theorem.

a In each diagram below, imagine that you started at the point \(X\), facing \(A\), and walked around the figure back to \(X\), passing through the vertices \(A, B, C, \ldots\) in alphabetical order. Through how many complete revolutions would you turn during your journey? Count clockwise rotation as negative, and anticlockwise revolutions as positive.

i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-33(1)}
\end{center}

iii

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-33(4)}
\end{center}

ii

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-33}
\end{center}

iv

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_aeea47671fc5a3105b5bg-33(2)}
\end{center}

b Use these ideas to develop a proof that the sum of the exterior angles of a convex polygon is \(360^{\circ}\).

4 Suppose that you make a journey on the surface of the globe. You start facing east at Singapore, which lies on the Equator. Then you walk east one-quarter of the way around the globe, then turn \(90^{\circ}\) north and walk to the North Pole, then turn \(90^{\circ}\) towards Singapore and walk back to Singapore, and finally turn facing east just as you started. Through how many revolutions have you turned during this journey, and what is the sum of the exterior angles of your triangular path?

5 The diagonals of a convex quadrilateral dissect the quadrilateral into four triangles.

a Prove that: If the quadrilateral is a trapezium, then a pair of opposite triangles have equal area.

b Conversely, prove that: If a pair of opposite triangles have equal area, then the quadrilateral is a trapezium.

6 a Let \(A B C D\) be a square. Let \(Q\) be a point on and \(R\) a point on \(B C\), and let \(A Q \perp D R\). Prove that \(A Q=D R\).

b Let \(A B C D\) be a square. Let \(\ell\) and \(m\) be perpendicular lines such that \(\ell\) intersects the sides \(A B\) and \(C D\) of the square, and \(m\) intersects the sides \(A D\) and \(B C\). Show that the intervals cut off on \(\ell\) and \(m\) by the square are equal in length.

7 Let \(A B C D\) be a square of side length 1 . Extend the sides \(A B\) to \(E, B C\) to \(F, C D\) to \(G\) and \(D A\) to \(H\) so that \(B E=C F=D G=A H=1\).

a Prove that \(E F G H\) is a square, and find its area.

b Show that the centre \(O\) of \(A B C D\) is also the centre of \(E F G H\).

8 Prove that: Any line through the intersection of the diagonals of a parallelogram dissects the parallelogram into two figures of equal area.

9 Prove that: The perpendicular bisector of the base of an isosceles triangle passes through its vertex.

\(10 \quad H\) and \(K\) are the midpoint of the sides \(A B\) and \(A C\) of \(\triangle A B C\). Points \(H\) and \(K\) are joined and the line produced to \(X\) so that \(H K=K X\). Prove that:

a \(C X\) is equal and parallel to \(B H\).

b \(H K=\frac{1}{2} B C\) and \(H K\) is parallel to \(B C\).


\end{document}