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\title{Review of congruence }

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\maketitle
\section*{and similarity}
This chapter reviews our knowledge of geometry. In particular, we review congruence tests and similarity tests for triangles.

Congruence and similarity are extremely useful tools in geometrical arguments. Both congruence and similarity have many applications and you will meet some of these in this chapter.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-01}
\end{center}

\section*{Review of triangles}
An initial discussion of the properties of triangles appeared in ICE-EM Mathematics Year 8 and ICE-EM Mathematics Year 9. We briefly review them here.

\section*{Triangles}
We recall:

\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*{Isosceles and equilateral 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}

\section*{Polygons}
\begin{itemize}
  \item The angle sum of a quadrilateral is \(360^{\circ}\).
  \item The sum of the interior angles of a convex polygon is \((n-2) 180^{\circ}\).
  \item The sum of the exterior angles of a convex polygon is \(360^{\circ}\).
\end{itemize}

\section*{Exercise 8A}
1 Find the values of \(x, y, \alpha, \beta\) and \(\gamma\).

a \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-02}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-02(1)}
\end{center}

b
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-02(3)}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-02(2)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-02(4)}
\end{center}

2 Find the values of \(x, y, \alpha, \beta, \gamma\) and \(\theta\). Give reasons in your solutions. Points marked \(O\) are the centres of circles.

a \(R\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(2)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(7)}
\end{center}

j

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03}
\end{center}

l

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(6)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(10)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(1)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(12)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(3)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(8)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(9)}
\end{center}

\(\mathbf{k}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(11)}
\end{center}

m

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-03(4)}
\end{center}

3 The exterior angles of a regular polygon are each \(60^{\circ}\). How many sides does the polygon have?

4 Three angles of a pentagon are each \(156^{\circ}\) and the remaining angles are equal. Find the size of the two remaining angles.

\section*{Congruence}
In ICE-EM Mathematics Year 8 and Year 9 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 up every part of one figure with the corresponding part of the other, so that:
  \item matching angles have the same size
  \item matching intervals have the same length
  \item matching regions have the same area.
\end{itemize}

The congruence arguments used in this chapter involve only congruent triangles. In ICE-EM Mathematics Year 8 and Year 9 we developed four tests for two triangles to be congruent, as follows.

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

SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or

AAS: two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle, or

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

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.

The statement 'Triangle \(A B C\) is congruent to triangle \(P Q R\) ' is written as:

\[
\triangle A B C \equiv \triangle P Q R
\]

where the vertices are written in matching order.

When a congruence test is used to justify the congruence, the test's initials are placed in brackets after the congruence statement, as in Example 1 on the next page.

\section*{Example 1}
If the two triangles are congruent, write down a congruence statement and the congruence test used to justify it.

If they are not, explain why not.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-05(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-05(2)}
\end{center}

b
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-05(3)}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-05}
\end{center}

\section*{Solution}
a \(\triangle A B C \equiv \triangle D F E(\mathrm{SAS})\)

b In \(\triangle P Q R, \angle R Q P=180^{\circ}-(42+110)^{\circ}\) \(=28^{\circ}\)

So \(\triangle L M N \equiv \triangle Q P R(\mathrm{ASA})\)

c In \(\triangle A B C, \angle A B C=80^{\circ}\). In \(\triangle X Y Z, \angle Z X Y=30^{\circ} . X Z \neq 15 \mathrm{~cm}\), since \(\triangle X Y Z\) is not isosceles. Hence, \(\triangle A B C\) is not congruent to \(\triangle X Y Z\), because \(A C \neq X Z\).

d \(\triangle P Q R \equiv \triangle U S T(\) RHS)

\section*{Quadrilaterals}
The sum of the interior angles of a quadrilateral is \(360^{\circ}\).

Congruence of triangles is used to establish properties of special quadrilaterals. A proof for each of the properties listed overpage was given in ICE-EM Mathematics Year 9.

\section*{Parallelograms}
A parallelogram is a quadrilateral whose opposite sides are parallel.

A parallelogram has the following properties:

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

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-06}
\end{center}

Here are four well known 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}

\section*{Rhombuses}
A rhombus is a quadrilateral with four equal sides. A rhombus is a parallelogram. (We also note that because the opposite sides of a parallelogram are equal, it is always sufficient to establish that just two adjacent sides are equal.)

The following are properties of a rhombus:

\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}

Here are two tests for whether a quadrilateral is a rhombus.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-06(2)}
\end{center}

\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}

\section*{Rectangles}
A rectangle is a quadrilateral in which all angles are right angles.

The following are properties of a rectangle:

\begin{itemize}
  \item A rectangle is a parallelogram.
  \item Its opposite sides are equal and parallel.
  \item Its diagonals bisect each other.
  \item The diagonals of a rectangle are equal.
\end{itemize}

Here are three tests for a rectangle:

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-06(1)}
\end{center}

\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}

\section*{Kites}
A kite is a quadrilateral with two pairs of adjacent equal sides.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-07}
\end{center}

\section*{Example 2}
Congruence is used to prove many results with quadrilaterals and triangles.

In the kite \(A B C D\),

\(D A=D C\) and \(B A=B C\).

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

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-07(4)}
\end{center}

\section*{Solution}
Join \(D\) to \(B\).

In the triangles \(A B D\) and \(C B D\),

\(B A=B C\) (given)

\(D A=D C\) (given)

\(D B\) is common,

so \(\triangle B D A \equiv \triangle B D C\) (SSS)

Hence, \(\angle B A D=\angle B C D\) (matching angles of congruent triangles)

\section*{Exercise 8B}
1 In each part, find a pair of congruent triangles. State the congruence test used.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-07(2)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-07(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-07(6)}
\end{center}

a
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-07(3)}
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-07(5)}
c

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08(2)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08(7)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08(4)}
\end{center}

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-08(1)}
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-08(9)}
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-08(6)}

2 In each part, it is known that \(\triangle A B C \equiv \triangle D E F\). Determine the unknown angles and side lengths. (Side lengths are given correct to one decimal place.)

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08(5)}
\end{center}

\(\mathbf{c}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08(3)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-08(8)}
\end{center}

d
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-08(10)}

\(4 P R S V\) is a square. The midpoint of \(P V\) is \(X\) and \(T\) is the midpoint of \(S V\).

a Draw a diagram and prove that \(R X=R T\).

b Join \(R V\) and prove that \(\angle T R V=\angle X R V\).

3 In the diagram at the right, \(A B C D\) is a square and \(D E=E C\).

a Draw a diagram and prove that \(\triangle A D E \equiv \triangle B C E\).

b Prove that \(A E=B E\).

5 The diagonals of a rectangle \(A B C D\) meet at \(O\) and \(\angle B O C=56^{\circ}\).

a Give reasons why \(O B=O C\).

b Use this to find:
i \(\angle A O D\)
ii \(\angle A O B\)
iii \(\angle O B C\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-09}
\end{center}

6 The diagonals of the parallelogram \(A B C D\) intersect at \(O\). A line through \(O\) meets the sides \(A B\) and \(C D\) at \(X\) and \(Y\), respectively. Prove that \(O X=O Y\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-09(3)}
\end{center}

7 In a parallelogram \(A B C D, P\) is the midpoint of \(B C\).

Both \(D P\) and \(A B\) are produced to meet at \(Q\).

Prove that \(A Q=2 A B\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-09(1)}
\end{center}

8 Two parallelograms, \(A B C D\) and \(A B X Y\), are on the same base, \(A B\). Prove that \(D C X Y\) is a parallelogram.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-09(2)}
\end{center}

9 The diagonals of a square \(A B C D\) meet at \(O\). The point \(K\) lies on \(A B\) such that \(A K=A O\).

Prove that \(\angle A O K=3 \angle B O K\).

10 Recall that a kite is a quadrilateral with two pairs of adjacent equal sides. Prove the following properties of a kite. You will need to draw a separate diagram for each point.

a If one diagonal of a quadrilateral bisects the two vertex angles through which it passes, then the quadrilateral is a kite.

b If one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite.

11 Draw a diagram and prove that, in a parallelogram, opposite sides are equal and opposite angles are equal.

12 Draw a diagram and prove that the diagonals of a parallelogram bisect each other.

13 Draw a diagram and prove that the diagonals of a rhombus are perpendicular.

14 Draw a diagram and prove that diagonals of a rectangle are equal.

\(15 A B C D\) is a rhombus.

The bisector of \(\angle A B D\) meets \(A D\) at \(K\).

Prove that \(\angle A K B=3 \angle A B K\).

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-09(4)}
\(16 A B C D\) is a rectangle. Equilateral triangles \(A B X\) and \(D A Y\) are drawn outside \(A B C D\). Draw a diagram and prove that triangle \(C X Y\) is equilateral.

17 Draw a diagram and prove that if each angle of a quadrilateral is equal to the opposite angle then the quadrilateral is a parallelogram.

18 Draw a diagram and prove that if each side of a quadrilateral is equal to the opposite side then the quadrilateral is a parallelogram.

\section*{8}
 Enlargements and similarity\section*{Enlargements}
\begin{itemize}
  \item An enlargement stretches a figure by the same factor in all directions.
  \item An enlargement transformation is specified by its centre of enlargement and its enlargement factor.
  \item When a figure is enlarged:
  \item matching lengths are in the same ratio and
  \item matching angles are equal.
\end{itemize}

The image is thus a scale drawing of the original figure.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-10(1)}
\end{center}

In the diagram at the right, \(O\) is the centre of enlargement. \(\triangle A^{\prime} B^{\prime} C^{\prime}\) is an enlargement by factor 2 of \(\triangle A B C\). Since the enlargement factor is \(2, \frac{A^{\prime} B^{\prime}}{A B}=\frac{B^{\prime} C^{\prime}}{B C}=\frac{C^{\prime} A^{\prime}}{C A}=2\).

\section*{Similarity}
Two figures are called similar if we can enlarge one figure so that its enlargement is congruent to the other figure. In simple terms, this means that by enlarging or shrinking one of two items, we get the other item, perhaps translated, rotated or reflected.

Thus, similar figures have the same shape, but not necessarily the same size, just as a scale drawing has the same shape as the original, but has a different size.
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-10}

\section*{Similar figures}
\begin{itemize}
  \item Two figures are called similar if there is an enlargement of one figure that is congruent to the other figure.
  \item Matching lengths in similar figures are in the same ratio, called the similarity ratio.
  \item Matching angles in similar figures are equal.
\end{itemize}

\section*{Similarity tests for triangles}
As with congruence, most problems involving similarity come down to establishing that two triangles are similar. In this section we review the four similarity tests for triangles. For each congruence test there is a corresponding similarity test.

\section*{The AAA similarity test}
\begin{itemize}
  \item If the angles of one triangle are respectively equal to the angles of another triangle, then the two triangles are similar.
\end{itemize}

Note: When using this test, it is sufficient to prove that just two pairs of angles are equal the third pair must then also be equal since the angle sum of any triangle is \(180^{\circ}\). Thus, the test is often called 'the AA similarity test'.

This similarity test corresponds to the AAS congruence test.

\section*{Example 3}
For each diagram, write a similarity statement beginning with ' \(\triangle A B C\) is similar to ...' and state the test you used.

Be careful to name the vertices in matching order.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-11(1)}
\end{center}

b \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-11}
\end{center}

\section*{Solution}
a \(\triangle A B C\) is similar to \(\triangle C P B\) (AAA)

b \(\triangle A B C\) is similar to \(\triangle B M C\) (AAA)

Note: A similarity statement should not only appeal to the test used, but also list the vertices of the triangles in matching order.

\section*{The SSS similarity test}
If we can match up the sides of one triangle with the sides of the other so that the ratio of matching lengths is constant, then the triangles are similar.
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-12}

The statement that the two triangles shown in the box above are similar is thus written as:

\section*{\(\triangle A B C\) is similar to \(\triangle P Q R\) (SSS)}
The similarity factor is \(\frac{3}{2}\).

The SSS similarity test corresponds to the SSS congruence test.

\section*{Ratios between triangles and ratios within triangles}
When two triangles are similar, we can read off the ratios of the matching lengths between the triangles in the box above. That is:

\[
\frac{P Q}{A B}=\frac{R Q}{C B}=\frac{P R}{A C}=\frac{3}{2}
\]

Alternatively, we can read off the ratios within the triangles. Thus, for the triangles above:

\[
\frac{P Q}{P R}=\frac{A B}{A C}=\frac{6}{5}
\]

and \(\frac{P Q}{R Q}=\frac{A B}{C B}=\frac{3}{2}\)

and \(\frac{R Q}{P R}=\frac{C B}{A C}=\frac{4}{5}\)

Note that \(\frac{a}{b}=\frac{x}{y}\) is equivalent to \(\frac{a}{x}=\frac{b}{y}\) because both statements are equivalent to \(a y=b x\).

That is, equal ratios between the triangles is equivalent to equal ratios within triangles.

It does not matter whether you use ratios between triangles or ratios within triangles.

\section*{Example 4}
a Prove that the two triangles in the diagram are similar.

b Which of the two marked angles are equal?

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

\[
\begin{aligned}
& \frac{A B}{C B}=2 \text { (given) } \\
& \frac{B C}{B D}=2 \text { (given) } \\
& \frac{C A}{D C}=2 \text { (given) }
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-13(1)}
\end{center}

so \(\triangle A B C\) is similar to \(\triangle C B D\) (SSS).

b Hence, \(\gamma=\theta\). (matching angles of similar triangles)

\section*{The SAS similarity test}
There are two ways of stating the test:

\begin{itemize}
  \item If the ratios of two pairs of matching sides are equal and the included angles are equal, then the two triangles are similar.
\end{itemize}

OR

\begin{itemize}
  \item If the ratio of the lengths of two sides of one triangle is equal to the ratio of the lengths of another triangle
\end{itemize}

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-13}
and the included angles are equal, then the two triangles are similar.

The statement that the two triangles in the box above are similar is thus written as:

\(\triangle A B C\) is similar to \(\triangle P Q R(\mathrm{SAS})\)

Consider \(\triangle P Q R\) and \(\triangle A B C\), as shown in the box above. The ratios of matching lengths are:

\[
\frac{A C}{P R}=\frac{A B}{P Q}=2
\]

The ratios within the triangles are:

\[
\frac{P R}{P Q}=\frac{A C}{A B}=\frac{5}{2}
\]

\section*{The RHS similarity test}
There are two ways of stating the test:

\begin{itemize}
  \item If the ratio of the hypotenuses and the ratio of one pair of sides of a right-angled triangle are equal, then the triangles are similar.
\end{itemize}

OR

\begin{itemize}
  \item If the ratio of the hypotenuse and one side of one right-angled triangle is equal to the ratio of the hypotenuse and one side of another right-angled triangle, then the two triangles are similar.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-14(1)}
\end{center}

The statement that the two triangles in the box above are similar is thus written as:

\(\triangle A B C\) is similar to \(\triangle P Q R\) (RHS)

Consider \(\triangle A B C\) and \(\triangle P Q R\) as shown in the box on the previous page. The ratios of matching lengths are:

\[
\frac{P R}{A C}=\frac{Q P}{A B}=\frac{3}{2}
\]

The ratios within the triangles are:

\[
\frac{B A}{A C}=\frac{Q P}{P R}=2
\]

\section*{Example 5}
Determine whether the triangles shown are similar and, if they are, state the appropriate similarity test.
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-14}

\section*{Solution}
In \(\triangle A B C\) and \(\triangle Y Z X\),

\(\angle A B C=\angle Y Z X=80^{\circ}\).

\(\frac{A B}{Y Z}=\frac{14}{10}=\frac{7}{5}\) and

\(\frac{B C}{Z X}=\frac{11.2}{8}=\frac{7}{5}\)

\(\triangle A B C\) is similar to \(\triangle Y Z X\) (SAS).
Alternatively,

\(\frac{B A}{B C}=\frac{14}{11.2}=1.25\)

\(\frac{Z Y}{Z X}=\frac{10}{8}=1.25\) and

\(\angle A B C=\angle Y Z X=80^{\circ}\)

so \(\triangle A B C\) is similar to \(\triangle Y Z X\) (SAS).

Similar figures can be used to calculate magnitudes of angles and lengths in practical situations, as in the following examples.

\section*{Example 6}
Some students estimate the height of an electricity pylon using the following method. One student holds a 3-metre pole vertical while another student sights from ground level. The pole is moved until the top of the pole lines up with the top of the pylon, as shown in the diagram.

The distances \(x\) metres and \(y\) metres are measured and it is found that \(x=4.2\) and \(y=75.6\).

Using similar triangles, calculate the approximate height of the pylon.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-15(1)}
\end{center}

\section*{Solution}
The two triangles are similar (AAA).

Thus, \(\frac{h}{3}=\frac{x+y}{x}\)

\[
\begin{aligned}
& =\frac{79.8}{4.2} \\
h & =3 \times \frac{79.8}{4.2}=57
\end{aligned}
\]

Hence, the height of the pylon is 57 metres.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-15}
\end{center}

\section*{Example 7}
In the figure shown,

\(\angle P T Q=\angle P R S=\alpha\).

Prove that \(P Q \times P R=P T \times P S\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-15(2)}
\end{center}

\section*{Solution}
In \(\triangle P T Q\) and \(\triangle P S R\),

\(\angle P T Q=\angle P R S\) (given)

\(\angle T P Q=\angle S P R\) (same angle)

so \(\triangle P T Q\) is similar to \(\triangle P R S\) (AAA).

Hence, \(\frac{P Q}{P S}=\frac{P T}{P R}\) (matching sides of similar triangles)

\(P Q \times P R=P T \times P S\)

\section*{Exercise \(8 \mathrm{C}\)}
1 Determine whether the triangles in each pair are similar. If they are similar, state the appropriate similarity test.

a
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-16(9)}

c \(S\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(2)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(5)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(8)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(7)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(3)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(4)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(6)}
\end{center}

j (Compare all three triangles)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-16(1)}
\end{center}

2 a State why these two triangles are similar.

b Calculate \(x\).
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-17}

3 a State why these two triangles are similar.

b Calculate \(y\).
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-17(1)}

4 a State why these two triangles are similar.

b Find \(\alpha\) and find \(a\), then \(b\) in terms of \(\beta\).

5 a In the diagram shown, name two triangles that are similar and state why they are similar.

b Write down the three equal ratios.

c Calculate the value of \(x\).

6 a List the three pairs of equal angles in the figure.

b Find the length of \(A B\).

c Find the length of \(D C\).
\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-17(2)}

7 a In the diagram shown, are the two triangles similar? If so, why?

b If \(A D=6 \mathrm{~cm}, D B=4 \mathrm{~cm}\) and \(A E=7 \mathrm{~cm}\), calculate \(A C\).

8 a State why \(\triangle L M P\) is similar to \(\triangle P M N\).

b If \(P M=10 \mathrm{~cm}\) and \(M N=6 \mathrm{~cm}\), calculate \(L M\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-18}
\end{center}

9 At a certain time of day, a flagpole casts a shadow \(15 \mathrm{~m}\) long, and at the same time a stick \(30 \mathrm{~cm}\) high casts a shadow \(24 \mathrm{~cm}\) long. Assuming that both the stick and the flagpole are perpendicular to the horizontal ground, find the height of the flagpole.

10 This diagram represents a river (shaded) with a tree on the bank at point \(A\). A man stands directly opposite \(A\), on the opposite bank, at point \(B\). He then walks \(100 \mathrm{~m}\) along the bank, to point \(C\), where he places a peg. He then walks a further \(50 \mathrm{~m}\) to point \(D\), turns \(90^{\circ}\) and walks \(65 \mathrm{~m}\) to point \(E\), where he finds that \(E, C\) and \(A\) are in a straight line. Find the width of the river.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-18(5)}
\end{center}

11 A line from the top of a church steeple to the ground just passes over the top of a pole \(3 \mathrm{~m}\) high, and meets the ground at a point \(A, 2 \mathrm{~m}\) from the base of the pole. If the distance of \(A\) from a point directly below the church steeple is \(22 \mathrm{~m}\), find the height of the steeple.

12 In the figure opposite, the line \(O C\) is perpendicular to line \(O D\) and \(O A=2 O C . B\) is a point on \(O D\) and \(A B \| C D\).

Prove that \(\triangle A O B\) is similar to \(\triangle C O D\) and hence prove that \(O B=2 O D\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-18(3)}
\end{center}

13 In the figure shown, \(A B C\) is a triangle, right-angled at \(B\), and \(B D \perp A C\). Prove that:

a \(\triangle A B D\) is similar to \(\triangle A C B\)

b \(\triangle B C D\) is similar to \(\triangle A C B\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-18(2)}
\end{center}

\(14 A B C D\) is a parallelogram and \(E\) is the midpoint of \(A D\). The intervals \(B E\) and \(A C\) intersect at \(P\). Prove that:

a \(\triangle A P E\) is similar to \(\triangle C P B\)

b \(A C=3 A P\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-18(4)}
\end{center}

15 In \(\triangle A B C, D\) lies on \(A B\) and \(E\) lies on the interval \(A C\) such that \(\angle E D B\) and \(\angle A C B\) are supplementary angles.

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

b Prove that \(\frac{A E}{A B}=\frac{A D}{A C}\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-18(1)}
\end{center}

16 In the figure shown, \(\triangle A B C\) is isosceles with \(A B=A C\). The point \(F\) lies on \(B C\) such that \(A F \perp B C\). The point \(P\) lies on \(B C\) and the point \(D\) lies on \(A B\) such that \(D P \perp A B\).

a Prove that \(\triangle P B D\) is similar to \(\triangle A C F\).

b Prove that \(\frac{F C}{D B}=\frac{A C}{P B}\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-19(1)}
\end{center}

17 In the diagram shown, \(A B=A C\) and \(L, M\) and \(N\) are midpoints of \(A B, B C\) and \(C A\), respectively. Prove that \(L M=N M\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-19(2)}
\end{center}

18 Complete the following proof of Pythagoras' theorem.

a Show that \(\triangle A B C\) is similar to \(\triangle A D B\).

b Show that \(a^{2}-a y=c^{2}\).

c Show that \(\triangle A B C\) is similar to \(\triangle B D C\).

d Show that \(a y=b^{2}\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-19(3)}
\end{center}

e Deduce that \(a^{2}=b^{2}+c^{2}\).

\section*{8 
 Similarity and intervals within a triangle}
Similarity is a useful tool to analyse an interval joining points on two sides of a triangle, as in the example below.

\section*{Example 8}
In triangle \(A B C\),

\(A P=14, P B=6, A Q=7\) and \(Q C=3\).

a Prove that \(\triangle A B C\) is similar to \(\triangle A P Q\).

b Prove that \(P Q\) is parallel to \(B C\).

c Find the ratio \(P Q: B C\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-19}
\end{center}

\section*{Solution}
a In triangles \(A P Q\) and \(A B C\),

\[
\begin{aligned}
& \frac{A B}{A P}=\frac{A C}{A Q}=\frac{10}{7} \\
& \angle B A C=\angle P A Q
\end{aligned}
\]

\(\triangle A B C\) is similar to \(\triangle A P Q\) (SAS).

b Since the triangles are similar, \(\angle A B C=\angle A P Q\).

Therefore, corresponding angles are equal.

Thus, \(B C \| P Q\).

c \(P Q: B C=7: 10\) (matching sides of similar triangles)

\section*{Exercise 8D}
1 Prove that the interval joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-20}
\end{center}

2 Prove that the line through the midpoint of one side of a triangle parallel to another side meets the third side of the triangle at its midpoint.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-20(1)}
\end{center}

3 Prove that the intervals joining the midpoints of the sides of a triangle dissect the triangle into four congruent triangles, each similar to the original triangle.

4 Prove that the midpoints of the sides of a quadrilateral form the vertices of a parallelogram.

\begin{itemize}
  \item Point \(N\) is the midpoint of \(B C\)
  \item Point \(Q\) is the midpoint of \(C D\)
  \item Point \(P\) is the midpoint of \(A D\)
  \item Point \(M\) is the midpoint of \(A B\)
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-20(2)}
\end{center}

5 Draw a diagram of a triangle \(A B C\) with a point \(P\) on \(A B\) and a point \(Q\) on \(A C\) such that \(A P=12, P B=9, A Q=4\) and \(Q C=3\).

Prove that \(P Q \| B C\).

6 The point \(S\) is the intersection point of two lines. Points \(A\) and \(B\) are the points of intersection of the first line with two parallel lines, such that \(B\) is further away from \(S\) than \(A\). Similarly, points \(C\) and \(D\) are the intersections of the second line with the two parallel lines, such that \(D\) is further away from \(S\) than \(C\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-21(4)}
\end{center}

Prove that:
a \(S A: A B=S C: C D\)
b \(S B: A B=S D: C D\)
c \(S A: S B=S C: S D\)

7 The point \(S\) is the intersection point of two lines. Points \(A\) and \(B\) are the points of intersection of the first line with two other lines, such that \(B\) is further away from \(S\) than \(A\). Similarly, points \(C\) and \(D\) are the intersection points of the second line with the two other lines, such that \(D\) is further away from \(S\) than \(C\). Prove that if \(S A: A B=S C: C D\),

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-21}
then the two intercepting lines \(A C\) and \(B D\) are parallel.

\section*{Review exercise}
1 Find the value of \(\alpha, \beta\) and \(\gamma\) in the diagram at the right.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-21(3)}
\end{center}

2 a Name the similar triangles in the diagram at the right.

b Find \(x\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-21(1)}
\end{center}

3 In the parallelogram \(A B C D, E\) is a point on \(C D\), and \(B E\) and \(A D\) are produced to meet at \(F\).

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

b Given that \(C D=3 E D, A B=6\) and \(B C=8\) :

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-21(2)}
i find \(E D\)
ii find \(D F\)

4 Find the value of the Greek letters in each diagram.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(1)}
\end{center}

b \(E\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(9)}
\end{center}

\(\mathbf{c}\)

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(2)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(4)}
\end{center}

5 In the diagram at the right, \(\triangle A B C\) is isosceles with \(A B=A C, \angle A B C=55^{\circ}\), \(A D\|E G\| F I\) and \(D I \| A C\).

a Find \(\angle B A C\) and \(\angle A C B\).

b Find \(\angle F I C\) and prove that \(\triangle F I C\) is isosceles.

c Prove that \(D B=D I\).

d If \(G D=G I\) and \(\angle F G I=34^{\circ}\), find \(\angle G I F\) and \(\angle E F G\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(8)}
\end{center}

6 For each pair of triangles below, write a congruence statement, including the appropriate congruence test.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(10)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(3)}
\end{center}

\(7 A B\) and \(D C\) are parallel sides of a trapezium \(A B C D\). The diagonals of the trapezium intersect at \(O\). Prove that \(\frac{B O}{O D}=\frac{A O}{O C}\).

\(8 B E\) and \(C F\) are altitudes of a triangle \(A B C\). Prove that \(\frac{B E}{C F}=\frac{A B}{A C}\).

9 Find the value of \(x\) in each diagram.

a

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-22(6)}
\end{center}

b

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-22(7)}
\(10 A B C D\) is a rectangle with \(A D=12\) and \(D C=5\), and \(B E\) and \(D F\) are perpendicular to \(A C\).

a Find \(A C\).

b Find \(E F\).

11 In triangle \(A B C, A B=8 \mathrm{~cm}, B C=5 \mathrm{~cm}\) and

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-23(1)}
\(C A=6 \mathrm{~cm}\). The side \(A B\) is produced to \(D\) so that \(B D=16 \mathrm{~cm}\), and \(A C\) is extended to \(E\) so that \(C E=26 \mathrm{~cm}\). Find \(D E\).

\section*{Challenge exercise}
1 Attic space in a particular house has the shape of a triangular prism.

Triangle \(A F C\) is isosceles with \(A F=A C=4 \mathrm{~m}\) and \(F C=3 \mathrm{~m}\).

A box in the form of a rectangular prism is placed in the attic, touching both sides.

A cross section is shown in the diagram below, where the face \(E G B D\) of the box is shown.

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-23}
\end{center}

The box is \(204 \mathrm{~cm}\) wide; that is, \(E D=G B=204 \mathrm{~cm}\).

a Find the length of \(A B\).

b Find the length of \(A X\), where \(X\) is the midpoint of \(F C\).

c Find the height, \(E G\), of the box.

d A box in the shape of a cube is to be placed on top of this box. Find the length, correct to the nearest \(\mathrm{cm}\), of an edge of the largest cube that could fit.

2 The diagonals of a square \(A B C D\) are \(A C\) and \(B D\), which intersect at \(O\). The bisector of \(\angle B A C\) cuts \(B O\) at \(X\) and \(B C\) at \(Y\). Prove that \(C Y=2 O X\). (Hint: Let \(A Y\) meet \(C D\) at \(Z\) and consider \(\triangle A C Z\).)
\(3 A, B, C\) and \(D\) are points on a straight line so that \(A B=B C=C D\). Also, \(B P Q C\) is a parallelogram. If \(B P=2 B C\), prove that \(P D\) is perpendicular to \(A Q\).

4 In triangle \(A B C, A B=A C\) and \(\angle A B C=2 \angle B A C\). The segment \(B C\) is produced to \(D\) so that \(2 \angle C A D=\angle B A C\). The point \(F\) lies on \(A B\) so that \(C F\) is perpendicular to \(A B\). Prove that \(A D=2 C F\). (Hint: Find an extra isosceles triangle and draw an altitude.)

\(5 X\) and \(Y\) are the midpoints of the sides \(P S\) and \(S R\) of a parallelogram \(P Q R S\). Prove that the area of triangle \(S X Y\) is one eighth the area of the parallelogram.

6 a In \(\triangle A B C, A D\) bisects \(\angle B A C\).

Prove that \(\frac{B D}{D C}=\frac{B A}{A C}\).

Hint: Construct \(C E\) parallel to \(D A\) to

meet \(B A\) extended at \(E\).

\begin{center}
\includegraphics[max width=\textwidth]{2024_02_22_11b81a2bc166020dbdd6g-24}
\end{center}

b The bisectors of the angle \(A\) and the angle \(C\) of a quadrilateral \(A B C D\) meet at point \(E\) on the diagonal \(B D\).

Prove that \(\frac{A D}{A B}=\frac{C D}{C B}\).

c The bisectors of the angles \(A, B\) and \(C\) of \(\triangle A B C\) meet the opposite sides at \(D, E\) and \(F\).

Prove that \(\frac{B D}{D C} \times \frac{C E}{E A} \times \frac{A F}{F B}=1\).

7 Triangle \(A B C\) is right-angled at \(C\). This question leads you through another proof of Pythagoras' theorem using enlargements.

a Enlarge triangle \(A B C\) by a factor of \(b\) to form triangle \(A^{\prime} B^{\prime} C^{\prime}\) and mark the side lengths of each

\includegraphics[max width=\textwidth, center]{2024_02_22_11b81a2bc166020dbdd6g-24(1)}
side on a diagram of triangle \(A^{\prime} B^{\prime} C^{\prime}\).

b Enlarge triangle \(A B C\) by a factor of \(a\) to form triangle \(A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\), and show the side lengths of each side on a diagram of triangle \(A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\).

c Join triangle \(A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}\) and \(A^{\prime} B^{\prime} C^{\prime}\) along sides \(B^{\prime} C^{\prime}\) and \(C^{\prime \prime} A^{\prime \prime}\) with \(C^{\prime \prime}\) and \(C^{\prime}\) coinciding.

d Show that the new triangle formed is similar to triangle \(A B C\).

e What is the enlargement factor that transforms triangle \(A B C\) to this triangle?

f Deduce Pythagoras' theorem.

8 Prove that the lines joining the midpoints of opposite sides of any quadrilateral bisect each other.


\end{document}