Sec 3 · G3 (Express)

Congruence and similarity

Practise congruence and similarity with step-by-step help. Free account.

1. Congruent triangles

Symbol	Meaning	Example
$\equiv$	congruent: same shape and same size	$\triangle ABC \equiv \triangle QRP$
vertex order	matching vertices in the same order	$A \leftrightarrow Q,\ B \leftrightarrow R,\ C \leftrightarrow P$
matching sides	equal	$AB = QR$
matching angles	equal	$\angle A = \angle Q$

Test	Equal parts
SSS	three sides
SAS	two sides and the angle between them
ASA / AAS	two angles and a corresponding side (ASA: the side between them)
RHS	right angle, hypotenuse and one other side

📖 Worked example

In triangles $ABC$ and $PQR$ , $BC = QR$ and $CA = RP$ . State one more condition that would make triangle $ABC$ congruent to triangle $PQR$ by the SAS test. Choose the condition.

Step 1 of 2

What does the SAS test need?

SAS needs two pairs of equal sides and the angles between them.

📖 Worked example

Triangles $ABC$ and $PQR$ are congruent, with the equal parts marked. Complete the statement $\triangle ABC \equiv \triangle \ldots$

Step 1 of 2

Find the part of PQR equal to each marked part of ABC

AB

(

8

cm) matches

\text{QR}

CA

(

6

cm) matches

\text{PQ}

\angle A

(

42^\circ

) matches

\angle \text{Q}

📖 Worked example

In the diagram, triangle $ABC$ is congruent to triangle $RPQ$ . Find $x$ .

Step 1 of 2

Match the letters in order

\triangle ABC \equiv \triangle RPQ

, so

BC

matches

PQ

Practise this →

2. Similar triangles

Symbol	Meaning	Example
$\sim$	similar: same shape, sides in the same ratio	$\triangle ADE \sim \triangle ABC$
ratio	matching sides divide to the same number	$\dfrac{DE}{BC} = \dfrac{AD}{AB}$

Test	Shown by
AA	two pairs of equal angles
SSS	three pairs of sides in the same ratio
SAS	two pairs of sides in the same ratio, the angle between them equal

Left: triangle ABC with D on AB and E on AC, DE parallel to BC; AD = 5, DB = 4, BC = 10. Right: lines AD and BC cross at X with AB parallel to DC.

DE ∥ BC: use the whole side AB = 5 + 4

\displaystyle \dfrac{DE}{BC} = \dfrac{AD}{AB}

e.g.

\displaystyle \dfrac{DE}{10} = \dfrac{5}{9}

\displaystyle DE = \dfrac{50}{9}

\displaystyle DE = 5.56\text{ (3 s.f.)}

AB ∥ DC, crossing at X: AX = 5, XD = 8, AB = 7

\displaystyle \dfrac{DC}{AB} = \dfrac{XD}{XA}

e.g.

\displaystyle \dfrac{DC}{7} = \dfrac{8}{5}

\displaystyle DC = 11.2

📖 Worked example

In the diagram, $D$ lies on $AB$ and $E$ lies on $AC$ , with $DE$ parallel to $BC$ . Find $DE$ , correct to 3 significant figures.

Step 1 of 3

DE ∥ BC, so triangle ADE is similar to triangle ABC

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

📖 Worked example

In the diagram, the lines $AD$ and $BC$ meet at $X$ , and $AB$ is parallel to $DC$ . Find $DC$ .

Step 1 of 2

AB ∥ DC, so triangle XAB is similar to triangle XDC

\angle XAB = \angle XDC

(alternate angles)

\angle AXB = \angle DXC

(vertically opposite angles)

Practise this →

3. Scale drawings and maps

Units
$1$ m	$100$ cm
$1$ km	$1000$ m $= 100\,000$ cm

Scale	With units	Simplest
$1 : 50\,000$	$1$ cm $: 50\,000$ cm	$1$ cm $: 0.5$ km
$1 : 200$	$1$ cm $: 200$ cm	$1$ cm $: 2$ m

Map to actual: 2.4 cm on a 1 : 50 000 map

\displaystyle 1\text{ cm} : 0.5\text{ km}

$2.4$ cm $: 2.4 \times 0.5$ km
$2.4$ cm $: 1.2$ km

Find the scale: 2 cm for 1 km

\displaystyle 2\text{ cm} : 1\text{ km}

$2$ cm $: 100\,000$ cm
$1 : 50\,000$

📖 Worked example

A map is drawn to a scale of $1 : 50\,000$ . Two towns are $2.4$ cm apart on the map. Find the actual distance, in kilometres, between the two towns.

Step 1 of 2

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 50\,000

\mathstrut 1\text{ cm} : 50\,000\text{ cm}

1\text{ cm} : 0.5\text{ km}

📖 Worked example

Two towns are $1$ km apart. On a map they are $2$ cm apart. Express the scale of the map in the form $1 : n$ .

Step 1 of 3

Write map : actual with units

\mathstrut 2\text{ cm} : 1\text{ km}

Scale	Length	Area
$1 : 50\,000$	$1\text{ cm} : 0.5\text{ km}$	$1\text{ cm}^{2} : 0.25\text{ km}^{2}$

📖 Worked example

A map is drawn to a scale of $1 : 20\,000$ . A park has an area of $15$ cm² on the map. Find the actual area of the park, in km².

Step 1 of 3

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 20\,000

\mathstrut 1\text{ cm} : 20\,000\text{ cm}

1\text{ cm} : 0.2\text{ km}

Practise this →

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Sec 3 · G3 (Express)

Congruence and similarity

Practise congruence and similarity with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Congruent triangles

Symbol	Meaning	Example
$\equiv$	congruent: same shape and same size	$\triangle ABC \equiv \triangle QRP$
vertex order	matching vertices in the same order	$A \leftrightarrow Q,\ B \leftrightarrow R,\ C \leftrightarrow P$
matching sides	equal	$AB = QR$
matching angles	equal	$\angle A = \angle Q$

Test	Equal parts
SSS	three sides
SAS	two sides and the angle between them
ASA / AAS	two angles and a corresponding side (ASA: the side between them)
RHS	right angle, hypotenuse and one other side

📖 Worked example

In triangles $ABC$ and $PQR$ , $BC = QR$ and $CA = RP$ . State one more condition that would make triangle $ABC$ congruent to triangle $PQR$ by the SAS test. Choose the condition.

Step 1 of 2

What does the SAS test need?

SAS needs two pairs of equal sides and the angles between them.

📖 Worked example

Triangles $ABC$ and $PQR$ are congruent, with the equal parts marked. Complete the statement $\triangle ABC \equiv \triangle \ldots$

Step 1 of 2

Find the part of PQR equal to each marked part of ABC

AB

(

8

cm) matches

\text{QR}

CA

(

6

cm) matches

\text{PQ}

\angle A

(

42^\circ

) matches

\angle \text{Q}

📖 Worked example

In the diagram, triangle $ABC$ is congruent to triangle $RPQ$ . Find $x$ .

Step 1 of 2

Match the letters in order

\triangle ABC \equiv \triangle RPQ

, so

BC

matches

PQ

Practise this →

2. Similar triangles

Symbol	Meaning	Example
$\sim$	similar: same shape, sides in the same ratio	$\triangle ADE \sim \triangle ABC$
ratio	matching sides divide to the same number	$\dfrac{DE}{BC} = \dfrac{AD}{AB}$

Test	Shown by
AA	two pairs of equal angles
SSS	three pairs of sides in the same ratio
SAS	two pairs of sides in the same ratio, the angle between them equal

DE ∥ BC: use the whole side AB = 5 + 4

\displaystyle \dfrac{DE}{BC} = \dfrac{AD}{AB}

e.g.

\displaystyle \dfrac{DE}{10} = \dfrac{5}{9}

\displaystyle DE = \dfrac{50}{9}

\displaystyle DE = 5.56\text{ (3 s.f.)}

AB ∥ DC, crossing at X: AX = 5, XD = 8, AB = 7

\displaystyle \dfrac{DC}{AB} = \dfrac{XD}{XA}

e.g.

\displaystyle \dfrac{DC}{7} = \dfrac{8}{5}

\displaystyle DC = 11.2

📖 Worked example

In the diagram, $D$ lies on $AB$ and $E$ lies on $AC$ , with $DE$ parallel to $BC$ . Find $DE$ , correct to 3 significant figures.

Step 1 of 3

DE ∥ BC, so triangle ADE is similar to triangle ABC

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

📖 Worked example

In the diagram, the lines $AD$ and $BC$ meet at $X$ , and $AB$ is parallel to $DC$ . Find $DC$ .

Step 1 of 2

AB ∥ DC, so triangle XAB is similar to triangle XDC

\angle XAB = \angle XDC

(alternate angles)

\angle AXB = \angle DXC

(vertically opposite angles)

Practise this →

3. Scale drawings and maps

Units
$1$ m	$100$ cm
$1$ km	$1000$ m $= 100\,000$ cm

Scale	With units	Simplest
$1 : 50\,000$	$1$ cm $: 50\,000$ cm	$1$ cm $: 0.5$ km
$1 : 200$	$1$ cm $: 200$ cm	$1$ cm $: 2$ m

Map to actual: 2.4 cm on a 1 : 50 000 map

\displaystyle 1\text{ cm} : 0.5\text{ km}

$2.4$ cm $: 2.4 \times 0.5$ km
$2.4$ cm $: 1.2$ km

Find the scale: 2 cm for 1 km

\displaystyle 2\text{ cm} : 1\text{ km}

$2$ cm $: 100\,000$ cm
$1 : 50\,000$

📖 Worked example

A map is drawn to a scale of $1 : 50\,000$ . Two towns are $2.4$ cm apart on the map. Find the actual distance, in kilometres, between the two towns.

Step 1 of 2

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 50\,000

\mathstrut 1\text{ cm} : 50\,000\text{ cm}

1\text{ cm} : 0.5\text{ km}

📖 Worked example

Two towns are $1$ km apart. On a map they are $2$ cm apart. Express the scale of the map in the form $1 : n$ .

Step 1 of 3

Write map : actual with units

\mathstrut 2\text{ cm} : 1\text{ km}

Scale	Length	Area
$1 : 50\,000$	$1\text{ cm} : 0.5\text{ km}$	$1\text{ cm}^{2} : 0.25\text{ km}^{2}$

📖 Worked example

A map is drawn to a scale of $1 : 20\,000$ . A park has an area of $15$ cm² on the map. Find the actual area of the park, in km².

Step 1 of 3

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 20\,000

\mathstrut 1\text{ cm} : 20\,000\text{ cm}

1\text{ cm} : 0.2\text{ km}

Practise this →

Try a question: Congruence and similarity

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise congruence and similarity with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free