# Similar Triangles Test – O Level Exam Preparation Guide

In this Similar Triangles revision note, you will learn the properties of Similar Triangles and how to test for them.

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## Similar Triangles

When two triangles are similar, they have the same shape but different size.

When a triangle undergoes an enlargement or reduction, the shape remains the same but the sizes are different. Hence they are similar triangles.

## Similar triangles Properties

ΔABC is similar to ΔXYZ (Note the order)

if and only if…

the ratio of corresponding sides are equal,

AB/XY = AC/XZ =  BC/YZ

and corresponding angles are equal.

∠A = ∠X , ∠B = ∠Y and ∠C = ∠Z

## Similar Triangles Test

There are 3 methods to test whether 2 triangles are similar.

1. 2 pairs of corresponding angles are equal

i.e. ∠A = ∠D, ∠B = ∠E (AA)

∆ABC and ∆DEF are similar.

Note: When 2 angles are the same, the 3rd angle will be the same too because the sum of angles is always 180 degrees

In ∆ABC and ∆DEC,

∠ACB = ∠DCE (common)

∠ABC = ∠DEC (corresponding angles)

∠BAC = ∠EDC (corresponding angles)

∴ ∆ABC is similar to ∆DEC. (AA Similarity)

2. 3 pairs of corresponding sides are in the same ratio

i.e. AB/DE =  BC/EF =  AC/DF (SSS)

In ∆ABC and ∆DEF,

AB/DE = 12/18 = 2/3, BC/EF = 8/12 = 2/3, AC/DF = 10/15 = 2/3

∴ ∆ABC is similar to ∆DEF. (SSS Similarity)

3. 2 pairs of corresponding sides are in the same ratio and a pair of included angles is equal

i.e. AB/DE = AC/DF and ∠A = ∠D (SAS)

In ∆ABC and ∆DEF,

∠BAC = ∠EDF = 50°

AB/DE = 3/5,  AC/DF = 12/20 = 3/5

∴ ∆ABC is similar to ∆DEF (SAS Similarity)

## Similar Triangles Example

In the diagram below, PQ is parallel to BC. APB and AQC are straight lines. Given that AP = x cm, BC = 18 cm, PB = 6 cm and PQ = 10 cm, find
(a) x.
(b) the ratio of AQ : AC.

Solution

Angle A is a common angle, Angle APQ = Angle ABC (corresponding angle).

∆APQ and ∆ABC are similar triangles (AA Similarity).

Therefore, the ratios of the corresponding sides are equal.

AP/AB =PQ/BC = AQ/AC

(a) To find x

AP/AB = PQ/BC

x/(x + 6) cm = 10cm/18cm

10(x + 6) = 18 x

10x + 60 = 18 x

8 x = 60

x = 7.5 cm

(b) To find the ratio of AQ : AC

Since the triangles are similar, the ratio of AQ : AC is the same as its corresponding sides PQ: BC.

AQ/ AC = PQ/ BC = 10 cm/18 cm = 5 : 9

## Check out our exam guide on other topics here!

Secondary Math Revision Notes

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