5.02 Similar triangle tests

There are four tests that we can use to test similarity between two triangles. If any one of these tests is satisfied then the two triangles must be similar. These four tests are:

• AAA: Three pairs of equal angles

• SSS: Three pairs of sides in the same ratio

• SAS: Two pairs of sides in the same ratio and an equal included angle

• RHS: Both have right angles, and the hypotenuses and another pair of sides are in the same ratio

These similarity tests can be proved to work in the same way that the congruency tests work, except with sides in fixed ratio rather than needing to be equal.

It is worth noting that, since all triangles have a fixed angle sum of 180^{ \circ{}}, having two matching angles is equivalent to having three matching angles.

To determine which sides are corresponding in two potentially similar triangles, we can use their positions with respect to any matching angles. If there is a common sized angle in both triangles, then the sides opposite those angles will be corresponding.

In this case, the sides labelled x and y are corresponding since they are opposite the same sized angle.

Similarly, if side lengths are given and two pairs of sides are in a fixed ratio between the two triangles, the angles between these pairs of sides will be corresponding.

In this case, since the side pairs 5 & 8 and 10 & 16 are in the fixed ratio of 1:2, the angles between these pairs are corresponding and therefore equal.

Examples

Example 1

Consider the two triangles in the diagram below:

Are \triangle ABC and \triangle DEF similar?

A

Yes, they satisfy AAA.

B

Yes, they satisfy SSS.

C

Yes, they satisfy SAS.

D

Yes, they satisfy RHS.

E

No, they are definitely not similar.

F

Unknown, there is not enough information.

Worked Solution

Create a strategy

Check the ratio of the given sides.

Apply the idea

Based on the given diagram, two triangles have equal angles of 84^{\circ{}}.

Sides AB and DE are corresponding, and sides AC and DF are corresponding.

\displaystyle \dfrac{DE}{AB}	\displaystyle =	\displaystyle \dfrac{10}{5}	Find the ratio of the sides
	\displaystyle =	\displaystyle 2	Evaluate
\displaystyle \dfrac{DF}{AC}	\displaystyle =	\displaystyle \dfrac{16}{8}	Find the ratio of the sides
	\displaystyle =	\displaystyle 2	Evaluate
	\displaystyle =	\displaystyle \dfrac{DE}{AB}	Compare the ratios

Now we know two pairs of sides are in the same ratio, and the included angles are equal.

This means that the two triangles satisfy SAS. Option C is the correct answer.

Example 2

Consider the following diagram:

a

Are \triangle ABC and \triangle ADE similar?

Worked Solution

Create a strategy

Determine if they have any equal angles.

Apply the idea

\triangle ABC and \triangle ADE have a common angle at A.

DE and BC are parallel, so \angle ADE = \angle ABC since they are alternate angles on parallel lines.

Similarly, \angle AED = \angle ACB.

The two triangles are similar since they have three pairs of equal angles, AAA

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b

Given that DE=7, what will the length of BC be?

Worked Solution

Create a strategy

Equate the ratios of corresponding sides.

Apply the idea

\displaystyle \dfrac{BC}{DE}	\displaystyle =	\displaystyle \dfrac{AC}{AE}	Equate the ratios of the sides
\displaystyle \dfrac{BC}{7}	\displaystyle =	\displaystyle \dfrac{16}{6}	Substitute the lengths
\displaystyle \dfrac{BC}{7} \times 7	\displaystyle =	\displaystyle \dfrac{16}{6} \times 7	Multiply both sides of equation by 7
\displaystyle BC	\displaystyle =	\displaystyle \dfrac{56}{3}	Evaluate the product

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