Similar Triangles Revision Notes for VCE SSCE General Mathematics

Similar Triangles

What are similar triangles?

Triangles are similar when they have identical shapes, though they may be different sizes. This means that matching angles in each triangle have the same measurement, and the ratios between matching sides are constant.

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The key to understanding similar triangles is recognizing that they maintain the same proportions even when scaled to different sizes. Think of it like a photograph being enlarged or reduced - the shape stays the same, only the size changes.

For example, look at these two triangles:

Both triangles have the same angles ( $35°$ and $55°$ ), but the larger triangle has sides that are exactly twice as long as the smaller triangle. The ratio between corresponding sides is $\frac{10}{5} = \frac{8}{4} = \frac{6}{3} = 2$ , which is constant.

Tests for triangle similarity

We can determine if triangles are similar by checking one of three conditions. You only need to prove one of these conditions to establish similarity.

AA test (Angle-Angle)

When two pairs of matching angles are equal in measure, the triangles must be similar.

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Critical concept: If two pairs of corresponding angles are equal, then the third pair must also be equal. This is because the angles in any triangle always sum to $180°$ . Therefore, you only need to check two angles to use this test.

SSS test (Side-Side-Side)

When all three pairs of matching sides have the same ratio, the triangles are similar. This constant ratio is called the scale factor.

In this example, we compare each pair of corresponding sides:

$\frac{5}{2.5} = \frac{4}{2} = \frac{2}{1} = 2$

Since all three ratios equal $2$ , the triangles are similar with a scale factor of $2$ . This means the larger triangle is exactly twice the size of the smaller one.

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The scale factor tells you how many times larger (or smaller) one triangle is compared to the other. A scale factor of 2 means the larger triangle has sides that are double the length of the corresponding sides in the smaller triangle.

SAS test (Side-Angle-Side)

When two pairs of matching sides have the same ratio, and the angle between those sides (called the included angle) is equal, the triangles are similar.

In this example:

The ratio of corresponding sides: $\frac{19.5}{6.5} = \frac{21}{7} = 3$
Both triangles have an included corresponding angle of $30°$

Since the two side ratios are equal and the included angles match, these triangles are similar by the SAS test, with a scale factor of $3$ .

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Common mistake: Make sure the equal angles are positioned between the two sides you're comparing! An angle that isn't between the compared sides cannot be used for the SAS test.

Worked example: checking if triangles are similar

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Worked Example: Determining Triangle Similarity

Question: Explain why triangle $ABC$ is similar to triangle $XYZ$ .

Solution:

First, we compare the ratios of corresponding sides:

$\frac{XZ}{AC} = \frac{21}{7} = \frac{3}{1}$

$\frac{YZ}{BC} = \frac{24}{8} = \frac{3}{1}$

Both ratios equal $\frac{3}{1}$ , so these pairs of corresponding sides are in the same ratio.

Next, we check the angles. Both triangles have an angle of $32°$ , which is the included angle (the angle between the two sides we just compared).

Conclusion: The triangles have two pairs of corresponding sides in the same ratio, and the included corresponding angles are equal. Therefore, triangle $ABC$ is similar to triangle $XYZ$ by the SAS test.

Exam tips

When working with similar triangles in exams, keep these important points in mind:

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Always identify which test you're using (AA, SSS, or SAS) in your answer
When calculating ratios, simplify them to make comparisons easier
For the SAS test, make sure the equal angles are between the sides you're comparing
Show all your working clearly, especially when calculating ratios
Remember that you only need to prove one of the three tests to establish similarity

Remember!

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Key Points to Remember:

Similar triangles have the same shape but may be different sizes, with equal corresponding angles and proportional corresponding sides
Three tests for similarity:
- AA (Angle-Angle): Two pairs of corresponding angles are equal
- SSS (Side-Side-Side): All three pairs of corresponding sides are in the same ratio
- SAS (Side-Angle-Side): Two pairs of corresponding sides are in the same ratio and the included angles are equal
The scale factor is the constant ratio between corresponding sides in similar triangles
You only need to prove one of the three tests to establish that triangles are similar

Similar Triangles (VCE SSCE General Mathematics): Revision Notes