Similar triangles proof

By Martin McBride, 2023-03-21
Tags: similar triangle proof
Categories: gcse geometry


The AA and SSS rules for similar triangles were covered here. In this article, we will prove those rules.

Similar triangles

Here are two similar triangles:

Similar triangles

For similar triangles, all corresponding angles are equal:

Angles are equal

And all corresponding sides are proportionate:

Sides are proportionate

To prove the AA and SSS rules, we need to show that the rules lead to those two conditions.

AA rule

If two triangles have two angles the same, they are similar by the AA rule. We will prove that here.

AA prove all angles are equal

We label the triangles such that these pairs of angles are equal:

AA proof

Since the angles in a triangle add up to 180°, we can find C:

AA proof

We can also find F:

AA proof

This uses the fact that A = D and B = E. It proves that C = F, so all corresponding angles are equal.

AA prove that all sides are proportionate

Here are the two triangles, redrawn to use the same angles A, B, and C for both triangles:

Similar triangles

We are going to use the sine rule to prove that the sides are proportionate.

Applying the sine rule to sides a and b of the small triangle:

AA proof

We can rearrange this:

AA proof

We repeat the same thing for the corresponding sides of the large triangle. Applying the sine rule to sides d and e:

AA proof

We can rearrange this:

AA proof

The previous formula for a / b gave the same value, so we know that:

AA proof

We multiply both sides by b / d:

AA proof

We can then cancel out some terms:

AA proof

This tells us that sides a and d are in the same proportions as sides b and e:

AA proof

We can do this again using sides a and c of the small triangle (and sides d and f of the large triangle). This results in:

AA proof

This proves that all corresponding sides are proportionate. So this proves the AA rule.

SSS rule

If two triangles have all corresponding sides that are proportionate, they are similar by the SSS rule. We will prove that next.

This shows the two triangles where the sides are proportional with a scale factor of s. So where small triangle has side a, the large triangle has an equivalent side s times a:

Similar triangles

We need to prove that all the corresponding angles are the same. We will use the cosine rule.

We can find the cosine of angle A of the small triangle, using the cosine rule, like this:

AA proof

We can also find the cosine of angle D of the large triangle, using the cosine rule, like this:

AA proof

For this triangle we have used s times a instead of a, similar for b and c. We can separate out the terms in s:

AA proof

Every term on the top and bottom has a factor of s squared, so we can cancel these out:

AA proof

This is exactly the same equation as we had for cos A, proving that angles A and D are the same.

We can use the same steps to prove that angles B and E are the same. And since the angles in a triangle add up to 180° this would then prove that C and F are the same. This proves the SSS rule.

Be careful of inverse trig functions

There is one important point that we have ignored so far in the SSS proof.

Inverse trigonometry functions do not give unique answers.

The cosine of 60° is 0.5. The cosine of 120° is also 0.5. In fact for any angle P:

AA proof

If we know the cosine of P, there will always be an acute angle and an obtuse angle satisfying the condition (unless P is 90° in which case the cosine is 0).

These two triangles illustrate this point:

AA proof

The cosine of A is equal to the cosine of D, but the angles are not the same because clearly one is acute and one is obtuse.

We won't prove it here, but the condition for the angle being acute is:

AA proof

And the condition for the angle being obtuse is:

AA proof

In the case of the SSS rule, we know that the two triangles have sides that are in the same proportions, so it is impossible for one triangle to meet the condition for the angle to be acute and the other to meet the condition for it to be obtuse. So if the cosines are equal, the angles are equal.

See also



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