6.4. Criteria for Similarity of Triangles

In the previous section, we stated that two triangles are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion). That is, in Δ ABC and Δ DEF, if (i) ∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F and AB/DE = BC/EF = CA/FD then the two triangles are similar (see Fig. 6.22).

Two triangles are considered similar if they have the same shape but can be different in size. This definition is made precise using two criteria. First, the corresponding angles of the triangles must be equal. For instance, if angle A in triangle ABC is equal to angle D in triangle DEF, and similarly for the other angles, then this condition is met. Second, the lengths of their corresponding sides must be in the same ratio. This means if you take the length of one side of triangle ABC and divide it by the corresponding side of triangle DEF, you should get the same ratio for all the sides.

Think about similar triangles in everyday life, such as miniature models of buildings. A model of a skyscraper may be much smaller than the actual building, but if it has all the same angles and the ratios of heights to widths are preserved, they are similar triangles.

Here, you can see that A corresponds to D, B corresponds to E and C corresponds to F. Symbolically, we write the similarity of these two triangles as ‘Δ ABC ~ Δ DEF’ and read it as ‘triangle ABC is similar to triangle DEF’. The symbol ‘~’ stands for ‘is similar to’.

In mathematics, we use symbols to express concepts concisely. The similarity of triangles is represented symbolically by the tilde (~) sign. For example, if triangle ABC is similar to triangle DEF, we write 'Δ ABC ~ Δ DEF'. This notation indicates that the corresponding angles of these triangles are equal and their sides have the same proportional lengths.

Consider a pair of shoes: if you have a small and a large size of the same style, they are similar in design (same shape), but different in size. Just like with our triangles, we could say the smaller pair is similar to the larger one using the same concept of similarity.

Now a natural question arises: For checking the similarity of two triangles, say ABC and DEF, should we always look for all the equality relations of their corresponding angles (∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F) and all the equality relations of the ratios (AB/DE, BC/EF, CA/FD)?

To confirm if two triangles are similar, ideally, we would check all corresponding angles and all corresponding side ratios. However, we can often find a shortcut. If we find even one pair of equal angles, we can prove the triangles are similar using the established criteria.

Imagine two triangular flags on the same pole; one is larger than the other. If you determine that the angle at the top of both flags is equal, you can conclude they are similar without measuring all angles and side ratios—the relationship works because of the properties of triangles.

This activity leads us to the following criterion for similarity of two triangles: Theorem 6.3: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. This criterion is referred to as the AAA (Angle–Angle–Angle) criterion of similarity of two triangles.

The AAA criterion suggests a very effective way to prove triangle similarity. If all corresponding angles of two triangles are equal, there’s no need to check the sides; they must be in proportion. This simplifies the process significantly, as dealing with angles is often simpler than measuring sides directly.

Think of shadows cast by two similar objects—if the angles formed at their bases are equal, then their heights must be proportionate. Thus, we can determine their relative heights without measuring them directly.

Theorem 6.4: If in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. This criterion is referred to as the SSS (Side–Side–Side) similarity criterion for two triangles.

The SSS criterion allows us to establish triangle similarity through side lengths alone. If the lengths of the sides of one triangle are in proportion to the lengths of the sides of another triangle, then the angles of those triangles must be equal. This is particularly useful in scenarios where measuring angles is impractical.

Imagine a set of similar triangular pizzas. If you know one pizza's size ratios are consistently larger than the smaller pizza's, you can confidently state they are similar without measuring all angles, just by looking at the slices.

Theorem 6.5: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. This criterion is referred to as the SAS (Side–Angle–Side) similarity criterion for two triangles.

The SAS criterion combines an angle measurement with side ratios. It allows us to establish similarity by showing that two triangles have one angle equal and the sides that are adjacent to this angle are in proportion. This is especially handy when angle measures are known, and one just wants to confirm or reveal the similarity.

Consider two triangular bridges supported by cables—if one angle of the triangle formed by cables is the same for both bridges and the lengths of the cables are in proportion, the bridges will be similar in shape.

--