Before trying to understand similarity of triangles it is very important to understand the concept of proportions and ratios, because similarity is based entirely on these principles.
Triangle similarity is another relation two triangles may have. We already learned about congruence, where all sides must be of equal length. In similarity, angles must be of equal measure with all sides proportional. Similarity is the relation of equivalence.
Two triangles $ABC$ and $DEF$ are similar, thus we write: $\bigtriangleup ABC \sim \bigtriangleup DEF$.
There are four theorems that we can use to determine if two triangles are similar.
AA theorem
Two triangles are similar if their two corresponding angles are congruent.
Let $ABC$ be the given triangle. So how can we construct a similar triangle?
We will expand segment lines $\overline{AB}$ and $\overline{AC}$ over the vertices $B$ and $C$, respectively. On the line $AC$ we choose the point $D$ and construct a line that is parallel to line $BC$ and that passes trough a point $D$. The intersection of the previously constructed line and line $AB$ is point $E$.
The resulting triangle $AFD$ is similar to the given triangle $ABC$, as shown below.
$\beta = \beta ’ , \gamma = \gamma ’ \Rightarrow \alpha = \alpha ’$ (if two corresponding angles are of equal measure, then the third is also equal and corresponding).
The following proportions are also true:
$\frac{\mid AB \mid}{\mid AF \mid} = \frac{\mid AC \mid}{\mid AD \mid} = \frac{\mid CB \mid}{\mid DF \mid}$.
SSS theorem
Two triangles are similar if the lengths of all corresponding sides are proportional.
Triangles $ABC$ and $DEF$ are the similar triangles if:
Example 1.
Are the following triangles
$$\triangle{ABC}: \quad a = 2, \quad b = 4, \quad c = 5,$$
$$\triangle{DEF}: \quad d = 4, \quad e = 8, \quad f = 10$$
similar?
Solution:
We have:
$$\frac{a}{d} = \frac{b}{e} = \frac{c}{f} \Rightarrow \frac{2}{4} = \frac{4}{8} = \frac{5}{10} \Rightarrow \frac{1}{2}=\frac{1}{2}=\frac{1}{2}.$$
It follows that triangles $ABC$ and $DEF$ are thus similar by the SSS theorem.
If we found a different value in any part, the triangles are not similar.
SAS theorem
Two triangles are similar if the corresponding lengths of two sides are proportional and the included angles are congruent.
Triangles $ABC$ and $DEF$ are similar if $\alpha = \alpha ’$ and $\frac{\mid DE \mid}{\mid AB \mid} = \frac{\mid AC \mid}{\mid DF \mid}$. It follows that all corresponding angles are congruent and the lengths of all sides are proportional.
Example 2.
Are the following triangles
$$\triangle{ABC}:\quad a = 20,\quad b = 5, \quad \alpha = 120^{\circ}$$
$$\triangle{WER}: \quad w = 10, \quad e = 2.5, \quad \beta = 120^{\circ}$$
similar?
Solution:
It must be: $\frac{a}{w} = \frac{b}{e}$.
We have:
$$\frac{20}{10} = \frac{5}{2.5} \Rightarrow 2 = 2.$$
These two corresponding sides are proportional and the included angles are of equal measure. It follows that the triangles $ABC$ and $WER$ are thus similar triangles according to the SAS theorem.
SSA theorem
Two triangles are similar if the lengths of two corresponding sides are proportional and their corresponding angles across the larger of these two are congruent.
Consider triangles $GIH$ and $JKL$. If we know that $\frac{h}{l} = \frac{g}{j}$ and if the angles across the larger ones are congruent, then triangles $GIH$ and $JKL$ are similar.
Example 3.
Are the following triangles
$$\triangle{ABC}: \quad a = 6, b = 3, \alpha = 70^{\circ},$$
$$\triangle{GHJ}: \quad a’ = 6, b’ = 12, \beta = 70^{\circ}$$
similar?
Solution:
We must compare all sides by their lengths.
In this case we have
$$\frac{a}{b’} = \frac{b}{a’} \Rightarrow \frac{6}{12} = \frac{3}{6} \Rightarrow \frac{1}{2} = \frac{1}{2},$$
which is true. The opposite angle to the side of the longest length in triangle $ABC $is $\alpha$ and opposite angle to the longest side in triangle $GHJ$ is $\beta$.
It follows that $\alpha = \beta$, which means that triangles $ABC$ and $GHJ$ are thus similar by the SSA theorem.
Triangle similarity worksheets
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