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Triangle similarity theorems

triangle similarity

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.

 

 

AAA similarity theorem

 

\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.

SSS similarity theorem

Triangles ABC and DEF are the similar triangles if:

condition similarity

 

 

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.

SAS similarity theorem

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.

 

SSA similarity theorem

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 ABCis \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

  Similar triangles (740.4 KiB, 482 hits)

  Similar right triangles (179.2 KiB, 303 hits)

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