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Triangle similarity theorems

triangle similarity

Before trying to understand similarity of triangles it is very important for you to understand proportions and ratios, because similarity is based on entirely that.
Triangle similarity is another relation two triangles may have. You already learned about congruence, where all sizes must be equal. In similarity, only angles must be equal, and lengths of sides proportional. There are four theorems that we use to determine if two triangles are similar.

Similarity is noted like this:
Two triangles ABC and DEF are similar, we write: \bigtriangleup ABC \sim \bigtriangleup DEF.

Theorem A-A-A: two triangles are similar if their matching angles are congruent.

You have a triangle ABC. How would you construct a similar triangle? Because it has to have equal angle measures, you can only draw parallel lines to the sides of given triangle, because parallel relation keeps the angle measure. This way, you will always get a similar triangle.


AAA similarity theorem

\beta = \beta ', \gamma = \gamma ' -> (because, if two angles in triangles match, that means that the third also matches) \alpha = \alpha '. From this we can conclude that triangles ABC and AFD are similar.
Which means that, besides the fact that their angles match, following proportions are true:

\frac{\mid AB \mid}{\mid AF \mid} = \frac{\mid AC \mid}{\mid AD \mid} = \frac{\mid CB \mid}{\mid DF \mid}

You have to be careful about setting these proportions. You should always first chose the triangle whose length of sides will be the numerators, and whose denominators. Don’t mix them up, or this proportion will not be true.

It’s easiest to remember that in the numerator you put the lengths of the smaller triangle, and in the denominator lengths of a bigger triangle. This way is less likely for you to make mistakes.

Theorem S-S-S: Two triangles are similar if all their pages are proportional.

SSS similarity theorem

Triangles ABC and DEF are similar triangles if:

condition similarity

If this is true, all the matching angles in these triangles have equal measure.
But here is very important to make sure you are taking the right sides. Because triangles can be rotated or mirrored and you lose the sense of which side goes with which. These things are easy to see, you match the smallest side of the first triangle with the smallest side of the second triangle, second largest with second largest, and largest with largest.

Example: Are following triangles similar?
Triangle ABC where a = 2, b = 4, c = 5,
Triangle DEF where d = 4, e = 8, f = 10.

Solution: the smallest pages are a and d, second largest are b and e, and largest are c and f. Now we put that into a proportion:

\frac{a}{d} = \frac{b}{e} = \frac{c}{f}

\frac{2}{4} = \frac{4}{8} = \frac{5}{10}

When we shorten these fractions we get \frac{1}{2} everywhere. And that means that triangles ABC and DEF are similar. If we got different value in any part, they would not be similar.

Theorem S-A-S: Two triangles are similar if their two sides are proportional and angles between them 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}. If this is true, that means that all the angles are congruent and all sides are proportional.

Example: Are following triangles similar?
ABC: a = 20, b = 5, angle between them \alpha = 120^{\circ}
WER: w = 10, e = 2.5, angle between them \beta = 120^{\circ}.

Solution: \frac{a}{w} = \frac{b}{e}? \frac{20}{10} = \frac{5}{2.5} 2 = 2. These two sides are proportional. And angles between them are of equal measure, that means that these two triangles are similar.

Theorem S-S-A: Two triangles are similar if their two sides are proportional, and angles across the larger one are congruent.


SSA similarity theorem

Here we have two triangles, triangle GIH and JKL. If we know that \frac{h}{l} = \frac{g}{j}, and we know which side of them is larger, if the angles across the larger ones are congruent than GIH and JKL are similar.

Example: Are these two triangles similar?
ABC: a = 6, b = 3, \alpha = 70^{\circ} (\alpha opposite angle of side a)
GHJ: a' = 6, b' = 12, \beta = 70^{\circ}. (\beta opposite angle of side b')

Solution: Do not be fooled by different markings. You cannot simply put that \frac{a}{a'} must be equal to \frac{b}{b'}.
Remember, you have to compare sides by their lengths.

In this case it has to be \frac{a}{b'} = \frac{b}{a'} \Rightarrow \frac{6}{12} = \frac{3}{6} which is true. The opposite angle to the largest side in triangle ABC is \alpha and opposite angle to the largest side in triangle is \beta. \alpha = \beta which means that triangles ABC and GHJ are similar.


Triangle similarity worksheets

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