Two triangles are similar if they have:
But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough. There are three ways to find if two triangles are similar: AA, SAS and SSS: AAAA stands for "angle, angle" and means that the triangles have two of their angles equal.
If two triangles have two of their angles equal, the triangles are similar. So AA could also be called AAA (because when two angles are equal, all three angles must be equal). SASSAS stands for "side, angle, side" and means that we have two triangles where:
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
 In this example we can see that:
So there is enough information to tell us that the two triangles are similar. Using TrigonometryWe could also use Trigonometry to calculate the other two sides using the Law of Cosines:
In Triangle ABC:
In Triangle XYZ:
Now let us check the ratio of those two sides: a : x = 22.426... : 14.950... = 3 : 2 the same ratio as before! Note: we can also use the Law of Sines to show that the other two angles are equal. SSSSSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
In this example, the ratios of sides are:
These ratios are all equal, so the two triangles are similar. Using TrigonometryUsing Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:
In Triangle ABC:
In Triangle XYZ:
So angles A and X are equal! Similarly we can show that angles B and Y are equal, and angles C and Z are equal. Copyright © 2017 MathsIsFun.com
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length. The side lengths of two similar triangles are proportional. That is, if Δ U V W is similar to Δ X Y Z , then the following equation holds: U V X Y = U W X Z = V W Y Z This common ratio is called the scale factor . The symbol ∼ is used to indicate similarity.
Example: Δ U V W ∼ Δ X Y Z . If U V = 3 , V W = 4 , U W = 5 and X Y = 12 , find X Z and Y Z . Draw a figure to help yourself visualize.
Write out the proportion. Make sure you have the corresponding sides right. 3 12 = 5 X Z = 4 Y Z The scale factor here is 3 12 = 1 4 . Solving these equations gives X Z = 20 and Y Z = 16 . The concepts of similarity and scale factor can be extended to other figures besides triangles. Triangles are similar if their corresponding sides are proportional.
By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you know only that all sides are proportional, that is enough information to know that the triangles are similar. This is called the SSS Similarity Theorem. SSS Similarity Theorem: If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar. If \(\dfrac{AB}{YZ}=\dfrac{BC}{ZX}=\frac{AC}{XY}\), then \(\Delta ABC\sim \Delta YZX\). What if you were given a pair of triangles and the side lengths for all three of their sides? How could you use this information to determine if the two triangles are similar? For Examples 1 and 2, use the following diagram:
Example \(\PageIndex{1}\) Is \(\Delta DEF\sim \Delta GHI\)? Is \(\dfrac{15}{30}=\dfrac{16}{33}=\dfrac{18}{36}\)? Solution \(\dfrac{15}{30}=\dfrac{1}{2}\), \(\dfrac{16}{33}=\dfrac{16}{33}\), and \(\dfrac{18}{36}=\dfrac{1}{2}\). \(\dfrac{1}{2}\neq \dfrac{16}{33}\), \(\Delta DEF\) is not similar to \(\Delta GHI\).
Example \(\PageIndex{2}\) Is \(\Delta ABC\sim \Delta GHI\)? Is \(\dfrac{20}{30}=\dfrac{22}{33}=\dfrac{24}{36}\)? Solution \(\dfrac{20}{30}=dfrac{2}{3}\), \(\dfrac{22}{33}=\dfrac{2}{3}\), and \(\dfrac{24}{36}=\dfrac{2}{3}\). All three ratios reduce to \(\dfrac{2}{3}\), \(\Delta ABC\sim \Delta GHI\).
Example \(\PageIndex{3}\) Determine if the following triangles are similar. If so, explain why and write the similarity statement. Solution We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with the longest sides and work down to the shortest sides. \(\begin{aligned} \dfrac{BC}{FD}&=\dfrac{28}{20}=\dfrac{7}{5} \\ \dfrac{BA}{FE}&=\dfrac{21}{15}=\dfrac{7}{5} \\ \dfrac{AC}{ED}&=\dfrac{14}{10}=\dfrac{7}{5}\end{aligned}\) Since all the ratios are the same, \(\Delta ABC\sim \Delta EFD\) by the SSS Similarity Theorem.
Example \(\PageIndex{4}\) Find \(x and \(y, such that \(\Delta ABC\sim \Delta DEF\). Solution According to the similarity statement, the corresponding sides are: \(\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF}\). Substituting in what we know, we have \(\dfrac{9}{6}=\dfrac{4x−1}{10}=\dfrac{18}{y}\). \(\begin{aligned} \frac{9}{6} &=\frac{4 x-1}{10} & \frac{9}{6} &=\frac{18}{y} \\ 9(10) &=6(4 x-1) & & 9 y=18(6) \\ 90 &=24 x-6 & & 9 y=108 \\ 96 &=24 x & & y=12 \\ x &=4 & \end{aligned}\)
Example \(\PageIndex{5}\) Determine if the following triangles are similar. If so, explain why and write the similarity statement. Solution We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with the longest sides and work down to the shortest sides. \(\begin{aligned} \dfrac{AC}{ED}&=\dfrac{21}{35}=\dfrac{3}{5} \\ \dfrac{BC}{FD}&=\dfrac{15}{25}=\dfrac{3}{5} \\ \dfrac{AB}{EF}&=\dfrac{10}{20}=\dfrac{1}{2} \end{aligned}\) Since the ratios are not all the same, the triangles are not similar.
Fill in the blanks.
Use the following diagram for questions 3-5. The diagram is to scale.
Fill in the blanks in the statements below. Use the diagram to the left.
Use the diagram to the right for questions 10-15.
Find the value of the missing variable(s) that makes the two triangles similar.
To see the Review answers, open this PDF file and look for section 7.6.
Video: Congruent and Similar Triangles Activities: SSS Similarity Discussion Questions Study Aids: Polygon Similarity Study Guide Practice: SSS Similarity Real World: Crazy Quilt |
What theorem states that two triangles are similar if the corresponding sides of two triangles are in proportion?
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