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.
In this lesson we’ll look at how to prove triangles are similar to one another.
In math, the word “similarity” has a very specific meaning.
Outside of math, when we say two things are similar, we just mean that they’re generally like one another.
But in math, to say two figures are similar means that they have exactly the same shape, but that they’re different sizes. Here are examples of similar squares, similar pentagons, and similar triangles:
Example
Are the triangles similar? Which theorem proves that they’re similar and complete the similarity statement.
???\triangle ABC\sim \triangle???____
In a pair of similar triangles, all three corresponding angle pairs are congruent and corresponding side pairs are proportional.
Example
Are the triangles similar? Which theorem proves that they’re similar and complete the similarity statement.
???\triangle WXY\sim \triangle???____
We know from the figure that ???\angle W\cong\angle V=59^\circ???. We also have a pair of vertical angles at ???Y???, and remember that vertical angles are congruent to one another.
Putting all this together, we can see say that the triangles are similar by Angle Angle (AA). When we match up the corresponding parts, the similarity statement is ???\triangle WXY\sim \triangle VZY???.