Similar triangles have corresponding sides and angles

 
 
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Similar triangles have corresponding angles and corresponding sides

In this lesson we’ll look at the ratios of similar triangles to find out missing information about similar triangle pairs.

Similar triangles

In a pair of similar triangles, corresponding sides are proportional and all three angles are congruent. This means if you know two triangles are similar to one another you can use the information to solve for missing parts.

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Corresponding angles

In a pair of similar triangles the corresponding angles are the angles with the same measure. In the diagram of similar triangles, the corresponding angles are the same color.

 
corresponding angle pairs
 

Corresponding sides

In a pair of similar triangles, the corresponding sides are proportional. Corresponding sides touch the same two angle pairs.

When the sides are corresponding it means to go from one triangle to another you can multiply each side by the same number.

 
corresponding sides of triangles
 

In the diagram of similar triangles the corresponding sides are the same color.

Naming similar triangles

To show two triangles are similar, you use the symbol ???\sim???. You need to match the letters from the first triangle to the angles with the corresponding vertices on the second triangle.

 
naming similar triangles
 

Here we can say that ???\triangle XYZ\sim \triangle BCA???.

 
 

When triangles are similar, you can use that fact to solve for the value of angles and sides


 
 
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Solving for the variable in two similar triangles

Example

If the two triangles in the diagram are similar, solve for the variable.

 
solving for the variable when two triangles are similar
 

In similar triangles, corresponding sides are proportional. In the diagram, ???x??? corresponds to ???8???, and ???42.5??? corresponds to ???17???. So we say

???\frac{x}{8}=\frac{42.5}{17}???

???17x=8(42.5)???

???17x=340???

???x=20???

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Corresponding sides touch the same two angle pairs.

How to solve for values in similar triangles

Example

If ???\triangle XVY\sim \triangle XWZ???, solve for ???x???.

 
solve for the variable in similar triangles
 

In similar triangles, corresponding sides are proportional. In the diagram, ???4x+12??? corresponds to ???88???, and ???91??? corresponds to ???143???.

???\frac{4x+12}{88}=\frac{91}{143}???

???(4x+12)143=91(88)???

???(4x+12)143=8,008???

???\frac{(4x+12)143}{143}=\frac{8,008}{143}???

???4x+12=56???

???4x=44???

???x=11???

 
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