Similar figures are the very same shape, however can be different sizes

In this lesson we’ll look at just how to prove triangles are comparable to one another.

You are watching: Which group contains triangles that are all similar

In math, words “similarity” has a very particular meaning.

Outside of math, as soon as we say two things are similar, we simply mean that they’re usually like one another.


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But in math, come say two figures are similar way that they have specifically the same shape, but that they’re various sizes. Right here are examples of comparable squares, similar pentagons, and comparable triangles:


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Similar triangles

Similar triangles are the very same shape yet not the exact same size. Remember the if 2 triangles room both specifically the exact same shape, and exactly the exact same size, then they room identical and we say they’re “congruent.”

In a pair ofsimilar triangles, all three equivalent angle pairs space congruent and also corresponding next pairs room proportional. The symbol for similarity is ???\sim???, for this reason if we want to say that triangles ???A??? and ???B??? are similar to one another, climate we deserve to write that together ???A\sim B???.

The triangles listed below are similar because the corresponding interior angles space congruent, and because the side lengths room proportional like this:

???\fracat=\fracbu=\fraccv???


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We’re going come look at 3 theorems that allow you come prove that triangles space similar.

Angle edge (AA)

If a pair of triangles have two matching angles that are congruent, then we have the right to prove that the triangles are similar. The reason is because, if you recognize two angles are congruent, then the third collection of equivalent angles have to be congruent as well because the angle in a triangle constantly sum come ???180^\circ???.


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Side side Side (SSS)

If a pair that triangles have actually three proportional matching sides, then we deserve to prove the the triangles space similar. The reason is because, if the matching side lengths are all proportional, then that will force matching interior angle procedures to be congruent, which method the triangles will be similar.

???\fracad=\fracbe=\fraccf???


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Side Angle next (SAS)

If a pair of triangles have one pair of equivalent congruent angles, sandwiched in between two pairs of proportional sides, then we deserve to prove the the triangles space similar.

???\fracad=\fracbe???


Theorems for proving that 2 triangles space similar


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Which organize proves that the triangles space similar?

Example

Are the triangle similar? i beg your pardon theorem proves that they’re similar and finish the similarity statement.

???\triangle ABC\sim \triangle???____


We understand from the figure that ???\angle A\cong\angle Z=76^\circ???. Therefore we have a pair of congruent angles, and also we have to see if we likewise have proportional side lengths.

???\frac\overlineZY\overlineAC=\frac183=6???

???\frac\overlineZX\overlineAB=\frac549=6???

We have the very same ratio in between corresponding next lengths.

Putting all this together, we deserve to see say that the triangle are comparable by next Angle next (SAS). Once we match up the matching parts, the similarity declare is ???\triangle ABC\sim \triangle ZXY???.


In a pair of similar triangles, all three matching angle pairs room congruent and also corresponding side pairs space proportional.


Example

Are the triangle similar? i beg your pardon theorem proves the they’re comparable and complete the similarity statement.

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???\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 also remember that vertical angles space congruent come one another.


Putting all this together, we have the right to see say the the triangle are comparable by angle Angle (AA). As soon as we match up the corresponding parts, the similarity statement is ???\triangle WXY\sim \triangle VZY???.


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learn mathKrista KingOctober 10, 2020math, learn online, online course, digital math, geometry, similarity, comparable triangles, similarity theorems, triangle similarity statements, triangle similarity theorems, AA, SAS, SSS, SSA, proportional sides, proportional next lengths, congruent angles, AA theorem, SSS theorem, SAS organize
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