There room three easy methods to prove similarity. These approaches are lot like those employed come prove congruence--they are approaches toshow that all corresponding angles are congruent and all corresponding sides room proportional without in reality needing to recognize the measure of every six parts of each triangle.

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AA (Angle-Angle)

If two pairs of equivalent angles in a pair that triangles are congruent, then the triangles are similar. We know this because if 2 angle pairs space the same, then the 3rd pair must also be equal. As soon as the 3 angle pairs space all equal, the 3 pairs of sides must likewise be in proportion. Picture three angles of a triangle floating around. If they space the vertices that a triangle, lock don"t identify the dimension of the triangle through themselves, because they can move farther away or closer to every other. However when lock move, the triangle castle create constantly retains its shape. Thus, castle always form similar triangles. The diagram listed below makes this much an ext clear.

Figure %: three pairs that congruent angle determine similar trianglesIn the above figure, angle A, B, and also C are vertices the a triangle. If one angle moves, the various other two must relocate in accordance to produce a triangle. For this reason with any type of movement, the three angles relocate in concert to develop a new triangle through the very same shape. Hence, any kind of triangles with three pairs of congruent angles will be similar. Also, keep in mind that if the 3 vertices are exactly the very same distance from each other, climate the triangle will certainly be congruent. In other words, congruent triangles room a subset of comparable triangles.

SSS (Side-Side-Side)

Another means to prove triangles are similar is by SSS, side-side-side. If the steps of matching sides room known, then their proportionality have the right to be calculated. If all three pairs are in proportion, climate the triangles space similar.

Figure %: If all three pairs of political parties of corresponding triangles room in proportion, the triangles space similar

SAS (Side-Angle-Side)

If two pairs of equivalent sides are in proportion, and also the had angle of each pair is equal, then the 2 triangles they kind are similar. Any type of time two sides of a triangle and their contained angle space fixed, climate all 3 vertices of that triangle room fixed. V all three vertices fixed and also two of the bag of political parties proportional, the 3rd pair of political parties must likewise be proportional.

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Figure %: 2 pairs that proportional sides and also a pair the equal included angles determines comparable triangles


These are the key techniques for proving congruence and also similarity. Through these tools, we can now perform two things. Given minimal information about two geometric figures, we may have the ability to prove their congruence or similarity. offered that numbers are congruent or similar, we deserve to deduce information about their matching parts that us didn"t formerly know. The link between the matching parts of a triangle and also the whole triangle is a two-way street, and also we can go in whichever direction we want.