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Geometry: Congruence

Proving Similarity of Triangles

Problems

Problems

There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.

AA (Angle-Angle)

If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear.

Figure %: Three pairs of congruent angles determine similar triangles
In the above figure, angles A, B, and C are vertices of a triangle. If one angle moves, the other two must move in accordance to create a triangle. So with any movement, the three angles move in concert to create a new triangle with the same shape. Hence, any triangles with three pairs of congruent angles will be similar. Also, note that if the three vertices are exactly the same distance from each other, then the triangle will be congruent. In other words, congruent triangles are a subset of similar triangles.

SSS (Side-Side-Side)

Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.

Figure %: If all three pairs of sides of corresponding triangles are in proportion, the triangles are similar

SAS (Side-Angle-Side)

If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional.

Figure %: Two pairs of proportional sides and a pair of equal included angles determines similar triangles

Conclusion

These are the main techniques for proving congruence and similarity. With these tools, we can now do two things.

  • Given limited information about two geometric figures, we may be able to prove their congruence or similarity.
  • Given that figures are congruent or similar, we can deduce information about their corresponding parts that we didn't previously know.
The link between the corresponding parts of a triangle and the whole triangle is a two-way street, and we can go in whichever direction we want.

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