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# Triangle Similarity Theorems

## Angle-Angle Similarity Theorem

If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

If and then

### Proof

Angle-Angle Similarity Theorem

Consider two triangles and whose two corresponding angles are congruent.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, can be dilated with the scale factor about forming the new triangle

Since a dilation is a similarity transformation, it can be concluded that and are similar triangles. Next, it has to be proven that a rigid motion that maps onto exists. The corresponding angles of similar figures are congruent, so and are congruent angles.
Additionally, since is congruent to by the Transitive Property of Congruence, is congruent to
The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.
In this case, the scale factor is Applying the Transitive Property of Equality, an equation can be formed and simplified.
It has been obtained that the two angles and the included side of are congruent to the corresponding two angles and the included side of
Therefore, by the Angle-Side-Angle (ASA) Congruence Theorem, the two triangles are congruent.
Since congruent figures can be transformed into each other using rigid motions, and and are congruent triangles, there is a rigid motion placing onto
The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps onto

Therefore, it can be concluded that and are similar triangles.

The proof is now complete.

## Side-Side-Side Similarity Theorem

If corresponding sides of two triangles are proportional, then the triangles are similar.

If then

### Proof

Consider two triangles and whose corresponding sides are proportional.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, can be dilated with the scale factor about forming the new triangle

Because dilation is a similarity transformation, it can be concluded that and are similar triangles. Now, it has to be proven that a rigid motion that maps onto exists. The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.
In this case, the scale factor is Since all of the sides of and are proportional, the scale factor can be expressed by any of the following ratios.
Applying the Transitive Property of Equality, three equations can be formed and simplified.
These relations imply that the three sides of are congruent to the three sides of Therefore, by the Side-Side-Side (SSS) Congruence Theorem, the two triangles are congruent.
Since congruent figures can be transformed into each other using rigid motions, and and are congruent triangles, there is a rigid motion placing onto

The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps onto

Therefore, it can be concluded that and are similar triangles.

The proof is now complete.

## Side-Angle-Side Similarity Theorem

If two sides of a triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

If and then

### Proof

Consider two triangles and whose two pairs of corresponding sides are proportional and the included angles are congruent.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, can be dilated with the scale factor about forming the new triangle

Because dilation is a similarity transformation, it can be concluded that and are similar triangles. Now, it has to be proven that a rigid motion that maps onto exists. The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.
In this case, the scale factor is Since and are proportional to and respectively, the scale factor can be expressed by any of the following ratios.
Applying the Transitive Property of Equality, three equations can be formed and simplified.
These relations imply that the two sides of are congruent to the corresponding two sides of Moreover, the included angles and are also congruent.
Therefore, by the Side-Angle-Side (SAS) Congruence Theorem, the two triangles are congruent.
Since congruent figures can be transformed into each other using rigid motions, and and are congruent triangles, there is a rigid motion placing onto

The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps onto

Therefore, it can be concluded that and are similar triangles.

The proof is now complete.