Proofs with Similar Triangles Topic Index | Geometry Index | Regents Exam Prep Center

 Definition:

Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent.

There are three accepted methods of proving triangles similar:

AA

To show two triangles are similar, it is sufficient to show that two angles of one triangle are congruent (equal) to two angles of the other triangle.

 Theorem: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

SSS
for similarity

BE CAREFUL!!  SSS for similar triangles is NOT the same theorem as we used for congruent triangles. To show triangles are similar, it is sufficient to show that the three sets of corresponding sides are in proportion.

 Theorem: If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.

SAS
for
similarity

BE CAREFUL!!  SAS for similar triangles is NOT the same theorem as we used for congruent triangles.  To show triangles are similar, it is sufficient to show that two sets of corresponding sides are in proportion and the angles they include are congruent.

 Theorem: If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.

Once the triangles are similar:

 Theorem: The corresponding sides of similar triangles are in proportion.

Dealing with overlapping triangles:

 Many problems involving similar triangles have one triangle ON TOP OF  (overlapping) another triangle.  Since is marked to be parallel to , we know that we have

 Topic Index | Geometry Index | Regents Exam Prep Center Created by Donna Roberts