SAS Similarity Theorem
SAS Similarity 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, then the triangles are similar.
Given: Two Triangles ABC and DEF such that angle A equals angle D, and AB/DE equals AC/DF.
Prove that: Triangle ABC ~ Triangle DEF
| Statements | Reasons |
| 1) AB = DP ; ∠A = ∠D and AC = DQ | 1) Given and by construction |
| 2) ΔABC ≅ ΔDPQ | 2) By SAS postulate |
| 3) AB AC ---- = ------ DE DF | 3) Given |
| 4) DP DQ ---- = ------ DE DF | 4) By substitution |
| 5) PQ || EF | 5) By converse of basic proportionality theorem |
| 6) ∠DPQ = ∠E and ∠DQP = ∠F | 6) Corresponding angles |
| 7) ΔDPQ ~ ΔDEF | 7) By AAA similarity |
| 8) ΔABC ~ ΔDEF | 8) From (2) and (7) |