SAS Similarity

Definition

AoPS

SAS stands for Side-Angle-Side, for two triangles to be similar triangles by SAS similarity, they must have a pair of congruent angle and the two sides next to the angle must be proportional.

Mathwords Definition

Side-angle-side similarity. When two triangles have corresponding angles that are congruent and corresponding sides with identical ratios as shown below, the triangles are similar.

Diagram

[asy] dot((0,0)); label("A",(0,0),SW); dot((5,0)); label("B",(5,0),SE); dot((3,4)); label("C",(3,4),N); draw((0,0)--(5,0)--(3,4)--cycle); markscalefactor = 0.1; draw(anglemark((5,0),(0,0),(3,4))); [/asy] [asy] size((8cm)); dot((0,0)); label("D",(0,0),SW); dot((5,0)); label("E",(5,0),SE); dot((3,4)); label("F",(3,4),N); draw((0,0)--(5,0)--(3,4)--cycle); markscalefactor = 0.0675; draw(anglemark((5,0),(0,0),(3,4))); [/asy] If $m\angle CAB = m\angle FDE$ and $\dfrac{CA}{FD} = \dfrac{AB}{DE}$, then the triangles are similar by SAS similarity.

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