SAS Similarity: Difference between revisions
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==Definition== | ==Definition== | ||
SAS stands for Side-Angle-Side, for two [[triangle]]s to be [[similar|similar triangles]] by SAS similarity, they must have a pair of congruent | ===AoPS=== | ||
SAS stands for Side-Angle-Side, for two [[triangle]]s to be [[similar|similar triangles]] by SAS similarity, they must have a pair of congruent angles 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 <math>m\angle CAB = m\angle FDE</math> and <math>\dfrac{CA}{FD} = \dfrac{AB}{DE}</math>, then the triangles are similar by SAS similarity. | |||
==See Also== | |||
*[[Similarity]] | |||
*[[Congruence]] | |||
==Categories== | ==Categories== | ||
[[Category:Geometry]] [[Category:Mathematics]][[Category:Stubs]] | |||
Latest revision as of 20:12, 28 January 2021
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 angles 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]](http://latex.artofproblemsolving.com/1/7/8/17873a662288147a81102d3f98300c4e3d7f4817.png)
![[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]](http://latex.artofproblemsolving.com/9/8/4/984781cf0bae47382fd022cb1da382c723e8fbdf.png)
If 
and 
, then the triangles are similar by SAS similarity.
