Difference between revisions of "SAS Similarity"
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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.
