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Geometry
6.4 – 6.5 – Prove Triangles Similar by AA, SSS and SAS
Learning Target: By the end of today’s lesson we will be able to successfully prove triangles are similar using AA, SSS,
and SAS postulates, where the corresponding angles are congruent and the corresponding sides are
proportional.
ANGLE –ANGLE (AA) SIMLARITY POSTULATE:
If two angles of one triangle are congruent to two angles
of another triangle, then the two triangles are similar.
∆JKL ~ ∆XYZ
1) Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning.
a)
b)
c)
2) Show that the two triangles are similar.
a) ∆RTV and ∆RQS
b) ∆LMN and ∆NOP
c) ∆BCD and ∆EFD
SIDE-SIDE-SIDE (SSS) SIMLARITY POSTULATE:
If the corresponding side lengths of two triangles are
proportional then the triangles are similar.
If
AB BC CA


, then ABC ~ RST.
RS
ST TR
3) Is either DEF or GHJ similar to ABC?
4) Find the value of x that makes ABC ~ DEF.
SIDE-ANGLE-SIDE (SAS) SIMLARITY POSTULATE:
If an angle of one triangle is congruent to an angle of a
second triangle and the lengths of the sides including these
angles are ___________________, then the triangles are similar.
If X  M and
ZX
XY

, then XYZ  MNP.
PM MN
5) A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow
that is 48 inches long. The chair casts a shadow that is 6 feet long. Are the triangles similar? If so, how tall is the chair?
90ooo
Triangle Similarity Postulate and Theorems:
AA Similarity Postulate: If A  D and B  E, then ABC ~ DEF.
(If 2 angles of 1 triangle = 2 angles of another triangle they are similar)
SSS Similarity Theorem: If
AB BC AC


, then ABC ~ DEF.
DE EF DF
(If all sides of 1 triangle are proportional to all sides of another triangle they are similar)
SAS Similarity Theorem: If A  D and
AB AC

, then ABC  DEF.
DE DF
(If all 2sides of 1 triangle proportional to 2 sides of another triangle and the included angles are =
then the triangles are similar)