SIMILARITY
THEOREMS
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Similarity Theorems in Triangles, Study notes of Descriptive Geometry
The Angle-Angle Similarity Postulate (AA~), Side-Angle-Side Similarity Postulate (SAS~), and Side-Side-Side Similarity Postulate (SSS~) for triangles, which establish the conditions for two triangles to be similar. The document also includes examples and exercises to help understand these concepts.
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2021/2022
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SIMILARITY
THEOREMS
Similarity in Triangles
Angle-Angle Similarity Postulate (AA~)-
If two angles of one triangle are
congruent to two angles of another
triangle, then the triangles are similar.
W
R
S
V
B
45
WRS BVS
because of the AA~
Postulate.
Similarity in Triangles
Side-Side-Side Similarity Postulate
(SSS~)- If the corresponding sides of two
triangles are proportional, then the
triangles are similar.
C
A
B
Q
R S
ABC QRS
because of the
SSS~ Postulate.
The scale factor is 1:5.
EXAMPLE
30 ° 30 °
Why aren’t these triangles
congruent?
What do we call these triangles?
So, how do we prove
that two triangles
really are congruent?
ASA (Angle, Side, Angle)
If two angles and the
included side of one
triangle are congruent
to two angles and the
included side of another
triangle,...
then the 2 triangles are CONGRUENT! F E D A C B
SAS (Side, Angle, Side)
If in two triangles, two
sides and the included
angle of one are
congruent to two sides
and the included angle
of the other,...
then the 2 triangles are CONGRUENT! F E D A C B
SSS (Side, Side, Side)
In two triangles, if 3
sides of one are
congruent to three sides
of the other,...
F E D A C B then the 2 triangles are CONGRUENT!
HA (Hypotenuse, Angle) If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent,... then the 2 triangles are CONGRUENT! F E D A C B
LA (Leg, Angle) If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent,... then the 2 triangles are CONGRUENT! A C B F E D
Example 1 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? F E D A C B
Example 2 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B F E D
The angle between two sides Included Angle HGI G
GIH
I
GHI
H
This combo is called side-angle-side, or just SAS. 19
Name the included angle: YE and ES ES and YS YS and YE Included Angle Y S E YES or E YSE or S EYS or Y The other two angles are the NON-INCLUDED angles. 20