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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
SV
B
4545
WRS BVS
because of the AA~
Postulate.
Similarity in Triangles
Side-Angle-Side Similarity Postulate
(SAS~)- If an angle of one triangle is
congruent to an angle of a second
triangle, and the sides including the
angles are proportional, then the
triangles are similar.
TEA CUP
because of the
SAS~ Postulate.
C
U P
T
E A
32
16 1232
28 21
The scale factor is 4:3.
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
BQ
R S
3
4
6
1530
20
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?
Ch
ris Gio
va
nello
, LB
US
D M
ath
Cu
rriculu
m O
ffice, 2
00
4
congruent polygons:
are polygons with congruent corresponding parts - their matching sides and angles
pg. 180
A X
B Y
ZC
D WPolygon ABCD Polygon XYZW
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
AAS (Angle, Angle, Side)
Special case of ASA
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-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 includedangle 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!
HL (Hypotenuse, Leg)
If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .
A
C
B
F
E
D
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
LL (Leg, Leg)
If both pair of legs 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
Ch
ris Gio
va
nello
, LB
US
D M
ath
Cu
rriculu
m O
ffice, 2
00
4
CPCTC:
corresponding parts of congruent triangles are
congruent
pg. 203
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
SY
E
YES or E
YSE or S
EYS or Y
The other two angles are the
NON-INCLUDED angles. 20
The side between two angles
Included Side
GI HI GH
This combo is called angle-side-angle, or just ASA. 21
Name the included side:
Y and E
E and S
S and Y
Included Side
SY
E
YE
ES
SY
The other two sides are the
NON-INCLUDED sides. 22
TRIANGLE PROPORTIONALITY
THEOREM
23
CONVERSE OF TRIANGLE
PROPORTIONALITY THEOREM
24
TRIANGLE MIDSEGMENT THEOREM
25
KEY CONCEPTS
A transversal is a line that cuts through two parallel
lines.
When a triangle contains a line that is parallel to one
of its sides, the two triangles formed can be proved
similar using the AA Similarity Postulate. Since
triangles are similar, their sides are proportional.
The three midsegments of a triangle form the
Midsegment Triangle.
26
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
F
G
H
K
J
Yes, FGH KJH because
of the AA~ Postulate
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.M
O R
G
H I6
10
3
4
No, these are not similar
because
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
A
X Y
B C
20
25
25
30
No, these are not similar
because
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
Yes, APJ ABC because of
the SSS~ Postulate.
A
P J
B C
3
5
2
3
8
3
Explain why these triangles are
similar. Then find the value of x.
3
5
4.5
x
These 2 triangles are similar
because of the AA~ Postulate.
x=7.5
Explain why these triangles are
similar. Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x=2.5
5
70 1103 3
x
Explain why these triangles are
similar. Then find the value of x.
22
1424
x
These 2 triangles are similar
because of the AA~ Postulate.
x=12
Explain why these triangles are
similar. Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x= 12
x
6
2
9
Explain why these triangles are
similar. Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x=8
15
4
x
5
Explain why these triangles are
similar. Then find the value of x.
These 2 triangles are similar
because of the AA~ Postulate.
x= 15
18
7.5 12
x
Please complete the Ways to
Prove Triangles Similar
Worksheet.
Side Splitter Theorem - If a line is parallel
to one side of a triangle and intersects the
other two sides, then it divides those sides
proportionally.
Similarity in Triangles
T
S U
R V
x 5
1610
You can either
use
or
Theorem
If three parallel lines intersect two
transversals, then the segments
intercepted are proportional.
a
b
c
d
Theorem
Triangle Angle Bisector Theorem -If a
ray bisects an angle of a triangle, then it
divides the opposite side on the triangle
into two segments that are proportional to
the other two sides of the triangle.
A
BC D