TRIANGLE THEOREMS
Theorem A
If 2 triangles have equal
altitudes, then the ratio
of their areas is equal to
the ratio of their bases.
Theorem B
2 triangles on the same base and between 2 parallels are
equal in area.
B
C
Theorem C
If a line is parallel to 1 side of a triangle and intersects the other 2 sides
in distinct points, then it divides the sides proportionally. Conversely,
if a line divides 2 sides of a triangle proportionally, then the line is
parallel to the third side.
Theorem
E
The
bisector of
an angle of
a triangle
separates
the
opposite
side into
segments
whose
lengths
are
proportion
al to the
lengths of
the other
2 sides.
ΔADC
B
⊥ BD
AC || ED
ΔABD
⊥ BD
A
AD is common base
BC || AD
ΔBDC ⊥
BD
B
D
E
AABD = AACD
Bases:
AC, AD, DC
A D C
AABC AC AABC AC AADB AD
=
,
=
,
=
AABD AD ABDC DC ACDB AC
A
D
C
AE DC
=
EB BD
AE DC
=
AB BC
EB BD
=
AB BC
B
A
BD CD
=
AB AC
Theorem F
Any 2 corresponding angle
bisectors of similar triangles
are proportional to the
R
corresponding sides.
D
Q
K
T
L
P
F
A
C
LP LM H
= RS B
RT
M
D
J
AD AB
=
FJ FG
G
S
Theorem I
The ratio of the perimeters of 2
similar triangles is equal to the
ratio
of
any pair of
E
corresponding
B sides.
D
C
A
Theorem J
If 2 triangles are similar, then the ratio of their areas equals the square of the ratio of the lengths of any 2 corresponding
sides.
F
Definition of Similar Triangles
2 triangles are similar if and
only if the corresponding
angles are congruent and the
E the corresponding
lengths of
sides are proportional.
A
Theorem G
Any 2 corresponding altitudes of similar triangles are proportional to the corresponding sides.
B
E
B
PABC AB
=
PDEF DE
A
ΔABC ~ ΔDEF
AABC AB
= DE
FA
DEF
D
C
AAA Similarity Theorem
If there exists a correspondence between the vertices of 2 triangles such that the 3 angles of 1 triangle are congruent to the
corresponding angles of the second triangle, respectively, then the 2 triangles are similar.
A
∠BAC ≅ ∠EDF
B
∠ABC ≅ ∠DEF
C∠BCA ≅ ∠EFD
AB BC AC
= =
DE EF DF
H
D
E
F
∠CAB ≅ ∠HFG
∠ABC ≅ ∠HFG
G ∠ACB ≅ ∠FHG
ΔABC ~ ΔFGH
Theorem H
Any
2
corresponding
medians
ofD
A triangles
similar
are
proportional to
the
C
corresponding
sides.
Theorem K
The ratio of the
areas
of
2
similar triangles
is equal to the
B
square of the
ratio of the
corresponding
perimeters.
A
SAS Similarity
Theorem
If 2 pairs of
corresponding
sides
of
2
triangles
are
S
proportional
and
the
included angles
are congruent,
then the 2
triangles
are
similar.
B
Z
W
Y
E
D
C
F
R
RS ST
=
XY YZ
∠RST ≅ ∠XYZ
ΔRST ~ ΔXYZ
SSS Similarity Theorem
If all 3 pairs of corresponding
sides of 2 triangles are
proportional, then theM 2
triangles are similar.
Right Triangle Similarity Theorem
In any right triangle, the altitude to the hypotenuse divides the right triangle into 2 right triangles, which are similar to each
other and to the given right triangle.
C
J
Q
C
C
P
A
K
D B A
D
D
ΔACD ~ ΔABC, ΔCBD ~ ΔABC,
ΔACD ~ ΔCBD
B
Geometric
Mean Theorem
The geometric
mean between
the 2 segments
is the altitude
drawn from the
vertex of the
T
X
right
triangle
into which the
foot of the
altitude divides
the
Hypotenuse.
The geometric
mean for 2
numbers a and
b is x such that x
= sqrt(ab)
JK
KL
LJ
= =
MP PQ QM
ΔJKL ~ ΔMPQ
L
C
x = √ab
a
b
x
A
Pythagorean Theorem
In any right triangle, the sum of
the squares of the lengths of
the legs is equal to the square
of the length of the
hypotenuse.
45-45-90 Triangle Theorem
In a 45-45-90 triangle, the length of the hypotenuse is equal to sqrt(2) times the length of a leg.
hypotenuse
45⁰
Hyp = leg√2
leg
45⁰
b
c2 = a2 + b2
c
leg
a
Hyp
= 2short
hypotenuse
Long = short√3
30⁰
60⁰
short
Theorem A
If 2 triangles have equal altitudes, then the
ratio of their areas is equal to the ratio of
their bases.
ΔADC ⊥ BD
ΔABD ⊥ BD
ΔBDC ⊥ BD
B
Bases: AC, AD, DC
A
D C
AABC AC AABC AC AADB AD
=
,
=
,
=
AABD AD ABDC DC ACDB AC
Ex.
Given: AD = 8; AABC = 20; AABD = 10
Find: DC
Sol’n:
20
=
10
AC
10AC
8
10
=
160
Theorem B
2 triangles on the same base and between
2 parallels are equal in area.
B
Ex.
Given: AB = 5; AC = 9; BC = 10
Find: AABC
Sol’n: AABC = √s(s-a)(s-b)(s-c)
a+b+c
Definition of Similar Triangles
2 triangles are similar if and only if the corresponding
angles are congruent and the lengths of the corresponding
sides are proportional.
E
a
c
50⁰
A
2
=
5+9+10
2
20
b
= 12
B
F
C
4
G
D
Ex.
Given: ΔABC ~ ΔDEF; PABC = 58
B
E
A
D
A
D
BD CD
=
AB AC
C
C
Ex.
Given: AE=6; EB= 5; BD=4
Find: DC
7
DC
2DC
28
Sol’n: =
=
2
4
2
2
Ans: DC = 14
F80⁰
50⁰
Theorem E
The bisector of an angle of a triangle
separates the opposite
side into
segments whose lengths are proportional
to the lengths of the other 2 sides.
B
Theorem G
Any 2 corresponding altitudes of similar triangles are
proportional to the corresponding sides.
Theorem J
If 2 triangles are similar, then the ratio of their areas
equals the square of the ratio of the lengths of any 2
corresponding sides.
AAA Similarity Theorem
If there exists a correspondence between the vertices of
2 triangles such that the 3 angles of 1 triangle are
congruent to the corresponding angles of the second
triangle, respectively, then the 2 triangles are similar.
5
C
D
BC || AD
AABD = AACD
Theorem F
Any 2 corresponding angle bisectors of similar triangles are
proportional to the corresponding sides.
Theorem I
The ratio of the perimeters of 2 similar triangles is equal to
the ratio of any pair of corresponding sides.
80⁰
AC || ED
AE DC
=
EB BD
AE DC
=
AB BC
EB BD
=
AB BC
AABC = √12(12-5)(12-9)(12-10)
Ans: AABC = 6√14
DC = AC – AD = 16-8
Ans: DC = 8
B
C
A
AD is common base
but s =
10
AC = 16
Theorem C
If a line is parallel to 1 side of a triangle
and intersects the other 2 sides in distinct
points, then it divides the sides
proportionally. Conversely, if a line
divides 2 sides of a triangle proportionally,
then the line is parallel to the third side.
A
E
D
30-60-90
Triangle
Theorem
In a 30-60-90
triangle,
the
length of the
hypotenuse is
twice the length
of the shorter
leg and the
length of the
longer leg is
sqrt(3)
times
the length of
the shorter leg.
Theorem H
Any 2 corresponding medians of similar triangles are
proportional to the corresponding sides.
Theorem K
The ratio of the areas of 2 similar triangles is equal to the
square of the ratio of the corresponding perimeters.
SAS Similarity Theorem
If 2 pairs of corresponding sides of 2 triangles are
proportional and the included angles are congruent, then
the 2 triangles are similar.
long
Find: a, b, c
a
b
Sol’n: = =
a
5
c
5
4
2) =
20
c
4
a
1) =
5
4
b
20
20a = 5b b = 4a
4a = 5c c = a 3)P = a + b + c = 58
4
5
a + 4a + a = 58
a = 10
5
5a + 20a + 4a = 290 b = 4a = 4(10)
29a 290
4
4
=
a = 10 c = a = (10)
29
29
5
5
Ans: a = 10 ; b = 40 ; c = 8
SSS Similarity Theorem
If all 3 pairs of corresponding sides of 2 triangles are
proportional, then the 2 triangles are similar.
Pythagorean Theorem
In any right triangle, the sum of the squares of the lengths
of the legs is equal to the square of the length of the
hypotenuse.
c c2 = a2 + b2
b
Right Triangle Similarity Theorem
In any right triangle, the altitude to the hypotenuse
divides the right triangle into 2 right triangles, which are
similar to each other and to the given right triangle.
45-45-90 Triangle Theorem
In a 45-45-90 triangle, the length of the hypotenuse is
equal to sqrt(2) times the length of a leg.
Geometric Mean Theorem
The geometric mean between the 2 segments is the
altitude drawn from the vertex of the right triangle into
which the foot of the altitude divides the Hypotenuse.
The geometric mean for 2 numbers a and b is x such that
x = sqrt(ab)
30-6-90 Triangle Theorem
In a 30-60-90 triangle, the length of the hypotenuse is
twice the length of the shorter leg and the length of the
longer leg is sqrt(3) times the length of the shorter leg.
C
ΔACD
(hypotenuse)
x30⁰
y
x
6 (long)
60⁰
6
60⁰
A √3
Hyp = 2short
(short) m Long = short
m D
a
Ex.
Find: Perimeter ; Area
Sol’n: x = 2m
6 = m√3
m=
6
30⁰
n
AB = 2x = 2(4√3) = 8√3
y = x√3 = 4√3 (√3) = 12
√3
( )= 2√3
√3 √3
x = 2m = 2(2√3) = 4√3
1
Area = (8√3)(6) Perimeter = 4√3 + 8√3 + 12
2
Ans: Area = 8√3 ; Perimeter = 12√3 + 12
B