Similar triangles are triangles that have the same shape but not necessarily the same size... PPP Similarity Theorem 3 pairs of proportional sides Six of those statements are true as a
Trang 2Similar triangles are triangles that have the same
shape but not necessarily the same size
A
C
B
D
F
E
When we say that triangles are similar there are several repercussions that come from it
∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
AB
DE BCEF ACDF
= =
Trang 31 PPP Similarity Theorem
3 pairs of proportional sides
Six of those statements are true as a result of the
similarity of the two triangles However, if we need to prove that a pair of triangles are similar how many of those statements do we need? Because we are working with triangles and the measure of the angles and sides are dependent on each other We do not need all six There are three special combinations that we can use to prove similarity of triangles
2 PAP Similarity Theorem
2 pairs of proportional sides and congruent angles between them
3 AA Similarity Theorem
2 pairs of congruent angles
Trang 41 PPP Similarity Theorem
3 pairs of proportional sides A
E
25
1 4
5 .
DF
m
AB
m = =
25
1 6
9
12 .
FE
m
BC
m = = 10 4 1 25
13 .
DE
m
AC
m = =
5
4 12
9.6
.4
Trang 52 PAP Similarity Theorem
2 pairs of proportional sides and congruent angles between them
G
L
66
0 5
7
LK
m
GH
m = =
66
0 5
10
KJ
m
HI
m = =
7
10.5
70°
70°
m ∠ H = m ∠ K
Trang 6The PAP Similarity Theorem does not work unless the congruent angles fall between the proportional
sides For example, if we have the situation that is
shown in the diagram below, we cannot state that the triangles are similar We do not have the information that we need
G
L
7
10.5
50°
50° Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively
Trang 73 AA Similarity Theorem
2 pairs of congruent angles M
Q
70°
70°
50°
50°
m ∠ N = m ∠ R
m ∠ O = m ∠ P ∆ MNO ∼ ∆ QRP
Trang 8It is possible for two triangles to be similar when
they have 2 pairs of angles given but only one of
those given pairs are congruent
87°
34°
34°
S
T
U
X Y
Z m ∠ T = m ∠ X
m∠S = 180°- (34° + 87°)
m∠S = 180°- 121°
m∠ S = 59°
m ∠ S = m ∠ Z
59°
59°
59°
34°
34°
Trang 9КОНЕЦ τ λο έ σ
fin
ito
la fin
sof
ﻥﺎﻴ ﺎﭙ
fin al
The end
K A T A P U S A N