Theorems for Similar Triangles

1
Theorems for Similar Triangles

• Lesson 7-5

2
Proportional Lengths

• Lesson 7-6

3
Todays Scripture

• " And ye shall measure from without the city on
  the east side two thousand cubits, and on the
  south side two thousand cubits, and on the west
  side two thousand cubits, and on the north side
  two thousand cubits and the city shall be in the
  midst this shall be to them the suburbs of the
  cities." 
• Num.355

4
Objectives

• Lesson 7-5
• Use the SAS Similarity Theorem and the SSS
  Similarity Theorem to prove.
• Lesson 7-6
• Apply the Triangle Proportionality Theorem and
  its corollary.
• State and apply the Triangle Bisector Theorem.

5
Theorem 7-1

• SAS Similarity Theorem
• If an angle of one triangle is congruent to an
  angle of another triangle and the sides including
  those angles are in proportion, then the
  triangles are similar.

6
Theorem 7-2

• SSS Similarity Theorem
• If the sides of two triangles are in proportion,
  then the triangles are similar.

7
Example 1

• Can the information given be used to prove ?PQR
  ?TSP? If so, how?

a. SP 8, TS 6, PT 12, QR 12, PQ 9, RP
18
b. SP 7, TS 6, PQ 9, QR 10.5, m?S 80,
m?QPR m?QRP 100
8
Solutions
PT
12
2

• Comparing the longest sides,
• Comparing the shortest sides,
• Comparing the remaining sides,
• , so ?PQR ?TSP by the
• SSS Similarity Theorem.

RP
18
3
TS
6
2


PQ
9
3
SP
8
2


QR
12
3
PT
TS
SP


RP
PQ
QR
9
12
18
Q
S
12
12
6
8
9
8
6
P
T
12
R
18
9

• b. Since m?QPR m?QRP 100, m?Q 80 m?S.
• Comparing the corresponding sides that
  include ?S and ?Q,
• (7 is 2/3
  of 10.5)
• and

SP
7
14
2



QR
10.5
21
3
80
10.5
9
SP
TS
TS
6
2



QR
PQ
PQ
9
3
80
7
6
So ?PQR ?TSP by the SAS Similarity Theorem
6
10
Individual Practice

• Can the two triangles shown be proved similar? If
  so, state the similarity and which similarity
  postulate or theorem you would use.

11
Solutions

• 1. SAS Similarity Theorem ?ABC ?EFD

AB
6
1


EF
12
2
m?B 90 m? F
CB
7
1


DF
14
2
12
Solutions

• 2. SSS Similarity Theorem ?GHI ?LKJ

HI
3
1


KJ
9
3
GH
6
1


KL
18
3
GI
8
1


LJ
24
3
13
Solutions

• 3. AA Similarity Post. ?MNO ?PQO

m?N 60 m?Q
m? NOM 40 m?QOP
14
Solutions

• 4. No We can not determine the VW ratio to ST or
  the m?R to m?U

15
Solutions

• 5. SAS Similarity Theorem ?XYZ ?BCA
• or ?XYZ ?CBA

6. SAS Similarity Theorem ?DHE ?DGF
16
Proportional Lengths

• Lesson 7-6

17
Proportional Lengths

• If , then AC and XZ are said to be
  divided proportionally.

AB
XY

BC
YZ
A
B
C
X
Y
Z
18
Theorem 7-3

• Triangle Proportionality Theorem
• If a line parallel to one side of a triangle
  interests the other two sides, then it divides
  those sides proportionally.

19

• The properties of proportions allow the Triangle
  Proportionality Theorem to justify many
  equivalent proportions.

DG
DH
larger piece
larger piece


smaller piece
GE
HF
smaller piece
GE
HF
smaller piece
smaller piece


DE
DF
whole side
whole side
DG
DH
larger piece
larger piece


DE
DF
Whole side
whole side
GE
DG
smaller piece
larger piece
whole side
DE




HF
DH
smaller piece
whole side
larger piece
DF
?DGH ?DEF, so some of these equivalent
proportions could have been justified by
similarity postulates or theorems. With so many
equivalent proportions, most exercises can be
done more than one way.
20
Example 1

• Find the value of x.

Solution
20
15
(or
)
x
28


8
x - 15
15
20
20x 420
x 21
21
Individual Practice

• Find the value of x.

22
1.
27
3

45
5
3
33

5
x 33
3x 99 165
3x 66
x 22
23
2.

• 9

3

21
7
x
3

16 x
7
7x 48 3x
4x 48
x 12
24
3.
10
5

8
16
5
7.5

8
7.5 x
37.5 5x 60
5x 22.5
x 4.5
25
4.
12
4

27
9
4
x

9
20 x
80 4x 9x
80 5x
16 x
26
5.
10
5

14
7
x
5

7
x 6
7x 5x 30
2x 30
x 15
27
6.
12
3

16
4
3
x

4
20
4x 60
x 15
28
Corollary
Remember If three parallel lines cut off
congruent segments on one transversal, then they
cut off congruent segments on every transversal.

• If three parallel lines intersect two
  transversals, then they divide the transversal
  proportionally

29
Example 2

• Find the value of x.

Solution
4
x

16
25 - x
1
x

4
25 - x
25 x 4x
25 5x
x 5
30
Individual Practice

• Find the value of x.

7.
8.
9.
31
Solutions

• 7.

12
x
8.
9.

27
x 10
5
x - 5

15
12.5

18
x
4
8
15x 150
27x 18x 180
4x - 20 40
15x 150
9x 180
4x 60
x 10
x 20
x 15
32
Theorem 7-4

• Triangle Angle-Bisector Theorem
• If a ray bisects an angle of a triangle, then it
  divides the opposite side into segments
  proportional to the other two sides.

33
Example 3

• Find the value of x.

Solution
27
x

40 - x
45
3
x

40 - x
5
5x 120 3x
8x 120
x 15
34
Individual Practice

• Find the value of x.

10.
11.
12.
35
Solutions
x
10
x
12.5
12.
x
24
11.

• 10.

15 - x
8
20
12
10
18
10x 100
x
1
x
4


15 - x
2
12
x 10
3
15 - x 2x
3x 48
15 3x
x 16
5 x
36
Homework

• (p.266-268) 1-10
• (p.272) 1-10 even