Download Proving Triangles Similar

Geometry CP2 (Holt 7-3)
K.Santos
Angle-Angle Similarity
Postulate (AA~)

If two angles of one triangle are congruent to two
angles of another triangle, then the triangles are similar.
Q
Given: <A ≅ < Q
A
<B ≅ < R
B
C
R
S
Then: ∆ABC ~ ∆QRS
Side-Side-Side Similarity
Theorem (SSS~)

If the three sides of one triangle are proportional to the
three corresponding sides of another triangle, then the
triangles are similar.
A
S
Q
Given:
𝐴𝐵
𝑄𝑅
=
𝐵𝐶
𝑅𝑆
=
𝐴𝐶
𝑄𝑆
B
C
R
Then: ∆ABC ~ ∆QRS
Side-Angle-Side Similarity
Theorem (SAS~)

If two sides of one triangle are proportional to two
sides of another triangle and their included angles are
congruent, then the triangles are similar.
A
S
R
Given: <A ≅ < Q
𝐴𝐵
𝑄𝑅
=
𝐴𝐶
𝑄𝑆
B
Then: ∆ABC ~ ∆QRS
C
Q
Methods for proving two triangles
are congruent

Definition of congruent polygons (triangles)
rarely use—would have to show 6 things
AA~ Postulate
SSS~ Theorem
SAS~Theorem
Example

Why must the triangles be similar?
Z 18 V
12
Y
24
𝑉𝑌
𝑉𝑍
𝑙𝑖𝑡𝑡𝑙𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
=
𝑉𝑊
𝑉𝑋
𝑏𝑖𝑔 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
12
18
1
=
(both
reduce
to
)
24
36
2
proportional sides:
included angles----congruent vertical angles
SAS ~ Theorem
Write a similarity statement.
∆VYZ ~ ∆VWX
W
36
X
Example—Find missing
side

Find the missing value of x in the figure.
12
6
x
8
6
𝑥
=
8
12
8x = 6(12)
8x = 72
x=9
𝑙𝑖𝑡𝑡𝑙𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝑏𝑖𝑔 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
Indirect Measurement
Example

In sunlight, a telephone pole casts a 9-ft shadow. At the same time
a person 6-ft tall casts a 4-ft shadow. Use similar triangles to find
the height of the telephone pole.
telephone
person 6-ft
pole x-ft
4-ft
9-ft
height shadow
6 𝑓𝑡
𝑥 𝑓𝑡
=
4𝑓𝑡
9 𝑓𝑡
4x = 6(9)
4x = 54
x = 13. 5
𝑝𝑒𝑟𝑠𝑜𝑛
𝑡𝑒𝑙𝑒𝑝ℎ𝑜𝑛𝑒 𝑝𝑜𝑙𝑒
so the telephone pole is 13.5-ft tall