Transcript 6.5 – Prove Triangles Similar by SSS and SAS

6.5 – Prove Triangles Similar
by SSS and SAS
Geometry
Ms. Rinaldi
Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of two triangles
are proportional, then the triangles are similar.
EXAMPLE 1
Is either
Use the SSS Similarity Theorem
DEF or
GHJ similar to
ABC?
SOLUTION
Compare ABC and DEF by finding ratios of
corresponding side lengths.
Remaining sides
Shortest sides
Longest sides
CA = 16 = 4
BC 12 = 4
AB 8
4
=
=
=
3
3
FD
12
EF
9
DE 6
3
ANSWER All of the ratios are equal, so
ABC ~
DEF.
EXAMPLE 1
Use the SSS Similarity Theorem (continued)
Compare ABC and
GHJ by finding ratios of
corresponding side lengths.
Shortest sides
AB 8 1
GH = 8 =
Longest sides
Remaining sides
CA 16 1
JG = 16 =
BC 12 = 6
=
HJ 10 5
ANSWER
The ratios are not all equal, so
not similar.
ABC and
GHJ are
EXAMPLE 2
Use the SSS Similarity Theorem
Which of the three triangles
are similar? Write a similarity
statement.
EXAMPLE 3
Use the SSS Similarity Theorem
ALGEBRA
Find the value of x that makes
ABC ~
DEF.
SOLUTION
STEP 1 Find the value of x that makes corresponding
side lengths proportional.
4
x –1
=
12
18
4 18 = 12(x – 1)
72 = 12x – 12
7=x
Write proportion.
Cross Products Property
Simplify.
Solve for x.
EXAMPLE 3
Use the SSS Similarity Theorem (continued)
STEP 2 Check that the side lengths are
proportional when x = 7.
DF = 3(x + 1) = 24
BC = x – 1 = 6
AB ? BC
DE = EF
6
4
12 = 18
AB ? AC
DE = DF
8
4
=
12 24
ANSWER
When x = 7, the triangles are similar by the SSS
Similarity Theorem.
Use the SSS Similarity Theorem
EXAMPLE 4
Find the value of x that makes
XYZ ~ PQR
Q
Y
20
30
x+6
21
X
12
Z
P
3(x – 2)
R
Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle
of a second triangle and the lengths of the sides
including these angles are proportional, then the
triangles are similar.
EXAMPLE 5
Use the SAS Similarity Theorem
Lean-to Shelter
You are building a lean-to shelter starting from a tree
branch, as shown. Can you construct the right end
so it is similar to the left end using the angle
measure and lengths shown?
EXAMPLE 5
Use the SAS Similarity Theorem (continued)
SOLUTION
Both m A and m F equal = 53°, so A ~ F. Next,
compare the ratios of the lengths of the sides that
include
A and
F.
Shorter sides
Longer sides
AB 9
3
AC 15 3
=
=
FG 6
2
FH = 10 = 2
The lengths of the sides that include
A and
proportional.
F are
ANSWER
So, by the SAS Similarity Theorem, ABC ~ FGH.
Yes, you can make the right end similar to the left end
of the shelter.
EXAMPLE 6
Choose a method
Tell what method you would use to show that the
triangles are similar.
SOLUTION
Find the ratios of the lengths of the corresponding sides.
Shorter sides
Longer sides
BC 9
CA 18 3
3
EC = 15 = 5
CD = 30 = 5
The corresponding side lengths are proportional. The
included angles ACB and DCE are congruent
because they are vertical angles. So, ACB ~ DCE
by the SAS Similarity Theorem.
Choose a method
EXAMPLE 7
Explain how to show that the indicated triangles are
similar.
A.
SRT ~
PNQ
Explain how to show that the indicated triangles are
similar.
B.
XZW ~
YZX