Download Triangle Congruence and Similarity

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Name
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California Standards
Geometry 5.0
Students prove that triangles are congruent or similar, and they are able
to use the concept of corresponding parts of congruent triangles.
Triangle Congruence and Similarity
In two congruent figures, all the parts of one figure are congruent to the
corresponding parts ofthe other figure.
Corresponding angles:
LA :=LF,LB:=
LE,LC:=
B
E
LD
Corresponding sides:
AB :=FE, BC:=ED,AC:=FD
~~
C
6ABC:=
A
F
0
6FED
When you write a congruence statement for two polygons,
always list the corresponding vertices in the same order.
Side-Side-Side (SSS) Congruence
Postulate
If three sides of one triangle are congruent to
three sides of a second triangle, then the two
triangles are congruent. 6ABC:= 6PQR
Side-Angle-Side (SAS) Congruence
Postulate
If two sides and the included angle of one
.triangle are congruent to two sides and the
included angle of a second triangle, then the
two triangles are congruent. 6DEF:= 6STU
B
A
Q
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If two angles and the included side of one
triangle are congruent to two angles and the
included side of a second triangle, then the two
triangles are congruent. 6DEF:= 6MNO
Hypotenuse-Leg (HL) Congruence
Theorem
.
c:
U
M~
F
J
If two angles and a non-included side of one
I
triangle are congruent to two angles and a non~
included side of a second triangle, then t4e two I G
triangles are congruent. 6 GHI:= 6 VWX
Geometry Standards
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Angle-Angle-Side (AAS) Congruence
Theorem
California Standards Review and Practice
Cl.
E.
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and a leg of a
second right triangle, then the two triangles are ~~
congruent. 6JKL:= 6XYZ
IK
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Angle-Side-Angle (ASA) Congruence
Postulate
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Triangle Similarity.,
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Date
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Example
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Angie-Angie (AA) Similarity
Postulate
!l!
PL yR
K
Q
If two angles of one triangle are congruent to
two angles of another triangle, then the two
triangles are similar. l1JKL ~ £"PQR
J
P
Side-Side-Side (SSS) Similarity
Theorem
B
If the corresponding side lengths of two
triangles are proportional, then the triangles
are similar.
AB
JK
K
~
A
J
4'
KL
If an angie 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.
SU
MO
Cl.
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Example 1
L
= BC = AC. l1ABC ~ l1JKL
Side-Angle-Side (SAS) Similarity
Theorem
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C
=
M
s~~
U
T
ST . l1STU ~ l1MNO
MN'
Prove Triangles Are Congruent
Write a proof.
GIVEN ~
A
B
M is the midpoint of AC.
M is the midpoint of BD.
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PROVE ~
l1AMB =: l1CMD
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Solution
-Statements
Reasons
1. M is the midpoint ~f AC.
1. Given
M is the midpoint of BD.
--
--
2.AM=:MC
2. Definition of Segment Midpoint
BM=:MD
3. LAMB =: L CMD
3. Vertical Angles Congruence Theorem
4. l1AMB =: l1CMD
4. SAS Congruence Postulate
California Standards Review and Practice
Geometry Standards
15
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Example 2
Date
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Determine Information to Show Congruence
State the third congruence that must be given to prove that LABC == LPQR using the
indicated postulate or theorem.
B
Q
a. LB == LQ, LC == LR
Use the ASA Congruence Postulate.
b.BC == QR, LB == LQ
Use the SAS Congruence Postulate.
c
Solution
R
a. Two angles in the first triangle are congruent to two angles in the second triangle. To
use the ASA Congruence Postulate, we need to know that the included side in the
first triangle is congruent to the included side in the second triangle, or BC == QR.
c. One side and one angle in the first triangle are congruent to one side and one angle
in the second triangle. To use the SAS Congruence Postulate, we need to know that
another side of the first triangle is congruent to the corresponding side of the second
triangle, such that the congruent angles are the included angles. So, AB == PQ.
Example 3
Determine Whether Triangles Are Congruent
Decide whether the congruence statement is true. Explain your reasoning.
w
a. LWYZ== LYWX
.:;>
A
z
. b. L VXY == L zxw
X
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V
X
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c. LJKL
== LMNO
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Solution
a. Yes, by the HL Congruence Theorem. L WYZ is a right angle by the Corresponding
. Angles Postulate. WY == WYby the Reflexive Property of Congrµent Segments, and
ZW == XY is given.
b. Yes, by the AAS Congruence Theorem. LX== LXby the Reflexive Property of
Congruent Angles and L V == L Z is given. ZW = ZT + TWand VY = VT + TY by
the Segment Addition Postulate. ZW = VYby the Transitive Property of Equality,
and ZW == VY by the Definition of Congruent Segments.
c. No; SSA is not one of the triangle congruence postulates or theorems.
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California Standards Review and Practice
Geometry Standards'
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Name
Example 4
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Use Corresponding
Date
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Parts
Write a congruence statement for the triangles. Identify an pairs of congruent
corresponding parts.
p
Solution
== 6.RPQ.
The diagram indicates that 6.ABC
Example 5
Corresponding
angles
LA
Corresponding
sides
AB
ShowTriangles
== LR, LB == LP, L C == L Q
== RP, BC == PQ, CA == QR
Are Similar
Show that the triangles are similar and write a similarity statement. Explain your
reasomng.
P 3
T
~R
4
S
16
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SQlution
D.-
Since we know the lengths of the sides, calculate the ratios of corresponding sides.
co
E
o
u
QS
PT -
.c=
~
~
;:
c
.8
8
4
QR
PR
10 - '5
= ;;
3
12
12 -
+
12
4
15 - '5
SR
16
16
4
TR - 4 + 16 - 20 - '5
= ~~. = ~, thus the triangles are similar by the SSS Similarity Theorem.
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6. TPR ~ 6.SQR by the SSS Similarity Theorem.
Answer
:::J
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:r:
Example.6
Prove Triangles Are Similar
GIVEN ~
PROVE ~
KP==LP,JL = 21,KM=
LQ = 24,NK= 16
14,
J
K
<
M
L
<;
J
6.JQL ~ 6.MNK
Solution
L QLJ
== L NKM by the Base Angles Theorem.
JL _ ~ -1.
KM
14
2
LQ
NK
.
Q
24_1.
16
2
The measures of the corresponding sides that include angles QLJ and NKM are
proportional, so the triangles are similar by the SAS Similarity Theorem.
CaliforniaStandardsReview and Practice
Geometry Standards
17
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Exercises
1. /:"IKL and /:"PQR are two triangles such that
L K := L Q. Which of the following is sufficient
5. In the figure below, HI bisects L KHI and L KII.
H
to prove the triangles are similar?
@ JK=PQ
©
K
LJ is right.
2. In the figure below,
wz II XY.
J
V
Which theorem or postulate can be used to prove
/:"HKI:= /:"HIJ?
I
.
". Z
X
y
@ ASA
®
AAS
©
®
SSS
SAS
6. In the figure below, L P := LX .
Which theorem or postulate can be used to prove
/:,.VWZ~
X
/:"VXY?
@ ASA
®
©
AAS
®
SAS
P
SSS
Z
3. In the figure below, /:,.ABE:= !iDCE.
R~
Ai"":"
B
oIL
Which of the following would be sufficient to
prove the triangles are similar?
-c
Which theorem or postulate can be used to prove
/:"CDB:= /:"BAC?
@ ASA
®
©
SSS
4. In the figure below, PQ
I
sv
©
PQ:=SR
®
AAS
AQ
•
I
®
RP
ZX
RQ
€
~
ZY
..r::
XY
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::c
Ci
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'6
co
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SR:=QR
California Standards Review and Practice
Geometry Standards
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Which theorem or postulate can be used to prove
/:"DEF:= /:,.GHF?
@ ASA
18
0..
®
would be enough
PS:=QR
co
PQ
I
®
®
>c
RP
ZX
7. In the, figure below, ED ..1 DF, HG..l GF, F is
the midpoint of DG.
II SR.
Which additional information
to prove /:"PQS:= /:,.RSQ?
@ PQ:=PS
SAS
y
Q
®
SSS
©
SAS
®
HL