G-09 Congruent Triangles and their parts

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G-09 Congruent Triangles and
their parts
“I can name corresponding sides and
angles of two triangles.”
Reflexive Property
AB = AB
Symmetric Property
• If A = B, then B = A
Transitive Property
• If A = B and B = C, then A = C
Addition, Subtraction, Multiplication,
Division Property (=)
Distributive Property
• If A(B + C), then AB + AC
Or
• If (B + C)A, then BA + CA
Substitution
• If A = B, then A can be substituted for any B in
the expression
Angle/Segment Addition Postulate
Definition of Congruence
• If AB = CD, then AB  CD
• Congruent segments are segments that have
the same length.
• Congruent angles are angles that have the
same measure.
Definition of Vertical Angles
• Vertical angles are two nonadjacent angles
formed by two intersecting lines.
• Vertical Angles are congruent
• 1 and 2 are vertical angles
Definition of Perpendicular Lines
• Perpendicular lines intersect to form 90
angles.
• Perpendicular lines are form congruent angles
If ABC  CBD
then ABC  90 and CBD  90
so ABC  CBD
Definition of
Complementary/Supplementary Angles
• Complementary Angles: 2 angles that add up
to be 90°
mA  mB  90
• Supplementary Angles: 2 angles that add up
to be 90°
mA  mB  180
Definition of Midpoint/Bisector
• The midpoint M of AB is the pt that bisects, or
divides, the segment into 2 congruent
segments.
• (segments) If M is the midpt of AB, then AM =
MB
• An angle bisector is a ray that divides an angle
into two congruent angles.
JK bisects LJM; thus LJK  KJM.
Definition of Right Angles
• All right angles are congruent
• If A and B are right angles, then A  B
Third Angle Theorem
Definition of Congruent Triangles
• If two or more triangles have corresponding
angles and sides that are congruent, then
those triangles are congruent.
• In a congruence statement, the order of the
vertices indicates the corresponding parts.
Helpful Hint
When you write a statement such as ABC 
DEF, you are also stating which parts are
congruent.
Example 1
A. Given: ∆PQR  ∆STU
Identify all pairs of corresponding congruent
parts.
• Angles:
• Sides:
Example 1
B. Given: ∆ABC  ∆DEF
Identify all pairs of corresponding congruent
parts.
• Angles:
• Sides:
Example 1
C. Given: ∆JKM  ∆LKM
Identify all pairs of corresponding congruent
parts.
• Angles:
• Sides:
Example 2
A. Given: polygon ABCD  EFGH
• BCD  ________
Example 2
B. Given: polygon ABCD  EFGH
• BAD  ________
Example 2
C. Given: polygon DEFGH  IJKLM
• MLK  ________
Example 3a:
• Given: K is the midpt. of JL,
JM  LM , JKM  LKM , J  L
Prove: MKJ  MKL
Statement
Reason
Statement
K is the midpt. of JL
JK  KL
JM  LM
KM  KM
JKM  LKM
JKM and LKM
are right angles
JKM  LKM
J  L
KMJ  KML
MKJ  MKL
Reason
Given
Definition of Midpoint
Given
Reflexive Property
Given
Definition of Perpendicular
lines
Right angles are congruent
Given
Third Angle Thm.
Definition of Congruent
Triangles
Example 3b
Given: YWX and YWZ are right angles.
YW bisects XYZ. W is the midpoint of XZ. XY  YZ.
Prove: ∆XYW  ∆ZYW
Example 3b:
Statement
Reason
YWX and YWZ are right angles.
Given
YWX  YWZ
YW bisects XYZ
XYW  ZYW
W is mdpt. of XZ
Given
Given
XW  ZW
YW  YW
X  Z
XY  YZ
∆XYW  ∆ZYW
Given
Example 3c
Given: AD bisects BE.
BE bisects AD.
AB  DE, A  D
Prove: ∆ABC  ∆DEC
Example 3c:
Statement
A  D
BCA  DCE
ABC  DEC
AB  DE
AD bisects BE, BE bisects AD
BC  EC, AC  DC
∆ABC  ∆DEC
Reason
Given
Given
Given
Example 3d
• Given: PR and QT bisect each other.
PQS  RTS, QP  RT
• Prove: ∆QPS  ∆TRS
Example 3d:
Statement
QP  RT
PQS  RTS
PR and QT bisect each other
QS  TS, PS  RS
QSP  TSR
QSP  TRS
∆QPS  ∆TRS
Reason
Given
Given
Given
Example 3e
Use the diagram to prove the following.
Given: MK bisects JL. JL bisects MK. JK  ML.
JK || ML.
Prove: ∆JKN  ∆LMN
Example 3e:
Statement
JK  ML
JK || ML
JKN  NML
JL and MK bisect each other.
JN  LN, MN  KN
∆JKN ∆LMN
Reason
Given
Given
Given
Vert. s Thm.
Third s Thm.
Def. of  ∆s
Example 4a
Given: ∆ABC  ∆DBC.
1. Find the value of x.
2. Find mDBC.
Example 4b
• Given: ∆ABC  ∆DEF
1. Find the value of x.
2. Find mF.
Example 4c
• Given: ∆ABD  ∆CBD
1. Find the value of x.
2. Find AD.
Example 4d
• Given: ∆RSU  ∆TSU
1. Find the value of x.
2. Find UT.
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