Transcript Understand corresponding parts of congruent triangles and prove

Chapter 4.3 Congruent Triangles
Objective: Understand corresponding parts of
congruent triangles and prove congruence by the
definition.
Check.4.38 Use the principle that corresponding parts of
congruent triangles are congruent to solve problems.
CLE 3108.4.8 Establish processes for determining
congruence and similarity of figures, especially as related to
scale factor, contextual applications, and transformations.
Spi.4.11 Use basic theorems about similar and congruent
triangles to solve problems.
Spi.4.12 Solve problems involving congruence, similarity,
proportional reasoning and/or scale factor of two similar
figures or solids.
Congruent Triangles
• ALL corresponding parts of congruent triangles are
congruent
D
• ABC  FDE
F
A
E
B
C
Name the corresponding congruent angles and sides
QRS  RTV
Q  T
QRS  TRV
S  V
QR  TR
QS  TV
SR  VT
Q
S
R
T
V
Properties of Triangle Congruence
• Reflexive
KLS  KLS
L
L
• Symmetric
K
S K
S
If KLJ  QPR then
QPR  KLJ
L
J
K
P
R
Q
L
• Transitive
If KLJ  QPR and
QPR  XYZ then
KLJ  XYZ
X
J
K
Q
Y
Z
P
R
Transformations of Congruent Triangles
• LMN  QRP
M
L
N
Verify that CDE  C’D’E’
C’
C
D
E
E’
D’
C(-5,7) D (-8,6) E(-3,3)
C’(5,7) D’(8,6) E’(3,3)
L
Prove the Transitive Property
Given: If KLJ  QPR and QPR  XYZ
Prove: KLJ  XYZ
Statements
1. KLJ  QPR
2. K Q, LP, JR,
KJQR, KLQP, LJPR
3. QPR  XYZ
4. Q X, PY, RZ,
QRXZ, QPXY, PRYZ
5. K X, LY, JZ
6. KJXZ, KLXY, LJYZ
7. KLJ  XYZ
Reasons
P
JQ
K
R
Y
X
Z
1. Given
2. Corresponding parts of congruent
angles are congruent (CPCTC)
3. Given
4. CPCTC
5. Transitive Property of angles
6. Transitive Property of segments
7. Def of congruent triangles
Constructions – Congruent Triangles Using Sides
•
•
•
•
•
•
Draw a triangle, label the vertices X, Y,
and Z
Elsewhere on the paper, use a straight
edge to construct segment RS Such that
RS  XZ
Using R as the center, draw and arc with
radius equal to XY
Using S as the center draw and arc with a
radius equal to YZ.
Let T be the point of intersection of the two
arcs.
T
Draw RT and ST to form RST
R
Y
Z
X
S
Constructions – Congruent Triangles using 2 sides
and an included angle
•
•
•
•
•
Draw a triangle, label the vertices A, B,
and C
Elsewhere on the paper, use a straight
edge to construct segment KL Such that
KL  BC
Construct and angle congruent to B using
KL as a side of the angle and K as the
vertex.
Construct JK such that JK  BA.
Draw JL to complete KJL
C
A
B
J
K
L
Practice Assignment
• Block Page 257 10-22 even and 28
• Honors: page 258 10 – 20 even, 24, 28, 32, 40