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Congruent Figures  Congruent Corresponding Parts
4.1: Congruent Figures
If two figures are congruent, then they have
corresponding parts that are congruent.
If the corresponding parts of two figures are
congruent, then the figures are congruent.
Congruent figures have congruent
corresponding parts.
Congruent Figures
ABC
QTJ. List the congruent
corresponding parts.
List the corresponding vertices in the same order.
Angles: A
Q
B
T
C
J
List the corresponding sides in the same order.
Sides: AB
QT
BC
TJ
AC
4-1
QJ
Congruent Figures
GEOMETRY LESSON 4-1
XYZ
KLM, mY = 67, and mM = 48. Find mX.
Use the Triangle Angle-Sum Theorem and the definition of congruent
polygons to find mX.
mX + mY + mZ = 180
Triangle Angle-Sum Theorem
mZ = mM
Corresponding angles of congruent
triangles that are congruent
mZ = 48
Substitute 48 for mM.
mX + 67 + 48 = 180
mX + 115 = 180
mX = 65
Substitute.
Simplify.
Subtract 115 from each side.
4-1
Congruent Figures
Can you conclude that
ABC
CDE in the figure below?
List corresponding vertices in the same order.
If
ABC
CDE, then BAC
DCE.
The diagram above shows BAC
DEC, not DCE.
Corresponding angles are not necessarily congruent, therefore you
cannot conclude that ABC
CDE.
Text Resource: Prentice Hall
4-1 Academy
Saint Agnes
Let’s investigate further. Are there two congruent triangles?
1
a. 1 2
b. B D
c. A E
d. AB  ED,
BC  DC
AC  EC
2
a. All vertical angles are congruent
b. All Right angles are congruent.
c. Given.
d. Given.
e. ABC   EDC
e. Def of Congruent Polygons
Proof of Theorem 4.1
Given: AQ
B T
Prove: C J
a. AQ and B T
a. Given
b. Triangle Sum Thm.
b. m A + mB + mC = 180
c. m Q + mT + mJ = 180
c. Triangle Sum Thm
d. m A + mB + mC = m Q + mT + mJ
d. Substitution
e. m Q + mT + mC = m Q + mT + mJ
e. Substitution
f. mC = mJ
f. Subtraction
QED
Theorem 4-1
If two angles of one triangle are congruent
to the corresponding angles of another
triangle, then the third pair of angles is
also congruent.
Two congruent pairs of angles  two congruent triangles
Show how you can conclude that
statements and reasons.
CNG
DNG. List
Congruent triangles have three congruent corresponding
sides and three congruent corresponding angles.
Examine the diagram, and list the congruent corresponding
parts for CNG and DNG.
a. CG DG
b. CN DN
c. GN GN
d. C D
e. CNG DNG
f. CGN DGN
g. CNG
DNG
Given
Given
Reflexive Property of Congruence
Given
Right angles are congruent.
If two angles of one triangle are congruent to two angles
of another triangle, then the third angles are congruent.
(Theorem 4-1.)
Definition of triangles
4-1
Use the information in the diagram. Tell why each statement
is true:
P
PQ bisects RT
S
R
T
Q
a. PR || TQ
b.  RPS  TQS
c. PR  QT , PS  QS
d.  PRS  QTS
e.  PSR   QST
f. PQ bisects RT
g. RS  TS
h. PRS  QTS
a. Given
b. Alt. Int. Angles
c. Given
d. Alt. Int Angles
e. Vertical Angles are 
f. Given
g. Def. Segment Bisector
h. Def of  Triangles