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Lesson 4.3
Exploring Congruent Triangles
Definition of Congruent
Triangles
• If ΔABC is congruent to ΔPQR, then there is a
correspondence between their angles and sides
such that corresponding angles are congruent and
corresponding sides are congruent. The notation
ΔABC  ΔPQR indicates the congruence and the
correspondence, as shown below
B
•
ΔABC  ΔPQR
Corresponding
angles are:
A   P
B  Q
C  R
Corresponding
sides are:
AB  PQ
BC  QR
CA  RP
C
A
Q
R
P
Congruent Triangles
Example 1
Naming Congruent Parts
• You and a friend have identical drafting
triangles, as shown below. Name all
congruent parts.
Example 1
Naming Congruent Parts
Corresponding
angles are:
D  R
E  S
F  T
∆DEF  ∆RST
Corresponding
sides are:
DE  RS
EF  ST
FD  TR
Which of the following expresses the correct
congruence statement for the figure below?
Classification of
Triangles by Sides
• An equilateral triangle has 3 congruent
sides
• An isosceles triangle has a least two
congruent sides
• A scalene triangle has no sides congruent
Classification of
Triangles by Angles
• An acute triangle has 3 acute angles. If
these angles are all congruent, the triangle
is also equiangular
• A right triangle has exactly one right angle
• An obtuse triangle has exactly one obtuse
angle
Obtuse
Vocabulary
• In ΔABC, each of the points
A, B, and C is a vertex of
the triangle
• The side BC is the side
opposite  A
• Two sides that share a
common vertex are adjacent
sides
Vocabulary
for right and isosceles
triangles
• In a right triangle, the sides
adjacent to the right angle are the
legs of the triangle.
• The side opposite the right angle is
the hypotenuse of the triangle
• An isosceles triangle can have 3
congruent sides. If is has only two,
the two congruent sides are the legs
of the triangle. The third side is the
base of the triangle.
Example 2
Proving Triangles are Congruent
• The outside structure of the Bank
of China is glass and aluminum and
consists of more than 50 congruent
triangles. Use the information given
below to prove that ΔAEBΔDEC.
• Statements
1. AB||CD
2.  EAB  EDC
3. ABE  DCE
4. AEB  CED
5.ABCD
6.E is midpoint of AD
7. AE ED
8. E is midpoint of BC
9. BE EC
10.ΔAEBΔDEC
• Reasons
1. Given
s are 
2. 2 lines ||  alt. int. 
s are 
3.2 lines ||  alt. int. 
4.Vertical angles are 
5.Given
6.Given
7. Def. of midpoint
8. Given
9. Def. of Midpoint
10.Def. of Congruent Triangles
Find the values of x and y given that
∆MAS ≌ ∆NER.
Solution:
Now we substitute 7 for x to solve for y:
Given:
Prove:
Theorem 4.1
Properties of Congruent Triangles
• 1. Every triangle is congruent to itself
• 2. If ΔABCΔPQR, then ΔPQR  ΔABC
• 3. If ΔABCΔPQR and ΔPQRΔTUV,
then ΔABCΔTUV