Download Section 4.2 Triangle Congruence Two triangles are congruent if and

Section 4.2
Triangle Congruence
Two triangles are congruent if and only if the corresponding sides and corresponding angles are
congruent.
E
B
A
C
F
D
∆ABC ≅ ∆DEF If and only if
AC ≅ DF
AB ≅ DE
BC ≅ EF
∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
Theorem 4.1 (SAS Congruence Postulate)
If two sides and the included angle of one triangle are congruent respectively to two sides and
the included angle of another triangle, then the two triangles are congruent.
E
B
A
C
D
F
Theorem 4.2
LL Congruence Theorem
If two legs of one right triangle are congruent respectively to two legs of another right triangle,
then the two triangles are congruent.
B
A
C
E
F
D
Theorem 4.3
HL Congruence Theorem
If the hypotenuse and leg of one right triangle are congruent respectively to the hypotenuse and
leg of another right triangle, then the two triangles are congruent.
B
A
C
E
D
F
Postulate 4.2
ASA Congruence Postulate
If two angles and the included side of one triangle are congruent respectively to two angles and
the included side of another triangle, then the two triangles are congruent.
E
B
C
A
F
D
Postulate 4.3
SSS Congruence
If three sides of one triangle are congruent respectively to three sides of another triangle, then the
two triangles are congruent.
E
B
A
C
D
F
Section 4.4
The Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of two sides of a triangle equals the square of the third
side, then the triangle is a right triangle.
A
c
b
B
a
C
CSCTC
Corresponding parts of congruent triangles are congruent.
Example 1
N
M
C
D
E
Let CD ≅ CM and CD ≅ CE . Prove that ∆CMN ≅ ∆CDE and MN ≅ DE
Statement
Reason
1) CD ≅ CM and CD ≅ CE .
Given
2) ∠MCN ≅ ∠DCE
If two lines intersect, then the vertical
angles formed are congruent
3) ∆MCN ≅ ∆DCE
SAS
4) MN ≅ DE
CPCTC
Example 2
A
N
C
Y
D
In ANDY, line segments AD and YN divide each in half at C. Prove that ∠DAN ≅ ∠DAY
Statement
Reason
1) Line segments AD and AD divide each
in half at C
Given
2) CA ≅ CD and CY ≅ CN
A midpoint divide a segment into two
congruent segments
3) ∠ACY ≅ ∠NCD and ∠ACN ≅ ∠DCY
If two lines intersect, then the vertical
angles formed are congruent.
4) ∆ACY ≅ ∆NCD and ∆ACN ≅ ∆DCY
SAS
5) AN ≅ BY and AY ≅ DN
CPCTC
6) AD ≅ AD
Reflective Property of Congruence
7) ∆DYA ≅ ∆AND
SSS
8) ∠DAN ≅ ∠DAY
CPCTC
Example 3
A
B
D
C
Given AB ≅ AC and AD is an altitude of ∆ABC , prove that BD ≅ DC
A
B
D
C
Statement
Reason
1) AB ≅ AC and AD is an altitude of ∆ABC
1) Given
2) ∠BDA and ∠CDA are right angles
2) Definition of an altitude of a triangle
3) ∠BDA ≅ ∠CDA
3) All right angles are congruent
4) ∠BDA and ∠CDA are right triangles
4) Definition of a right triangle
5) AD ≅ AD
5) Reflective Property of Congruence
6) ∆BDA ≅ ∆CDA
6) HL
7) BD ≅ DC
7) CPCTC