Transcript 4.4 – Prove Triangles Congruent by SAS

4.4 – Prove Triangles
Congruent by SAS
Geometry
Ms. Rinaldi
Included Angles
The included angle is the angle between two
sides.
Side-Angle-Side (SAS)
Congruence Postulate
If two sides and the included angle
of one triangle are congruent to
two sides and the included angle
of a second triangle, then the two
triangles are congruent.
B
C
A
S
T
If Side AB  RS
Angle A  R
Side CA  TR
Then
ABC  RST
R
EXAMPLE 1
Use SAS
Decide whether the congruence
statement is true.
Explain your reasoning.
ABC  DEF
SOLUTION
(S)
(A)
(S)
AC  DF
C  F
CB  FE
So, by SAS,
ABC  DEF
EXAMPLE 2
Use SAS
Decide whether the congruence
statement is true.
Explain your reasoning.
NJH  DFR
SOLUTION
Although there are two pairs of congruent sides, the
congruent angles are not included (between) the
congruent sides.
Therefore SAS does not apply and the triangles are not
congruent.
EXAMPLE 3
Use SAS
Decide whether the congruence
statement is true.
Explain your reasoning.
EPF  ERF
Hint: This is possible! Do not forget the side they
both share!
EXAMPLE 4
Use SAS
Decide whether the congruence
statement is true.
Explain your reasoning.
PTF  GTS
Hint: This is possible! You have the sides, you
only need a pair of congruent angles in between…
EXAMPLE 5
Use SAS
Decide whether the congruence
statement is true.
Explain your reasoning.
ABC  CDA
Hint: This is possible! Look for alternate interior
angles as well as a side that both triangles share.
EXAMPLE 6
Use SAS and properties of shapes
In the diagram, QS and RP pass through
the center M of the circle. What can you
conclude about
MRS and
MPQ?
SOLUTION
Because they are vertical angles, PMQ
RMS. All
points on a circle are the same distance from the center,
so MP, MQ, MR, and MS are all equal.
ANSWER
MRS and
MPQ are congruent by the SAS
Congruence Postulate.
EXAMPLE 7
Use SSS or SAS
State the third congruence that must be given to prove that
ABC  DEF using the indicated postulate.
B
Given:
AB  DE
E
CB  FE
Use the SSS Congruence Postulate.
C
A
D
SOLUTION
To use SSS, you need to know that
AC  DF
F
EXAMPLE 8
Use SSS or SAS
State the third congruence that must be given to prove that
ABC  DEF using the indicated postulate.
B
Given:
E
A  D CA  FD
Use the SAS Congruence Postulate.
C
A
D
SOLUTION
To use SAS, you need to know that
BA  ED
F
EXAMPLE 9
Use SSS or SAS
State the third congruence that must be given to prove that
ABC  DEF using the indicated postulate.
B
Given:
 B  E
E
AB  DE
Use the SAS Congruence Postulate.
C
A
D
F