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Objectives
• Define congruent polygons
• Prove that two triangles are congruent
using SSS, SAS, ASA, and AAS shortcuts
1
Definition of Congruence
• Congruent figures have the same shape and
size
• Congruent polygons have congruent
corresponding parts – matching sides and
angles
2
Proving Two Triangles Congruent
• Using definition: all corresponding sides
and angles congruent
• Using shortcuts: SSS, SAS, ASA, AAS
3
Side-Side-Side (SSS) Postulate
If the three sides of one triangle are
congruent to the three sides of another
triangle, then the two triangles are
congruent.
ΔGHF ≅ ΔPQR
4
Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle, then the
two triangles are congruent.
ΔBCA ≅ ΔFDE
5
Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one
triangle are congruent to two angles and
the included side of another triangle, then
the two triangles are congruent.
ΔHGB ≅ ΔNKP
6
Angle-Angle-Side (AAS) Theorem
If two angles and a nonincluded side of one
triangle are congruent to two angles and the
corresponding nonincluded side of another
triangle, then the triangles are congruent.
ΔCDM ≅ ΔXGT
7
SSS Example
Explain why
∆ABC  ∆CDA.
• AB  CD and BC  DA (Given)
• AC  AC (Reflexive Property of Congruence)
• ∆ABC  ∆CDA by SSS.
8
SAS Example
Explain why
∆XZY  ∆VZW
• XZ  VZ and WZ  YZ (Given)
• ∠XZY  ∠VZW
(Vertical angles are  )
• ∆XZY  ∆VZW
(SAS)
9
ASA Example
Explain why
∆KLN  ∆MNL
KL  MN
(Given)
KL || MN
(Given)
∠KLN  ∠MNL (Alternate Interior Angles are  )
LN  LN
(Reflexive Property)
∆KLN  ∆MNL (SAS)
10
AAS Example
Given: JL bisects
∠KJM. Explain why
∆JKL  ∆JML
JL bisects ∠KJM
∠KJL  ∠MJL
∠K  ∠M
JL  JL
∆JKL  ∆JML
(Given)
(Defn. of angle bisector)
(Given)
(Reflexive Property)
(AAS)
11