Objectives
Apply SSS and SAS to construct triangles and solve problems.
Prove triangles congruent by using SSS and SAS.
For example, you only need to know that two
triangles have three pairs of congruent corresponding
sides. This can be expressed as the following
postulate.
Remember!
Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of
congruent parts.
Example 1: Using SSS to Prove Triangle Congruence
Use SSS to explain why ∆ABC  ∆DBC.
It is given that AC  DC and that AB  DB. By the
Reflexive Property of Congruence, BC  BC.
Therefore ∆ABC  ∆DBC by SSS.
Check It Out! Example 1
Use SSS to explain why
∆ABC  ∆CDA.
It is given that AB  CD and BC  DA.
By the Reflexive Property of Congruence, AC  CA.
So ∆ABC  ∆CDA by SSS.
An included angle is an angle formed by two
adjacent sides of a polygon.
B is the included angle between sides AB and
BC.
Caution
The letters SAS are written in that order because the
congruent angles must be between pairs of
congruent corresponding sides.
Example 2: Engineering Application
The diagram shows part of the support
structure for a tower. Use SAS to
explain why ∆XYZ  ∆VWZ.
It is given that XZ  VZ and that YZ  WZ. By the Vertical
s Theorem. XZY  VZW. Therefore ∆XYZ  ∆VWZ by
SAS.
Check It Out! Example 2
Use SAS to explain why ∆ABC 
∆DBC.
It is given that BA  BD and ABC  DBC. By the
Reflexive Property of , BC  BC. So ∆ABC  ∆DBC by
SAS.
Objectives
Apply ASA, AAS, and HL to construct triangles and to solve problems.
Prove triangles congruent by using ASA, AAS, and HL.
An included side is the common side of two
consecutive angles in a polygon. The following
postulate uses the idea of an included side.
included side
Vocabulary
Methods to Establish Component Congruence, when congruence is not given
1.If a segment is bisected, you have two congruent segments
2. If a midpoint is given you have two congruent segments
3. If you have vertical angles you have two congruent angles.
4. If you are given an angle bisector, you have two congruent angles.
5. If you have parallel lines cut by a transversal, the alternate interior angle are
congruent.
6. If you have a perpendicular line, there are two congruent 90 degree angles.
7. If you are given a perpendicular bisectors, you have 4 congruent angles and two
congruent segments.
8. A common or shared side is considered a set of congruent sides.
9. A common or shared angle is considered a set of congruent angles.
Check It Out! Example 2
Determine if you can use ASA to prove NKL
 LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by
the Reflexive Property. No other congruence relationships can be
determined, so ASA cannot be applied.
You can use the Third Angles Theorem to prove another
congruence relationship based on ASA. This theorem is AngleAngle-Side (AAS).
Example 3: Using AAS to Prove Triangles Congruent
Use AAS to prove the triangles congruent.
Given: X  V, YZW  YWZ, XY  VY
Prove:  XYZ  VYW
Check It Out! Example 3
Use AAS to prove the triangles congruent.
Given: JL bisects KLM, K  M
Prove: JKL  JML
Example 4A: Applying HL Congruence
Determine if you can use the HL Congruence Theorem to prove
the triangles congruent. If not, tell what else you need to know.
According to the diagram, the
triangles are right triangles that
share one leg.
It is given that the hypotenuses
are congruent, therefore the
triangles are congruent by HL.
Check It Out! Example 4
Determine if you can use the HL
Congruence Theorem to prove ABC 
DCB. If not, tell what else you need to
know.
Yes; it is given that AC  DB. BC  CB by the Reflexive
Property of Congruence. Since ABC and DCB are right
angles, ABC and DCB are right triangles. ABC  DCB by
HL.