Math 2 Geometry Based on Elementary Geometry, 3rd ed, by Alexander

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Math 2 GeometryBased on Elementary Geometry,
3rd ed, by Alexander Koeberlein

• 3.1
• Congruent Triangles

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Definition

• Two triangles are congruent when the six parts
  of the first triangle are congruent to the six
  parts of the second triangle.

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Definition

• Two triangles are congruent when the six parts
  of the first triangle are congruent to the six
  parts of the second triangle.
• What are the six parts?

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Definition

• Two triangles are congruent when the six parts
  of the first triangle are congruent to the six
  parts of the second triangle.
• What are the six parts?
• How does this relate to our understanding of
  congruence as same shape and size?

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Converse of Definition

• If two triangles are congruent, then the six
  parts of one triangle are congruent to the
  corresponding parts of the other triangle.
• The converse of a definition is also true
• If the six parts of a triangle are congruent to
  the corresponding parts of another triangle, then
  the triangles are congruent.

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Consequence of the definition

• Converse of the definition
• If the six parts of a triangle are congruent to
  the corresponding parts of another triangle, then
  the triangles are congruent.
• To prove two triangles are congruent, we can
  show that the corresponding six parts of the two
  triangles are congruent.

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Properties of Congruent ?s

• ?ABC ? ?ABC (Reflexive Property of ?)

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Properties of Congruent ?s

• ?ABC ? ?ABC (Reflexive Property of ?)
• If ?ABC ? ?DEF then ?DEF ? ?ABC (Symmetric
  Property of ?)

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Properties of Congruent ?s

• ?ABC ? ?ABC (Reflexive Property of ?)
• If ?ABC ? ?DEF then ?DEF ? ?ABC (Symmetric
  Property of ?)
• If ?ABC ? ?DEF and ?DEF ? ?GHI, then ?ABC ? ?GHI
  (Transitive Property of ?)

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Construction

• Construct a triangle whose sides have the
  lengths of the segments provided in the figure.

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SSS (Side-Side-Side)Postulate 12

• Method for Proving Triangles Congruent
• If the three sides of one triangle are congruent
  to the three sides of a second triangle, then the
  triangles are congruent.

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SAS (Side-Angle-Side)Postulate 13

• Method for Proving Triangles Congruent
• If two sides and the included angle of one
  triangle are congruent to the two sides and the
  included angle of a second triangle, then the
  triangles are congruent.

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ASA (Angle-Side-Angle)Postulate 14

• Method for Proving Triangles Congruent
• If two angles and the included side of one
  triangle are congruent to two angles and the
  included side of a second triangle,

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AAS (Angle-Angle-Side)Theorem 3.1.1

• Method for Proving Triangles Congruent
• If two angles and a non-included side of one
  triangle are congruent to two angles and a
  non-included side of a second triangle, then the
  triangles are congruent.