Math 2 Geometry Based on Elementary Geometry, 3 rd ed, by Alexander & Koeberlein 3.1 Congruent Triangles

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

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?

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?

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.

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.

Properties of Congruent  s  ABC   ABC (Reflexive Property of  )

Properties of Congruent  s  ABC   ABC (Reflexive Property of  ) If  ABC   DEF then  DEF   ABC (Symmetric Property of  )

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  )

Construction Construct a triangle whose sides have the lengths of the segments provided in the figure. A A B B C C

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. A B C D E F

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. A B C D E F  A   F Seg AB  Seg FE Seg AC  Seg FD

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, A B C D E F  A   F  C   D Seg AC  Seg FD

AAS (Angle-Angle-Side) Theorem 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. A B C D E F  A   F  C   D Seg BC  Seg ED