6.0 Geometric Proofs Math 10Geometry Unit Lesson 1 Lesson 1

Definition of Congruent Figures are two figures that are identical in size and shape ex: a b c p q r

How Do We Prove that 2 triangles are congruent??? Look at the triangles and figure out what they are telling you. Look at the triangles and figure out what they are telling you. List all that they are telling you and the reason you believe it is true. List all that they are telling you and the reason you believe it is true. Write a congruency statement and what the postulate (proof) is (SSS, SAS, ASA, RHS, SAA) Write a congruency statement and what the postulate (proof) is (SSS, SAS, ASA, RHS, SAA)

Lets Review……. SSS  Side, Side, Side SSS  Side, Side, Side SAS  Side, Angle (contained), Side SAS  Side, Angle (contained), Side ASA  Angle, Side (contained), Angle ASA  Angle, Side (contained), Angle RHS (HL)  Right, Hypotenuse, Side (aka: Hypontenuse, Leg) RHS (HL)  Right, Hypotenuse, Side (aka: Hypontenuse, Leg) NEW - SAA  Side (not contained), Angle, Angle NEW - SAA  Side (not contained), Angle, Angle

Side-Side-Side (SSS) Postulate: If all three pairs of corresponding sides of two triangles are equal, the two triangles are congruent. If you know:then you know:and you know: AB = DE BC = EF AC = DF  BAC =  EDF  ABC =  DEF  ACB =  DFE

Side-Side-Side (SSS) Postulate: You Know…. Reason AB = DE Given BC = EF Given AC = DF Given Statement: SSS SSS

Side-Angle-Side (SAS) Postulate: If two pairs of corresponding sides and the corresponding contained angles of two triangles are equal, the two triangles are congruent. If you know:then you know:and you know: AB = DE  ABC =  DEF AC = DF  BAC =  EDF BC = EF  ACB =  DFE

You Know…. Reason… AB = DE given  ABC =  DEF given AC = DF given SAS Side-Angle-Side (SAS) Postulate:

Angle-Side-Angle (ASA) Postulate: If two angles and the contained side of one triangle are equal to two angles and the contained side of another triangle, the two triangles are congruent. If you know:then you know:and you know:  BCA =  EFD  ABC =  DEF AB = DE AC = DF  BAC =  EDF BC = EF

Angle-Side-Angle (ASA) Postulate: You know…. Reason…  BAC =  EDF Given  ABC =  DEF Given AB = DE given ASA

Right angle - Hypotenuse-Side (RHS)/ Hypotenuse Leg Postulate: If the hypotenuse and another side of one right triangle are equal to the hypotenuse and one side of a second right triangle, the two triangles are congruent. If you know: then you know: and you know:  BAC =  EDF = 90 o BC = EF AC = DF  ABC =  DEF  ACB =  DFE AB = DE

Right angle - Hypotenuse-Side (RHS) Postulate/ Hypotenuse Leg: You Know….. Reason…  BAC =  EDF = 90  Given BC = EF Given AC = DF Given RHS/ HL

Angle, Angle Side Postulate: If two angles and a non-contained side of one triangle are equal to two angles and the non- contained side of another triangle, the two triangles are congruent. If you know: then you know: and you know:  BAC =  EDF  ACB =  DFE AC = DF  ABC =  DEF BC= EF AB = DE

Side Angle Angle You Know….. Reason…  BAC =  EDF Given AC = DF Given  ABC =  DEF Given SAA

Reasons for congruence… Given (provided by diagram symbols) Given (provided by diagram symbols) Vertically Opposite angles theorem Vertically Opposite angles theorem Alternate interior angles (z - pattern) Alternate interior angles (z - pattern) Corresponding angles (F – pattern) Corresponding angles (F – pattern) Def. of an isosceles triangle Def. of an isosceles triangle Common (share and angle or a side) Common (share and angle or a side)

Example A B AC = EC given A B AC = EC given  ACB =  ECD opp.  ACB =  ECD opp.  BAC =  DEC alt. int.  BAC =  DEC alt. int. C ASA C ASA D E D E

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