Geometry 2.3 Proving Theorems

Intro Theorems are statements that are proved. Theorems are statements that are proved. They are deduced from postulates, statements that are accepted without proof. They are deduced from postulates, statements that are accepted without proof. POE’s are treated as postulates POE’s are treated as postulates Deductive Reasoning uses postulates defn.’s, thm.’s, and given information. Deductive Reasoning uses postulates defn.’s, thm.’s, and given information. Hint: deductive = definition

4 Reasons Used in Proofs Given Information Given Information Defn.’s Defn.’s Postulates(POE’s) Postulates(POE’s) Theorems from yesterday i.e.(that is) theorems that have been proven Theorems from yesterday i.e.(that is) theorems that have been proven

Midpoint Thm. If M is the midpoint of, then AM = (1/2)AB and MB = (1/2)AB How is this different from the midpoint defn.? Key: The midpoint theorem uses ½. A.A. M.M. B.B.

Statements 1) M is the midpoint of 2) A M = MB 3) A M + MB = AB 4) A M + AM = AB or 2AM = AB 5) A M = (1/2)AB 6) M B = (1/2)AB Now that the Midpoint Thm. has been proven, it may be used as a reason in a proof! Reasons 1) G iven 2) D efn. of Midpoint 3) S egment Add. Post. 4) S ubstitution (Steps 2 & 3) 5) D ivision POE 6) S ubstitution (Steps 2 & 5) Proof of the Midpoint Theorem G: M is the midpoint of P: AM = (1/2)AB; MB = (1/2)AB A.A. M.M. B.B.

Midpoint Defn. versus Midpoint Thm. (uses ½) If Y is the midpoint of, If Y is the midpoint of, …then what is true by the reason of midpoint defn.? Answer: AY = YB Answer: AY = YB …then what is true by the reason of midpoint thm.? Answer: AY = (1/2)AB or YB = (1/2)AB Answer: AY = (1/2)AB or YB = (1/2)AB.Y.Y.A.A.B.B

Angle Bisector Them. If BX is the bisector of, then If BX is the bisector of, then How is this different from the angle bisector defn.? Key: The theorem uses ½..C.C B.. X. A

Angle Bisector Thm. Versus the Angle Bisector Defn. If BX is the bisector of, then If BX is the bisector of, then is true by the reason of __________? is true by the reason of __________? Answer: Angle Bisector Defn. If BX is the bisector of, then If BX is the bisector of, then is true by the reason of __________? is true by the reason of __________? Answer: Angle Bisector Thm. Answer: Angle Bisector Thm..C.C B.B.. X. A

Please turn your books to P. 45 1) Angle Add. Post. 2) Segment Add. Post. 3) Angle Add. Post. 4) Midpoint Defn. 5) Midpoint Thm. 6) Segment Bisector Defn. 7) Segment Bisector Defn. 8) Angle Bisector Thm. 9) Angle Bisector Defn. 10) Reasons 1) Given 2) m<XBC or <XBC by Angle Bisector Defn. 3) Angle Add. Post. 4) Substitution (Steps 2 & 3) 5) Mult. POE 6) Substitution (Steps 2 & 5)

HW P. 41 #4-12 (4X) P. 41 #4-12 (4X) P.46 #1-19 Odd P. 51 CE #1-21 Odd Quiz on Wednesday