Geometry
This text covers the fundamentals of proving theorems in geometry, emphasizing the distinction between definitions, postulates, and theorems. It introduces deductive reasoning, outlining steps and reasons used to establish proofs, such as those for the Midpoint Theorem and the Angle Bisector Theorem. Key concepts include the definition and theorem of the midpoint, the relationship between segments, and the use of accepted postulates in calculations. Students will understand how to apply these theorems in geometry problems and use them in logical reasoning.
Geometry
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Presentation Transcript
Geometry 2.3 Proving Theorems
Intro • Theorems are statements that are proved. • They are deduced from postulates, statements that are accepted without proof. • POE’s are treated as postulates • Deductive Reasoning uses postulates defn.’s, thm.’s, and given information. Hint: deductive = definition
4 Reasons Used in Proofs • Given Information • Defn.’s • Postulates(POE’s) • 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 . M . B .
Statements M is the midpoint of AM = MB AM + MB = AB AM + AM = AB or 2AM = AB AM = (1/2)AB MB = (1/2)AB Now that the Midpoint Thm. has been proven, it may be used as a reason in a proof! Reasons Given Defn. of Midpoint Segment Add. Post. Substitution (Steps 2 & 3) Division POE Substitution (Steps 2 & 5) Proof of the Midpoint Theorem G: M is the midpoint of P: AM = (1/2)AB; MB = (1/2)AB A . M . B .
Midpoint Defn. versus Midpoint Thm.(uses ½) . A . Y . B • If Y is the midpoint of , …then what is true by the reason of midpoint defn.? Answer: AY = YB …then what is true by the reason of midpoint thm.? Answer: AY = (1/2)AB or YB = (1/2)AB
Angle Bisector Them. • If BX is the bisector of , then How is this different from the angle bisector defn.? Key: The theorem uses ½. . A . X . C B .
Angle Bisector Thm. Versus the Angle Bisector Defn. • If BX is the bisector of , then is true by the reason of __________? Answer: Angle Bisector Defn. • If BX is the bisector of , then is true by the reason of __________? Answer:Angle Bisector Thm. . A . X . C B.
Angle Add. Post. Segment Add. Post. Angle Add. Post. Midpoint Defn. Midpoint Thm. Segment Bisector Defn. Segment Bisector Defn. Angle Bisector Thm. Angle Bisector Defn. 10) Reasons Given m<XBC or <XBC by Angle Bisector Defn. Angle Add. Post. Substitution (Steps 2 & 3) Mult. POE Substitution(Steps 2 & 5) Please turn your books to P. 45
HW • P. 41 #4-12 (4X) P.46 #1-19 Odd P. 51 CE #1-21 Odd Quiz 2.1-2.3 on Wednesday
