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Algebra Proofs By Dylan Keretz 6th period Memory Jogger • Let a, b, and c be all real numbers. / = Division • Addition Property If a=b, then a+c=b+c • Subtraction Property If a=b, then a-c=b-c • Multiplication Property If a=b, then ac=bc • Division Property If a=b and c=0, then a/c=b/c • Reflexive Property For any real number a, a=a • Symmetric Property If a=b, then b=a • Transitive Property If a=b and b=c, then a=c • Substitution Property If a=b, then a can be substituted for b in any equation or expression Proving Statements About Segments • A true statement that • Properties of Segment follows as a result of other Congruence true statements is called a • s=segment c=congruent to. theorem. All theorems must Applies to all slides. be proved. You can prove a • Reflexive For any theorem using a two- segment AB, sAB c sAB column proof. A two-column • Symmetric If sAB c sCD, proof has numbered then sCD c sAB statements and reasons • Transitive If sAB c sCD, that show the logical order and sCD c sEF, then sAB of an argument. c sEF. Example 1 • Given: sPQ c sXY • Prove: sXY c sPQ Statements Reasons 1.sPQ c sXY 1. Given 2. PQ = XY 2. Def. of congruent segments 3. XY=PQ 3. Symmetric prop. Of equality 4. sXY c sPQ 4. Def. of congruent segments Exercise • Given: LK=5, JK=5, sJK c sJL • Prove: sLK c sJL 3. LK=JK Statements Reasons LK=LJ 1. JK=5 1. Given 2. LK=5 2. Given 3. ? 3. Transitive prop. of equality 4. sLK c sJK 4. Def. of congruent segments 5.sJK c sJL 5. Given 6. sLK c sJL. 6.Transitive Prop. Of Congruence Proving Statements About Angles • Properties of Angle Congruence • Angle congruence is reflexive, symmetric, and transitive. Here are some examples: NOTE: ^ = angle • Reflexive For any angle A, ^A c ^A. • Symmetric If ^A c ^B, then ^B c ^A. • Transitive If ^A c ^B and ^B c ^C, then ^A c ^C. Example • Given: ^A c ^B Prove: ^A c ^C ^B c ^C Statements Reasons 1. ^A c ^B, 1. Given ^B c ^C 2. m^A = m^B 2. Def. of congruent angles 3. m^B = m^C 3. Def. of congruent angles 4. m^A = m^C 4. Transitive prop. Of equality 5. ^A c ^C 5. Def. of congruent angles Exercise • Given: m^3=40 , ^1 c ^2, ^2 c ^3 • Prove: m^1 = 40 Statements Reasons 1.m^3 = 40 , ^1 c ^2, ^2 c ^3 1. Given 2.^1 c ^3 2. Transitive prop. of congruence 3. m^1 = m^3 3. ? 4.m^1 = 40 4.Substitution prop. of equality 3. Transitive prop. Of congruence Def. of congruent angles Substitution prop. Of equality To view the program again, please click here!