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Building a System of Geometry Knowledge 2.4

Building a System of Geometry Knowledge 2.4. Algebraic properties. Homework. Addition Property. If a = b , then a + c = b + c. Subtraction Property. If a = b , then a – c = b – c. Homework. Multiplication Property. If a = b , then ac = bc. Division Property.

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Building a System of Geometry Knowledge 2.4

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  1. Building a System of Geometry Knowledge 2.4

  2. Algebraic properties

  3. Homework Addition Property • If a = b , then a + c = b + c Subtraction Property • If a = b , then a – c = b – c

  4. Homework Multiplication Property • If a = b , then ac = bc Division Property • If a = b and c ≠ 0 , then a/c = b/c

  5. Homework Substitution Property • If a = b, you may replace a with b in any true equation that has a in it, and the resulting equation will still be true.

  6. Homework Equivalence Properties • Reflexive Property for any number a, a = a • Symmetric Property if a = b , b = a • Transitive Property if a = b and b = c, then a = c

  7. Homework Overlapping SegmentsTheorem Overlapping segment theorem: Given a segment with points A, B, C, and D arranged as shown, the following statements are true: If AB = CD then AC = BD If AC = BD then AB = CD

  8. Homework Overlapping AnglesTheorem Given AED with points B and C in its interior as shown, the following statements are true: 1. If mAEB=mCED, then mAEC=mBED. 2. If mAEC =mBED, then mAEB=mCED.

  9. Homework Equality and Congruency • For all these properties, you can change the equal sign to a congruent sign and they are still true. • In the Reasons column of the proof write Definition of Congruence.

  10. Homework Two Column Proof Given: AB = CD Prove: AC = BD Statement Reason 1. AB = CD 1. Given 2. Addition Property 2. AB + BC = BC + CD 3. AB + BC = AC 3. Segment Addition Postulate 4. BC + CD = BD 4. Segment Addition Postulate 5. AC = BD 5. Substitution Property

  11. Homework Two Column Proof Given: a = b Prove: a + c = b + c Statement Reason 1. a = b 1. Given 2. a + c = a + c 2. Reflexive Property of Equality 3. a + c = b + c 3. Substitution Property of Equality

  12. Homework Two Column Proof Given: 2x − 5 = 3Prove: x = 4 Statement Reason 1. 2x − 5 = 3 1. Given 2. Addition Property of Equality 2. 2x − 5 + 5 = 3 + 5 3. 2x = 8 3. Simplify 4. 2x ÷ 2 = 8 ÷ 2 4. Division Property of Equality 5. x = 4 5. Simplify

  13. Homework Two Column Proof Given: B  CProve: x = 7 Statement Reason 1. B  C 1. Given 2. Definition of Congruence 2. mB = mC 3. 5x + 12 = 47 3. Substitution Property of Equality 4. 5x = 35 4. Subtraction Property of Equality 5. x = 7 5. Division Property of Equality

  14. Homework Two Column Proof Given: m1 + m3 = 180 Prove:1  2 Statement Reason 1. m1 + m3 = 180 1. Given 2. Linear Pair Property 2. m2 + m3 = 180 3. m1 + m3 = m2 + m3 3. Substitution Property of Equality 4. m1 = m2 4. Subtraction Property of Equality 5. 1  2 5. Definition of Congruence

  15. Homework Two Column Proof Given: mHGK = mJGL Prove:1  3 Statement Reason 1. mHGK = mJGL 1. Given 2. Angle Addition Postulate 2. m∠HGK = m∠1 + m∠2 3. Angle Addition Postulate 3. m∠JGL = m∠3 + m∠2 4. Substitution Property of Equality 4. m∠1 + m∠2 = m∠3 + m∠2 5. Subtraction 5. m∠1 = m∠3 6. ∠1  ∠3 6. Definition of Congruence

  16. Homework Given: PQ = 2x + 5 QR = 6x – 15 PR = 46 Prove: x = 7 Two Column Proof P Q R Statement Reason 1. PQ = 2x + 5, QR = 6x – 15, PR = 46 1. Given 2. PQ + QR = PR 2. Segment Addition Postulate 3. 2x + 5 + 6x – 15 = 46 3. Substitution Property of Equality 4. 8x – 10 = 46 4. Simplify 5. Addition Property of Equality 5. 8x = 56 6. Division Property of Equality 6. x = 7

  17. PR PR PR 2 2 Homework Two Column Proof PR Given: Q is the midpoint of PR. Prove: PQ = and QR = 2 2 Statement Reason 1. Q is midpoint of PR 1. Given 2. PQ = QR 2. Definition of midpoint 3. PQ + QR = PR 3. Segment Addition Postulate 4. QR + QR = PR & PQ + PQ =PR 4. Substitution Property of Equality 5. Simplify 5. 2QR = PR 2PQ = PR 6. Division Property of Equality 6. QR = PQ =

  18. Homework Two Column Proof Given: 2(3x + 1) = 5x + 14 Prove: x = 12 Statement Reason 1. 2(3x + 1) = 5x + 14 1. Given 2. 6x + 2 = 5x + 14 2. Distributive property 3. x + 2 = 14 3. Subtraction Property of Equality 4. Subtraction Property of Equality 4. x = 12

  19. Homework Two Column Proof Given: 55z – 3(9z + 12) = –64. Prove: z = –1 Statement Reason 1. 55z – 3(9z + 12) = – 64 1. Given 2. 55z – 27z – 36 = – 64 2. Distributive Property 3. 28z – 36 = – 64 3. Simplify 4. 28z = –28 4. Addition Property of Equality 5. Division Property of Equality 5. z= – 1

  20. Homework Summary of Properties

  21. Assignment Geometry: 2.4A and 2.4B Section 9 - 24

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