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Deductive Reasoning

Learn about deductive reasoning, geometric vocabulary, properties of equality, biconditional statements, congruence properties, and theorems in geometry. Master proofs, postulates, and theorems about angles, perpendicular lines, midpoint, and angle bisector. Explore complementary angles, supplementary angles, and perpendicular lines. Plan and execute geometric proofs effectively.

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Deductive Reasoning

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  1. Deductive Reasoning Geometry Chapter 2

  2. Vocabulary • Converse-formed by interchanging the hypothesis and the conclusion Statement: If p, then q Converse: If q, then p

  3. Vocabulary • Counterexample-an example that can be found for which the hypothesis is true and the conclusion is false.

  4. Conditional • If-then Statements/Conditional Statements • “If B is between A and C, then AB+BC=AC • If Katie eats a lot, then Katie is fat. Hypothesis is in Red Conclusion is in Blue

  5. Biconditional • A statement that contains the words “if and only if”. • 3x=12 if and only if, x=4 • Katie gets hyper in the morning if and only if she drinks coffee

  6. Properties from Algebra Geometry Ch.2 Section 2

  7. Properties of Equality Addition Property If a=b and c=d, then a+c=b+d Subtraction Property If a=b and c=d then a-c=b-d Multiplication Property If a=b, then ca=cb Division Property If a=b and c≠0, then a/c=b/c

  8. Properties from Algebra Substitution Property If a=b, then either a or b may be substitute for the other in any equation (or inequality). Reflexive Property a=a Symmetric Property If a=b, then b=a Transitive Property If a=b Distributive Property a(b+c)=ab+ac

  9. Properties of Congruence • Reflexive Property – • Symmetric Property- • Transitive Property-

  10. Proving Theorems Geometry Ch.2 Lesson 3

  11. Vocabulary • Theorem-statements that are proved • Postulates-statements that are accepted without proof

  12. Midpoint Theorem • If M is the midpoint of line AB, then AM=1/2AB and MB=1/2AB

  13. Angle Bisector Theorem If ray AD is the bisector of <CAB, then m<CAD=1/2m<CAB and m<DAB=1/2m<CAB

  14. Proofs/Deductive Reasoning

  15. Theorems about Angles and Perpendicular Lines Geometry Ch. 2 Lesson 4

  16. Vocabulary • Complementary Angles-two angles whose measures have the sum of 90 degrees. • Supplementary Angles-two angles whose measures have the sum of 180 degrees • Vertical Angles-two angles such that the sides of one angle are opposite rays to the sides of the other angle.

  17. Theorem • Vertical Angles are congruent

  18. Perpendicular Lines Geometry Chapter 2 Lesson 5

  19. Vocabulary • Perpendicular Lines-two lines that intersect to form right angles (90 degree angles).

  20. Theorem • If two lines are perpendicular, then they form congruent adjacent angles

  21. Theorem • If two lines form congruent adjacent angles, then the lines are perpendicular.

  22. Theorem • If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.

  23. Planning A Proof Chapter 2 Lesson 6

  24. Theorem • If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.

  25. Theorem • If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.

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