Properties of Congruence

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Properties of Congruence

  1. 1. "Mathematicsteaches youto think"
  2. 2. Properties ofCongruence
  3. 3. An example of congruence. The twofigures on the left are congruent, while the third is similar to them. The last figure isneither similar nor congruent to any of the others. Note that congruences alter some properties, such as location and orientation, but leave othersunchanged, like distance and angles. The unchanged properties are called invariants.
  4. 4. Congruenceof triangles
  5. 5. DeterminingCongruence
  6. 6. The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS),two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles.
  7. 7. Sufficient evidence forcongruence between twotriangles in Euclidean spacecan be shown through thefollowing comparisons:
  8. 8. Angle-Angle-Angle In Euclidean geometry, AAA (Angle- Angle-Angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does notprovide information regarding the sizeof the two triangles and hence proves only similarity and not congruence in Euclidean space.
  9. 9. However, in sphericalgeometry and hyperbolicgeometry (where the sumof the angles of a triangle varies with size) AAA issufficient for congruence on a given curvature of surface.
  10. 10. If two triangles arecongruent, then each part of the triangle (side or angle) iscongruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved twotriangles are congruent, you canfind the angles or sides of one of them from the other.
  11. 11. "Corresponding Parts of Congruent Triangles are Congruent"CPCTC is intended as an easy wayto remember that when you have two triangles and you have proved they are congruent, theneach part of one triangle (side, or angle) is congruent to the corresponding part in the other.
  12. 12. JustificationUsingProperties ofEquality andCongruence
  13. 13. Properties OfEquality ForReal Numbers
  14. 14. Properties OfCongruence
  15. 15. EXAMPLES:
  16. 16. Statement Reason1. 15y + 7 = 12 - 20y 1. Given2. 35y + 7 = 12 2. Additive Property3. 35y = 5 3. Subtractive Property4. Y = 1/7 4. Division Property
  17. 17. Statement Reason Statement Reason1. m ∠1 + m ∠2 =100 1. Given2. m∠ 1 = 80 2. Given3. 80 + m∠ 2 = 100 3. Substitution Property4. m ∠2 = 20 4. Subtraction Property
  18. 18. Statement Reason Statement Reason1. m∠ 1 + m∠ 3 = 80 1. Given2. m∠ 1 = 40 2. Given3. m∠ 3 = 40 3. Subtraction Property4. m∠ 4 + m∠ 2 = 80 4. Given5. m∠ 2 = 40 5. Given6. m∠ 4 = 40 6. Subtraction Property7. m∠ 3 = m∠ 4 7. Transitive Property
  19. 19. Statement Reason Statement Reason1. m∠ 1 + m∠ 2 = 180 1. Given2. m∠ 2 + m∠ 3 = 180 2. Given3. m∠ 1 + m∠ 2 = m∠ 2 + m∠ 3 3. Transitive Property4. m∠ 2 = m∠ 2 4. Reflexive Property5. m∠ 1 = m∠ 3 5. Subtraction Property
  20. 20. Proofs are the heart of mathematics. Ifyou are a math major, then you must come to terms with proofs--you must be able to read, understand and write them. What is the secret? What magic do you need to know? The short answer is: there is no secret, nomystery, no magic. All that is needed is some common sense and a basic understanding of a few trusted and easy to understand techniques.
  21. 21. PROOFS
  22. 22. Ending a proof Sometimes, the abbreviation "Q.E.D." is writtento indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", whichis Latin for "that which was to be demonstrated". A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" in an oral presentation on a board.
  23. 23. EXAMPLE
  24. 24. THE END.

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