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A Lesson on Basic Geometry Proof

A Lesson on Basic Geometry Proof

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Geometry lesson Geometry lesson Presentation Transcript

  • Geometry Lesson
    Bob Roach
    ED538
    Fall 2010
  • Geometry Grade 9-11
    California Geometry Standards Grades 8 through 12
    2.0 Students write geometric proofs, including proofs by contradiction.
  • Warm Up Exercise
    On a piece of paper to be handed in, copy each of the three statement sets then draw (write) a conclusion if possible:
    1) All rectangles have congruent diagonals. ABCD is a rectangle.
    2) All squares have four congruent sides. GHIJ has four congruent sides.
    3) If l is perpendicular to m, then ABC is a right angle. If ABC is a right angle then BCD is complementary to BDA. l is perpendicular to m.
  • Warm Up Exercise Answer (1)
    All rectangles have congruent diagonals. ABCD is a rectangle. We conclude that ABCD has congruent diagonals.
    Statements
    Reasons
    Given
    Given
    Law of Detachment
    All rectangles have congruent diagonals
    ABCD is a rectangle
    ABCD has congruent diagonals
  • Warm Up Exercise Answer (2)
    All squares have four congruent sides. GHIJ has four congruent sides.
    Statements
    Reasons
    Reasons
    Given
    Given
    p -> q ; q
    a) All squares have four congruent sides
    b) GHIJ has four congruent sides.
    c) No conclusion
  • Warm Up Exercise Answer (3)
    Statements
    Reasons
    Given
    Given
    Given
    Syllogism using conditional a and b
    p -> q
    q -> r
    ___________
    p -> r
    e) Law of Detachment c and d
    a) l ⟂ m -> ABC is a right angle
    b) ABC is a right angle -> BCD is complementary to BDA
    c) l ⟂ m
    d) l ⟂m-> BCD is complementary to BDA
    e) BCD is complementary to BDA
  • Homework Questions?
  • Five Essential Parts of a Proof
    State the theorem to be proved
    List the given information
    If possible draw a diagram to illustrate
    State what it is to be proved.
    Develop the proof using deductive reasoning
  • What is a Theorem?
    Statements that are proved by using:
    Definitions
    Postulates
    Undefined terms (line, point, plane,...)
    Deductive reasoning
    are called Theorems
    We then use proved theorems to prove other theorems
  • Verifying Segment Relationships
    Theorem 2-1: Congruence of segments is reflexive, symmetric, and transitive
    AB ≅ AB (Reflexive)
    If AB ≅ CD then CD ≅ AB (Symmetric)
    If AB ≅ CD and CD ≅ EF then AB ≅ EF (Transitive)
    Proofs: Use properties of real numbers with the definition of segment congruence (have same length)
  • Proof of Theorem 2-1 (Symmetric Part)
    P
    Q
    Given: PQ ≅ RS
    Prove: RS ≅ PQ
    S
    R
    Statements
    Reasons
    Given
    Definition of ≅ line segments
    Symmetric property(=)
    Definition of ≅ line segments
    a) PQ ≅ RS
    b) PQ = RS
    c) RS = PQ
    d) RS ≅ PQ
  • Example of Line Segment Proof
    Given: PQ ≅ XY
    QR ≅ YZ
    Prove: PR ≅ XZ
    Z
    X
    Y
    R
    P
    Q
    Statements
    Reasons
    a) PQ ≅ XY
    QR ≅ YZ
    Given
    Definition of ≅ line segments
    Addition property(=)
    Segment addition postulate
    Substitution property (=)
    Definition of ≅ line segments
    QR = YZ
    b) PQ = XY
    c) PQ + QR = XY + YZ
    c) PQ + QR = XY + YZ
    d) PR = PQ + QR
    XZ = XY + YR
    e PR = XZ
    f) PR ≅ XZ
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