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Math Help - Geometric Proofs Using Numbers...Possible?

  1. #1
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    Smile Geometric Proofs Using Numbers...Possible?

    The Statement vs. Reason chart normally used for proofs in high schools across the USA is a common thing.

    COROLLARY:

    "An exterior angle of a triangle is greater than either remote interior angle."

    Can I prove this corollary using numbers instead of the statement vs reasons chart?

    Sample:

    Let angle A = exterior angle

    Let BCD = a triangle

    Let angle B = 70 degrees

    Let angle C = 10 degrees

    Let angle D = 100 degrees

    Then angles A and angle D form a linear pair on a straight line.

    Then the measure angle A = 180 degrees - the measure of angle D, which is 100 degrees. The number 180 comes from the fact that a straight line measures 180 degrees.

    Of course, 180 - 100 = 80.

    So, the measure of angle A = 80 degrees, which is indeed GREATER THAN than EITHER of the two remote interior angles of 10 degrees and 70 degress.

    Can this accepted as the prove for the above corollary without using the statement vs reasons chart that often confuses students far and near?

    Also, can this method with numbers be used for any geometric proofs situation?





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    Quote Originally Posted by magentarita View Post
    The Statement vs. Reason chart normally used for proofs in high schools across the USA is a common thing.

    COROLLARY:

    "An exterior angle of a triangle is greater than either remote interior angle."

    Can I prove this corollary using numbers instead of the statement vs reasons chart?

    Sample:

    Let angle A = exterior angle

    Let BCD = a triangle

    Let angle B = 70 degrees

    Let angle C = 10 degrees

    Let angle D = 100 degrees

    Then angles A and angle D form a linear pair on a straight line.

    Then the measure angle A = 180 degrees - the measure of angle D, which is 100 degrees. The number 180 comes from the fact that a straight line measures 180 degrees.

    Of course, 180 - 100 = 80.

    So, the measure of angle A = 80 degrees, which is indeed GREATER THAN than EITHER of the two remote interior angles of 10 degrees and 70 degress.

    Can this accepted as the prove for the above corollary without using the statement vs reasons chart that often confuses students far and near?

    Also, can this method with numbers be used for any geometric proofs situation?




    You have used a specific example to verify that the corollary is true. To show that it is universally true, you need to prove the corollary for all exterior angles of every triangle. Showing something is true by establishing a pattern of truths through examples and events is inductive. We must use deductive reasoning here.
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    Masters is right. The way I would prove this statement is to show that an exterior angle of a triangle is equal to the sum of the two remote interior angles. Since no angle can be equal to zero, neither of the two remote interior angles will be as large as the exterior angle.
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    Then...

    Quote Originally Posted by masters View Post
    You have used a specific example to verify that the corollary is true. To show that it is universally true, you need to prove the corollary for all exterior angles of every triangle. Showing something is true by establishing a pattern of truths through examples and events is inductive. We must use deductive reasoning here.
    In other words, there is no way to escape using deductive reasoning to solve geometric proofs, right?

    Most people do not use deductive reasoning throughout their lives, right?

    Your reply?
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    Tell me......

    Quote Originally Posted by icemanfan View Post
    Masters is right. The way I would prove this statement is to show that an exterior angle of a triangle is equal to the sum of the two remote interior angles. Since no angle can be equal to zero, neither of the two remote interior angles will be as large as the exterior angle.
    Can you show me how to prove this using the statements vs reasons chart?

    In this form of prove, we state what is given in the statement section and then in the reason section we write GIVEN.

    What comes after that?

    Statements...................Reason
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    Quote Originally Posted by magentarita View Post
    Can you show me how to prove this using the statements vs reasons chart?

    In this form of prove, we state what is given in the statement section and then in the reason section we write GIVEN.

    What comes after that?

    Statements...................Reason
    See diagram.
    Reasons are in red.

    1) m\angle 1+m\angle 2=180 Angles 1 and 2 make up a linear pair which total 180 degrees.

    2) m\angle 2+m\angle 3+m\angle 4=180 The sum of the measures of the angles of a triangle = 180 degrees.

    3) m\angle 1=180-m\angle 2 Subtraction using Step 1.

    4) m\angle 3+m\angle 4 = 180-m\angle2 Subtraction using Step 2.

    5) m\angle 1=m\angle 3 + m\angle 4 Substitution, using Steps 3 and 4.

    6) \therefore m\angle 1 > m\angle 3 \ \ and \ \ m\angle 1 > m\angle 4 A sum is greater than each of its parts.
    Attached Thumbnails Attached Thumbnails Geometric Proofs Using Numbers...Possible?-ineq.bmp  
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    Quote Originally Posted by magentarita View Post
    In other words, there is no way to escape using deductive reasoning to solve geometric proofs, right?

    Most people do not use deductive reasoning throughout their lives, right?

    Your reply?
    Wrong, deductive reasoning is used quite extensively if you think. And we think. Deductive reasoning is one of the two basic forms of valid reasoning. While inductive reasoning argues from the particular to the general, deductive reasoning argues from the general to a specific instance.
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