Results 1 to 8 of 8

Math Help - The Three Isometries

  1. #1
    Member
    Joined
    Sep 2010
    Posts
    98
    Thanks
    1

    The Three Isometries

    Two triangles can be congruent if one side and the adjacent angles of one are congruent to one side and the adjacent angles of another.

    I know the proof to this I just don't know the three isometries that are needed. Does anyone know these?

    The proof I have it this:
    Suppose we are given triangle ABC and triangle DEF. These two triangles have angles CAB and FDE which are congruent to each other. These triangles also have angles CBA and FED which are congruent to each other as well. Line segments AB and DE re congruent to each other. Now if either line segment AC is congruent to DF or line segment CB is congruent to EF, we would be finished by using SAS postulate. Therefore, we will assume that line segment AC is not congruent to line segment DF, and in particular that AC is greater than DF. Since AC is greater than DF, there exists a point C' which is not equal to C on line segment AC. Line segment AC' is congruent to line segment DF and , by SAS, triangle ABC' is congruent to triangle DEF. Thus angles ABC' is congruent to angle DEF. Now, by transitive property of angles congruent angle ABC' is congruent to Angles DEF. Now, by the transitive property of angles congruent angle ABC' is congruent to angle ABC, which contradicts the angles construction postulate. Therefore, line segment AC in congruent to line segment DF and triangle ABC is congruent to triangle DEF by SAS.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    909
    Thanks
    27

    proof of triangle congruency

    Quote Originally Posted by matgrl View Post
    Two triangles can be congruent if one side and the adjacent angles of one are congruent to one side and the adjacent angles of another.




    I know the proof to this I just don't know the three isometries that are needed. Does anyone know these?



    The proof I have it this:


    Suppose we are given triangle ABC and triangle DEF. These two triangles have angles CAB and FDE which are congruent to each other. These triangles also have angles CBA and FED which are congruent to each other as well. Line segments AB and DE re congruent to each other. Now if either line segment AC is congruent to DF or line segment CB is congruent to EF, we would be finished by using SAS postulate. Therefore, we will assume that line segment AC is not congruent to line segment DF, and in particular that AC is greater than DF. Since AC is greater than DF, there exists a point C' which is not equal to C on line segment AC. Line segment AC' is congruent to line segment DF and , by SAS, triangle ABC' is congruent to triangle DEF. Thus angles ABC' is congruent to angle DEF. Now, by transitive property of angles congruent angle ABC' is congruent to Angles DEF. Now, by the transitive property of angles congruent angle ABC' is congruent to angle ABC, which contradicts the angles construction postulate. Therefore, line segment AC in congruent to line segment DF and triangle ABC is congruent to triangle DEF by SAS.


    these are three theorems of congruency

    SAS ASA SSS Which one is your statement


    bjh
    Last edited by bjhopper; October 6th 2010 at 04:05 PM. Reason: misspelling
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2010
    Posts
    98
    Thanks
    1
    SAS is in my statement. Should I be incorporating all 3?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    909
    Thanks
    27
    Hello matgrl,
    One will do it but pick the right one based on the givens


    bjh
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Sep 2010
    Posts
    98
    Thanks
    1
    Is it than SAS?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    909
    Thanks
    27
    The first few sentences of your proof define triangle congruency but the theorem is ASA


    bjh
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Sep 2010
    Posts
    98
    Thanks
    1
    Do you know the proof for this?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member oldguynewstudent's Avatar
    Joined
    Oct 2009
    From
    St. Louis Area
    Posts
    241
    After reading your proof several times, I am getting the feeling that you are trying to prove the ASA theorem for congruencies of triangles. Is this correct?

    If so, I believe that your reasoning is sound. However, if this is the case, then you did not finish proving that ASA is true. You need to finish by stating that the proof shows that for any triangle with ASA, then the triangles are congruent.

    Also if this is the case, you should add a few words to

    Two triangles can be congruent if one side and the adjacent angles of one are congruent to one side and the adjacent angles of another.

    showing that this is the statement which you are trying to prove. If I am wrong about what you are trying to prove, then I apologize. Otherwise, very good reasoning.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isometries of R2 and R3
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 10th 2012, 06:02 AM
  2. Isometries of R2
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 4th 2012, 08:17 PM
  3. Isometries of R3
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 4th 2012, 03:38 AM
  4. Isometries
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 2nd 2010, 06:21 PM
  5. Isometries
    Posted in the Geometry Forum
    Replies: 1
    Last Post: March 1st 2010, 03:02 AM

Search Tags


/mathhelpforum @mathhelpforum