Geometry for elementary academy The Side-Side-Side accordance assumption

 11 September 16:58   

    In this chapter, we will alpha the altercation of accordance and accordance theorems.

    We say the two triangles are coinciding if they accept the aforementioned shape.

    The triangles $riangle\; ABC$ and $riangle\; DEF$ accordance if and alone if all the afterward action hold:

    # The ancillary $overline$ equals $overline$.





    # The ancillary $overline$ equals $overline$.





    # The ancillary $overline$ equals $overline$.





    # The bend $angle\; ABC$ equals $angle\; DEF$.





    # The bend $angle\; BCA$ equals $angle\; EFD$.





    # The bend $angle\; CAB$ equals $angle\; FDE$.





    Note that the adjustment of vertices is important.

    It is accessible that $riangle\; ABC$ and $riangle\; ACB$ won’t accordance admitting it is the aforementioned triangle.

    Congruence theorems accord a set of beneath altitude that are acceptable in adjustment to appearance that two triangles congruence.

    The first accordance assumption we will altercate is the Side-Side-Side theorem.

    Given two triangles $riangle\; ABC$ and $riangle\; DEF$ such that their abandon are equal, hence:

    # The ancillary $overline$ equals $overline$.





    # The ancillary $overline$ equals $overline$.





    # The ancillary $overline$ equals $overline$.





    Then the triangles are coinciding and their angles are according too.





    In adjustment to prove the assumption we charge a new postulate.

    The advance is that one can move or cast any appearance in the even with out alteration it.

    It particular, one can move a triangle after alteration its abandon or angles.

    Note that this advance in true in even geometry but not in general.

    If one considers geometry over a ball, the advance is no best true.

    Given the postulate, we will appearance how can we move one triangle to the additional triangle area and appearance that they coincide.

    Due to that, the triangles are equal.

    # ancillary $overline$ to the point D.

    # $circ\; D,overline$.

    # The amphitheater $circ\; D,overline$ and the articulation $overline$ bisect at the point E appropriately we accept a archetype of $overline$ such that it coincides with $overline$.

    # with $overline$ as its base, $overline$, $overline$ as the abandon and the acme at the ancillary of the acme F. Alarm this triangle triangles $riangle\; DEG$

    The triangles $riangle\; DEF$ and $riangle\; ABC$ congruent.

    # The credibility A and D coincide.

    # The credibility B and E coincide.

    # The acme F is an circle point of $circ\; D,overline$ and $circ\; E,overline$.

    # The acme G is an circle point of $circ\; D,overline$ and $circ\; E,overline$.

    # It is accustomed that $overline$ equals $overline$.

    # It is accustomed that $overline$ equals $overline$.

    # Therefore, $circ\; D,overline$ equals $circ\; D,overline$and $circ\; E,overline$ equals $circ\; E,overline$.

    # However, circles of altered centers has at alotof one circle point in one ancillary of the articulation the joins their centers.

    # Hence, the credibility G and F coincide.

    # , Accordingly $overline$ coincides with $overline$ and $overline$ coincides with $overline$.

    # Therefore, the $riangle\; DEG$ coincides with $riangle\; DEF$ and accordingly congruent.

    # Due to the advance $riangle\; DEG$ and $riangle\; ABC$ are according and accordingly congruent.

    # Hence, $riangle\; DEF$ and $riangle\; ABC$ congruent.

    # Hence, $angle\; ABC$ equals $angle\; DEF$, $angle\; BCA$ equals $angle\; EFD$ and $angle\; CAB$ equals $angle\; FDE$.

    The Side-Side-Side accordance assumption appears as [http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI8.html Book I, prop 8] at the Elements.

    The affidavit actuality is in the spirit of the aboriginal proof.

    In the aboriginal affidavit Euclid claims that the vertices F and G haveto accompany but doesn’t appearance why.

    We acclimated the acceptance that “circles of altered centers has at alotof one circle point in one ancillary of the articulation the joins their centers”.

    This acceptance is true in even geometry but doesn’t follows from Euclid’s aboriginal postulates.

    Since Euclid himself , we adopted to accord a added detaild proof, admitting the added assumption.