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Math Help - Proving - All the angles of a rectangle are right angles.

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    Proving - All the angles of a rectangle are right angles.

    Hey Guys,

    I need some help trying to prove that all the angles of a rectangle are right angles.
    My teacher asks us to go in depth so we have to list a ton of reasons.

    So first let's have the Given

    Given: Rectangle ABCD with right angle A
    Reason: Given, & the it's also the definition of a rectangle.

    1) Statement: Diagonal line segment AC and line segment DB
    1)Reason: I constructed a diagonal line across the rectangle

    2)Statement: DA || BC and AB || DC
    2)Reason: Definition of parallelogram

    3)Statement: AC is congruent to AC and BD is congruent to BD
    3)Reason: Identity

    4)Statement: (M) is for midpoint where the two diagonals meet. So
    Triangle DMA is congruent to Triangle MCB
    4)Reason: SSS

    and here I am lost. I'm not sure if my statements and reasons are correct and in order. I will appreciate the help. Thank you, forum.
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    Re: Proving - All the angles of a rectangle are right angles.

    A rectangle is DEFINED to be a four sided polygon with all angles congruent.

    The angle sum of a polygon of "n" sides is \displaystyle \begin{align*} (n - 2) \cdot 180^{\circ} \end{align*}. So for a quadrilateral, the angle sum is \displaystyle \begin{align*} (4 - 2)\cdot 180^{\circ} = 360^{\circ} \end{align*}.

    Since all the angles are equal, that means if "a" is one of the angles, then \displaystyle \begin{align*} 4a= 360^{\circ} \implies a = 90^{\circ} \end{align*}.

    Therefore, the angles in a rectangle are all right angles.
    Last edited by Prove It; July 10th 2013 at 02:46 AM.
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    Re: Proving - All the angles of a rectangle are right angles.

    Hey Cake.

    You might want to give us an indication of how much detail is required from your teacher.

    The reason I ask is that many people would just take this fact as a given where-as you are on the other hand, required to go into a lot of detail.

    Do you have a list of axioms that you need to use and if so, can you list them here?
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    Re: Proving - All the angles of a rectangle are right angles.

    Quote Originally Posted by Prove It View Post
    A rectangle is DEFINED to be a four sided polygon with all angles congruent.

    The angle sum of a polygon of "n" sides is \displaystyle \begin{align*} (n - 2) \cdot 180^{\circ} \end{align*}. So for a quadrilateral, the angle sum is \displaystyle \begin{align*} (4 - 2)\cdot 180^{\circ} = 360^{\circ} \end{align*}.

    Since all the angles are equal, that means if "a" is one of the angles, then \displaystyle \begin{align*} 4a= 360^{\circ} \implies a = 90^{\circ} \end{align*}.

    Therefore, the angles in a rectangle are all right angles.
    Hey Prove it,

    Yes, I am aware of all the facts you have listed. I just have to list the in between theorems and postulates.
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    Re: Proving - All the angles of a rectangle are right angles.

    Quote Originally Posted by chiro View Post
    Hey Cake.

    You might want to give us an indication of how much detail is required from your teacher.

    The reason I ask is that many people would just take this fact as a given where-as you are on the other hand, required to go into a lot of detail.

    Do you have a list of axioms that you need to use and if so, can you list them here?
    Hey Chiro,

    Well, I have to list a bunch of in between theorems and postulates. Right triangle theorems as well as parallelogram theorems, and finish it off with a rectangle theorem stating all angles of a rectangle are right angles. It's sort of answering a proof in paragraph form except they just have to be bullet points. I think it's way too much to answer online, considering the amount of exact detail that my teacher goes by.
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