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Math Help - Intersecting lines

  1. #1
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    Intersecting lines

    In the figure below, transversal r crosses both p and q, and angle a and b are measures of the indicated angles, both between 0 and 180. Lines p and q will cross somewhere to the left of transversal r (that is, on the side opposite the indicated angles).
    Question: Why is a + b > 180 a true relationship between a and b for all possible positions of transversal r?
    Attached Thumbnails Attached Thumbnails Intersecting lines-lines.jpg  
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  2. #2
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    Hello, dannyc!

    \text}In the figure below, transversal }r\text{ crosses both }p\text{ and }q,
    \text{and angles }a\text{ and }b\text{ are measures of the indicated angles,}
    \text{both between }0^o\text{ and }180^o.

    \text{Lines }p\text{ and }q\text{ intersect somewhere to the left of transversal }r.

    \text{Why is }a + b \,>\, 180^o\text{ true for all possible positions of transversal }r\,?

    Code:
                                       r
                                      /       p 
                                   K /    *
                                    o
                             *     / a
                      *           /
               *                 /
      J o                       /
               *               /
                      *       / b
                             o
                            / L    *
                                         *
                                             q

    Let: . \begin{Bmatrix} p \cap q &=& J \\<br />
p \cap r &=& K \\ q \cap r &=& L \end{Bmatrix}

    We have: . \begin{Bmatrix}\angle JKL \,=\,180^o - a \\ \angle JLK \,=\,180^o - a \end{Bmatrix}

    Hence: . \angle J \;=\; 180^o - (180^o - a) - (180^o - b) \;=\;(a+b) - 180^o


    Since \angle J is a positive angle: . (a+b)-180^o \:>\:0^o

    Therefore: . a + b \:>\:180^o

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  3. #3
    Junior Member
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    Quote Originally Posted by Soroban View Post
    Since \angle J is a positive angle: . (a+b)-180^o \:>\:0^o

    Thanks Soroban! I think I got it now! (Btw, JLK = 180 - b above)
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