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Thread: Oblique Triangle

  1. #1
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    Oblique Triangle

    On the attached image we have three angles, d $\displaystyle \phi$ and $\displaystyle \theta$ with r2 as a side of the triangle
    I know d and $\displaystyle \phi$ but I am trying to find $\displaystyle \theta$.
    r2 is the normal to a surface and angle d is created by refraction. Since it is measured anti-clockwise it results in a negative angle of -19.77.
    Angle $\displaystyle \phi$ is equal to 35 degrees.
    The author states to determine angle $\displaystyle \theta$ you add d to $\displaystyle \phi$ and this is what I do not understand.
    I would subtract 180 from (145+19.77).
    Both ways give the same result, but I am trying to understand the other way

    Oblique Triangle-img_1142.jpg
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  2. #2
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    Re: Oblique Triangle

    Quote Originally Posted by bmccardle View Post
    The author states to determine angle $\displaystyle \theta$ you add d to $\displaystyle \phi$ and this is what I do not understand.
    Let's label the remaining interior angle of the triangle a for convenience.
    Then the following is true.
    180 = $\phi$ + a (straight line)
    180 = d + a + $\theta$ (sum of angles in triangle)

    Thus
    $\phi$ + a = d + a + $\theta$
    $\phi$ = d + $\theta$
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  3. #3
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    Re: Oblique Triangle

    Quote Originally Posted by bmccardle View Post
    On the attached image we have three angles, d $\displaystyle \phi$ and $\displaystyle \theta$ with r2 as a side of the triangle
    I know d and $\displaystyle \phi$ but I am trying to find $\displaystyle \theta$.
    r2 is the normal to a surface and angle d is created by refraction. Click image for larger version. 

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    This is a straightforward application if the exterior angle theorem.
    The angle $\phi$ is an exterior angle so that $m(\angle\phi)=m(\angle d)+m(\angle\theta)$
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  4. #4
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    Re: Oblique Triangle

    Quote Originally Posted by MacstersUndead View Post
    Let's label the remaining interior angle of the triangle a for convenience.
    Then the following is true.
    180 = $\phi$ + a (straight line)
    180 = d + a + $\theta$ (sum of angles in triangle)

    Thus
    $\phi$ + a = d + a + $\theta$
    $\phi$ = d + $\theta$
    Quote Originally Posted by Plato View Post
    This is a straightforward application if the exterior angle theorem.
    The angle $\phi$ is an exterior angle so that $m(\angle\phi)=m(\angle d)+m(\angle\theta)$
    Thank you both I really appreciate the help.

    I have another question where the oblique triangle has two exterior angles. I looked for a solution, but I did not find one.

    In the attached image angle $\displaystyle \angle u_2$ and $\displaystyle \angle c$ are exterior angles and the author states that:
    $\displaystyle \angle u_2 = \angle \theta_2 - \angle c$

    Is there a good resource for these theorem's?
    Thanks again!

    Oblique Triangle-img_1154.jpg
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  5. #5
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    Re: Oblique Triangle

    "u" and "$\displaystyle \theta_2$" are two angles in a triangle. Since the angles in a triangle sum to 180 degrees (or $\displaystyle 2\pi$ radians) the third, unlabeled, angle in that triangle is $\displaystyle 180- u- \theta_2$ degrees ($\displaystyle 2\pi- u- \theta_2$ radians). By the "vertical angle theorem" angle c is the same [tex]c= 80- u- \theta_2[tex] degrees.
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