# Thread: Oblique Triangle

1. ## 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

2. ## Re: Oblique Triangle

Originally Posted by bmccardle
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 + $\theta$

3. ## Re: Oblique Triangle

Originally Posted by bmccardle
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.
This is a straightforward application if the exterior angle theorem.
The angle $\phi$ is an exterior angle so that $m(\angle\phi)=m(\angle d)+m(\angle\theta)$

4. ## Re: Oblique Triangle

Originally Posted by MacstersUndead
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 + $\theta$
Originally Posted by Plato
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!

5. ## 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.