# Find the slope of intercecting line given angle

My question is very similar to this one, but I don't know how to modify the answer for an angle not relative to the origin. It's been way to long since math class.

If I have the line that passes through two points, lets say 1,3 and 5,3, and intersects a line at 5,3 with an angle θ, how do I calculate the slope of the second line? The formula given (tan(arctan(y/x)−θ)) assumes the angle has a relation to the x axis, but I need a more general formula.

This image should illustrate the case.

I have known line, and a point at which it intersects another line. If I know the angle formed by the two lines, how can I find the slope of the second line.

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Draw a picture! –  David H Apr 5 at 19:16
@DavidH Done. Let me know if that's clear –  Tyrsius Apr 6 at 0:03

θ is given by $tan θ = | \frac {m – M} {1 + mM} |$

From θ = 100 degrees, tan θ is a known quantity H, say.

M, the slope of the line passing through (0, 10) and (10, 0) = … = –1

Therefore, $H = \frac {m + 1} {1 – m}$

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$m = … = \frac {H – 1} {H + 1}$

Note: In case θ = 100 degrees, it is necessary to use the supplementary angle (80 degrees) instead to make sure that both sides are positive.

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Sorry, I didn't follow m-M. What is M? The slope of the first line? Is the formula m = ( tan(θ) -1) / ( tan(θ) + 1)? –  Tyrsius Apr 6 at 6:12
M is the slope of the line passing through (0, 10) and (10, 0). To find M, use the formula M = (0 - 10)/(10 - 0) and you get M = -1. –  Mick Apr 6 at 9:10
L(1), the line passing through (0, 10) and (10, 0) has a slope = M = ... =–1. L(2) is your dotted line and it has a slope = m. If L(1) and L(2) cut each other at angle θ, then the relation between (m, M, θ) is given by the formula as quoted. ‘m – M’ simply means ‘n minus M’. Since M and θ are known quantities, through that formula, we can obtain the value of m exactly like what you have quoted in your comment. You have to let θ be 80 degrees though. –  Mick Apr 6 at 9:27
So then I just do the same thing a second time for another 20 degrees then, to get the 100 degree line slope? –  Tyrsius Apr 7 at 4:21
Also, for a more general formula, is it m = ( tan(θ) + M) / ( tan(θ) - M)? –  Tyrsius Apr 7 at 4:45