Is it possible to derive explicit formula for unknown angle in an irregular quadrilateral, where 4 sides and 1 angle is known?

Question

Hi,

I've got an irregular quadrilateral where all 4 sides and 1 angle is known. Is it possible to derive an explicit formula for calculating one of the unknown angles adjacent to the known angle?

I've managed to derive an implicit formula. I used the method of splitting the quadrilateral across a diagonal, then using the cosine rule with a=diagonal for both triangles. These 2 cosine expressions can then be equated, which gives a formula relating opposing angles.

The same process can then be carried out for the other diagonal. This gives 2 expressions, each relating opposing angles in the quadrilateral.

By then using the fact that the sum of all 4 angles is 360 degrees, and substituting the 2 derived expressions in for 2 of the angles, gives a formula relating the unknown angle to the known angle. However this equation is implicit as I can't rearrange it to get the unknown angle on one side.

Any help would be greatly appreciated, and if it's not possible to get an explicit formula, then please let me know; many thanks.

Answer 1

Yes, you are on the right track!!! Good job!

Re-draw the quadrilateral like an "arrow" shape, with two sides meeting at

one point, the other two sides meeting at another (different) point and a

signal diagonal connecting them.

It looks like a pair of wings, or an arrow.

Specifically, for quadrilateral ABCD,

AB and AC are on one side and BD and CD are on the other.

The two triangles ABC and BCD share the common side

BC and A and D are the wingpoints.

There are two unknowns, so we need 2 equations.

The diagonal BC is unknown and so is the other angle

at the opposite wingpoint, namely angle BDC

As you have stated, you can find the length of the

diagonal BC using the law of cosines.

Specifically, BC^2 = AB^2 + AC^2 - 2(AB)(AC)* cos (BAC)

where AB, AC, and angle BAC are known.

The second equation is:

BC^2 = BD^2 + CD^2 - 2(BD)(CD)*cos(BDC)

where BD and CD are known

You can solve this non-linear system setting the

equations equal to each other, since they both equal BC^2.

You will have to find the inverse cosine or arc-cosine to

get the missing angle.

Once found, you can get the length of the diagonal

by plugging it back in