Prove the SAS Triangle Similarity Theorem from Trigonometry

[Lazerlike42]

Homework Statement

As given in the title. The law of cosines, the law of sines, or any other aspect of trigonometry may be used. Ultimately, I need to show that when two triangles have two pairs of proportional sides and the included angles congruent, that they are similar - that is, the remaining pair of sides are proportional and the other two angles are congruent. I could do it all if I could just prove that one of the other pairs of angles were congruent.

Homework Equations

The Attempt at a Solution

I've spent 3+ hours now messing around with the law of sines, the law of cosines, and anything else I could think of. I honestly have no idea; nothing I've tried seems remotely close.

 

[Mark44]

The Law of Cosines comes to mind. Suppose the triangles are oriented the same way, with angle C to the left, and with sides a and b joining to form angle C on one triangle, and sides ka and kb meeting to form angle C on the other triangle. Suppose that the unknown side on the first triangle is c1 and the the corresponding side on the other triangle is c2.

What does the Law of Cosines say about c1 and c2?

After you have shown that the third pair of sides are in the same proportion as the other pairs of sides, use the Law of Sines or the Law of Cosines again to find one of the other unknown angles on each triangle. At that point you will have found all three sides and two angles of each triangle, so finding the third angle of each triangle will be easy.

 

[Lazerlike42]

The Law of Cosines comes to mind. Suppose the triangles are oriented the same way, with angle C to the left, and with sides a and b joining to form angle C on one triangle, and sides ka and kb meeting to form angle C on the other triangle. Suppose that the unknown side on the first triangle is c1 and the the corresponding side on the other triangle is c2.

What does the Law of Cosines say about c1 and c2?

After you have shown that the third pair of sides are in the same proportion as the other pairs of sides, use the Law of Sines or the Law of Cosines again to find one of the other unknown angles on each triangle. At that point you will have found all three sides and two angles of each triangle, so finding the third angle of each triangle will be easy.

Thanks so much. It really was very simple. I never think of proportions in triangles in the sense of common multiples, so I didn't think of that.