Prove the SAS Triangle Similarity Theorem from Trigonometry

In summary, the conversation discusses the use of the Law of Cosines and the Law of Sines to prove that two triangles with two pairs of proportional sides and congruent included angles are similar. It is suggested to use the Law of Cosines to find the third pair of proportional sides, and then use the Law of Sines or the Law of Cosines again to find the remaining angles of the triangles. The conversation concludes with the realization that proportions in triangles can be thought of in terms of common multiples.
  • #1
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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.
 
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  • #2
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.
 
  • #3
Mark44 said:
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.
 

1. What is the SAS Triangle Similarity Theorem?

The SAS Triangle Similarity Theorem states that if two triangles have two pairs of corresponding sides that are proportional and the included angles are congruent, then the triangles are similar.

2. How can trigonometry be used to prove the SAS Triangle Similarity Theorem?

Trigonometry can be used to prove the SAS Triangle Similarity Theorem by using the Law of Sines and the Law of Cosines. The Law of Sines states that the ratio of the sine of an angle to the length of its opposite side is constant for all sides of a triangle. The Law of Cosines states that the sum of the squares of the lengths of two sides of a triangle is equal to the square of the third side minus twice the product of the two sides and the cosine of the included angle.

3. Can you provide an example of how trigonometry can be used to prove the SAS Triangle Similarity Theorem?

Yes, consider two triangles ABC and DEF, where ∠A = ∠D, ∠B = ∠E, and AB/DE = BC/EF. Using the Law of Cosines, we can set up the following equations: AB² = AC² + BC² - 2(AC)(BC)cos(∠A) and EF² = DE² + DF² - 2(DE)(DF)cos(∠D). Since ∠A = ∠D and ∠B = ∠E, we can substitute these values into the equations. We can also substitute AB/DE for BC/EF. Simplifying the equations, we get AC/DE = BC/EF, which shows that the two triangles are similar by definition.

4. Are there other methods to prove the SAS Triangle Similarity Theorem?

Yes, there are other methods to prove the SAS Triangle Similarity Theorem, such as using the Side-Angle-Side (SAS) Congruence Theorem and the Angle-Angle (AA) Similarity Theorem. However, using trigonometry is a more straightforward and efficient method to prove this theorem.

5. Why is the SAS Triangle Similarity Theorem important?

The SAS Triangle Similarity Theorem is important because it allows us to determine if two triangles are similar without having to measure every side and angle. This theorem is also used in many real-world applications, such as in surveying, engineering, and navigation. It is also a fundamental concept in geometry and is often used in more complex geometric proofs.

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