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Chapter 6: Problem 24

Use the previous problem to show that in every triangle, the sum of the lengths of any two sides is greater than the length of the third side.

Short Answer

Expert verified
Using the angles within a triangle, we can prove the Triangle Inequality Theorem that states the sum of the lengths of any two sides is greater than the length of the third side. We can show that: 1. If angle A and angle B add up to less than 180掳, then a + b > c 2. If angle B and angle C add up to less than 180掳, then b + c > a 3. If angle A and angle C add up to less than 180掳, then c + a > b Hence, this proves the Triangle Inequality Theorem for any triangle with sides a, b, and c.

Step by step solution

01

Identify the sides of the triangle

To start with, let's name the sides of the triangle as a, b, and c. We will prove that the sum of the lengths of any two sides of the triangle is greater than the length of the third side.
02

Apply the previous problem's information

Using the information from the previous problem, we know that if angle A is opposite to side a, angle B is opposite to side b, and angle C is opposite to side c, the angles within the triangle must satisfy the following relationships: angle A + angle B + angle C = 180掳
03

Prove that a + b > c

Let's start by proving that a + b > c. We know that angle C is opposite to side c, so for the sum of angles A and B, we have: angle A + angle B = 180掳 - angle C Since angles A and B add up to less than 180掳, the sum of their corresponding sides must be greater than side c. Symbolically, we can write this as: a + b > c
04

Prove that b + c > a

Similarly, let's prove that b + c > a. Since angle A is opposite to side a, for the sum of angles B and C, we have: angle B + angle C = 180掳 - angle A Again, since angles B and C add up to less than 180掳, the sum of their corresponding sides must be greater than side a. Symbolically, we can write this as: b + c > a
05

Prove that c + a > b

Lastly, let's prove that c + a > b. Since angle B is opposite to side b, for the sum of angles A and C, we have: angle A + angle C = 180掳 - angle B Yet again, since angles A and C add up to less than 180掳, the sum of their corresponding sides must be greater than side b. Symbolically, we can write this as: c + a > b
06

Conclusion

From our proofs in steps 3, 4, and 5, we have shown that for any triangle with sides a, b, and c: a + b > c b + c > a c + a > b And with these relationships established, we have successfully proven that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side, demonstrating the Triangle Inequality Theorem.

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