If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the greater side lies opposite the greater angle.
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Inequalities
some Important Facts
Rules useful in handling problems involving inequalities.
To understand why this statement is true, we need to remember a few properties of triangles.
1. Triangle Angle Sum Property: The sum of the three angles in a triangle is always 180 degrees.
2. Triangle Inequality Property: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
Now, let's analyze the given statement:
"If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the greater side lies opposite the greater angle."
Let's consider a triangle ABC:
A
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c/_____\b
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/_________\
B a C
Here, angle A is unequal to angle B.
Proof:
1. If angle A is unequal to angle B, then angle A + angle B must be less than 180 degrees since the sum of the angles in a triangle is always 180 degrees.
2. Let's assume that side a is greater than side b. We'll prove that angle A is greater than angle B.
3. According to the triangle angle sum property, angle C = 180 - (angle A + angle B). Since angle A + angle B < 180, angle C > 0.
4. By using the triangle inequality property, we can conclude that side a + side b > side c. If side a > side b, then side a + side b > side b + side b = 2*side b. So, side a + side b > 2*side b > side c.
5. Now, let's assume that angle A is less than angle B. Since side a > side b and angle A < angle B, we can use the triangle inequality property to say that side a + side b > side b + side b. However, this contradicts our previous deduction that side a + side b > side c.
6. Hence, our assumption that angle A is less than angle B is incorrect. Therefore, angle A must be greater than angle B.
Consequently, we can conclude that if two angles of a triangle are unequal, the sides opposite these angles are unequal, and the greater side lies opposite the greater angle.