Inequalities in Two Triangles Calculator

Side-Side-Side Inequality (Hinge Theorem):

\[ \text{If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.} \]

Triangle 1 - Side 1:

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Triangle 1 - Side 2:

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Triangle 1 - Included Angle:

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Triangle 2 - Side 1:

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Triangle 2 - Side 2:

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Triangle 2 - Included Angle:

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Result:

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1. What is the Side-Side-Side Inequality?

The Side-Side-Side Inequality (also known as the Hinge Theorem) compares the lengths of sides in two triangles when two sides of one triangle are congruent to two sides of another triangle, but the included angles are different.

2. How Does the Calculator Work?

The calculator applies the Hinge Theorem and Law of Cosines:

\[ c = \sqrt{a^2 + b^2 - 2ab\cos(C)} \]

Where:

\( a, b \) — Two sides of the triangle
\( C \) — Included angle between sides a and b
\( c \) — Third side opposite angle C

Hinge Theorem: If two sides of one triangle are equal to two sides of another triangle, the triangle with the larger included angle will have the longer third side.

3. Importance of Triangle Inequalities

Details: Understanding triangle inequalities helps in proving geometric relationships, solving construction problems, and analyzing triangle properties in various applications.

4. Using the Calculator

Tips: Enter lengths of two sides and their included angle for both triangles. The calculator will compare the third sides based on the Hinge Theorem.

5. Frequently Asked Questions (FAQ)

Q1: What if the corresponding sides aren't equal?
A: The Hinge Theorem only applies when two sides of one triangle are equal to two sides of another triangle.

Q2: Can this be used for any type of triangle?
A: Yes, the theorem applies to all triangles - acute, right, and obtuse.

Q3: What's the relationship between angle size and side length?
A: Larger angles open wider, forcing the opposite side to be longer to connect the endpoints.

Q4: How is this different from the Triangle Inequality Theorem?
A: The Triangle Inequality Theorem states that the sum of any two sides must be greater than the third side, while the Hinge Theorem compares two triangles.

Q5: What are practical applications of this theorem?
A: Used in engineering, architecture, and physics when analyzing forces or structures with triangular components.