Inequalities Involving Two Triangles
Overview
This lesson compares two triangles that are not congruent. When two sides of one triangle are congruent to two sides of another, everything hinges on the angle between those sides: open the hinge wider and the third side stretches longer. The SAS Inequality Theorem (the hinge theorem) runs the comparison from the included angle to the third side, and the SSS Inequality Theorem runs it in reverse, from the third side back to the included angle.
Vocabulary
- Included Angle
- The angle formed by two sides of a triangle; the angle between the two sides.
- SAS Inequality Theorem (Hinge Theorem)
- If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle has a greater measure than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.
- SSS Inequality Theorem
- If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle has a greater measure than the included angle of the second triangle.
- Congruent Sides
- Sides with the same length; the hinge theorems only apply when the two sides forming the angle are congruent in both triangles.
Main Ideas
- Picture the congruent sides as a hinge: the wider the angle opens, the longer the side across from it becomes.
- SAS Inequality: greater included angle means longer third side (angle compared first, side concluded).
- SSS Inequality: longer third side means greater included angle (side compared first, angle concluded) — the reverse of the hinge theorem.
- Neither theorem applies unless the two sides forming the compared angle are congruent in both triangles.
- With expressions, a comparison such as 'longer' becomes a strict inequality, not an equation.
Lesson Notes
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Lesson Video
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Practice Problems
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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.