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Inequalities Involving Two Triangles - Interactive Activity

Two triangles can share two pairs of sides that are the same length and still be different triangles. What makes them different is the angle between those sides, and that angle decides the length of the side across from it. This activity lets you open and close that hinge, compare two triangles side by side, and work comparisons step by step.

Definitions and Main Ideas

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 Theorem (Hinge Theorem): 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.

Hinge Explorer

What to do: First, move the First triangle slider to open or close its hinge and watch its third side stretch or shrink. Then switch on Compare two triangles. Then move the Second triangle slider to set the second angle, and notice which triangle has the longer third side.

First triangle — included angle 40°

This slider opens or closes the first triangle's hinge. Watch its third side stretch or shrink.

Compare two triangles shows a second triangle beside the first, so you can see which included angle gives the longer third side.

Which Is Longer?

What to do: Look at each pair of triangles. Decide which third side is longer, which included angle is greater, or whether the hinge theorems can be used at all. Choose an answer. A first miss gives a hint and another try.

These buttons are your answer choices for the pair shown.

The Next pair button cycles through all eight pairs in order.

Build the Comparison

What to do: Work each problem one step at a time. First choose the theorem, then write the inequality or equation, then solve it, then check your answer. The diagram marks the part each step is about, in both triangles.

Choose the statement, inequality, or value for this step.

Continue moves to the next step. New Problem cycles through all five problems in order.

Your Turn

What to do: Answer each question. A first miss gives a hint and another try. A second miss shows the full explanation. Your score counts first-try answers only.

Previous and Next move between the six questions in order.

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