The triangle inequality and the Hinge Theorem
- The hinge sweep — 3 tasks
- Which sum can fail — 3 tasks
- The range of the third side — 3 tasks
- The dial — 3 tasks
- Bigger angle, bigger triangle? — 3 tasks
- Practice — 26 tasks
- Summary — 2 tasks
- The range, and what turns the dial — 3 tasks
Classroom activities
Things a class does rather than writes on. Each one is a finished PDF: print it, cut it up if it says so, and hand it round.
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Will These Three Sides Close? The Triangle Inequality — Cut & Glue Sort 20–25 minutes, including the cutting · 4 sheets
12 sets of three side lengths, printed out of order. Students sort the numbers, add the two smaller ones, and glue each piece in one of three rows: makes a triangle, leaves a gap, or lies flat. The flat row is the case tests ask about: two sides that add to exactly the third make no triangle.
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Does the Hinge Theorem Fit? — Answer Maze 10–15 minutes · 4 sheets
8 boxes on one page, each with two small triangles or two walks. Half the boxes ask which way the Hinge Theorem goes: the theorem, its converse, and two walkers who turn by different angles. Half ask whether it fits at all: a shared side, an angle that is not between the equal sides, a side with no number, and equal angles. The row of box numbers at the foot is the check.
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Can You Be Sure Which Side Is Longer? The Hinge Theorem — Mystery Picture 25–35 minutes · 4 sheets
25 squares, each two triangles that share their middle side, with the two arms and two angles printed: fill the square if the Hinge Theorem shows that the left bottom side is longer. The white squares are the mistakes students really make: using angles that are not between the equal sides, arms that are not equal, the wrong direction, and equal angles. When every square is right, a picture appears.
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How Long Can the Third Side Be? The Triangle Inequality — Scavenger Hunt a full class period · 13 sheets
16 cards around the room, each giving two sides of a triangle whose third side is a whole number. Some ask for the largest third side, some for the smallest, and some for the largest perimeter. Using the sum or the difference itself, which makes the three sides lie flat, lands on a card that sends the group back, and a clean run is 8 lines on the record sheet.