TwiddleLab / Math / Shapes and Proof
Same Shape, Same Size
What you will learn
- Three side lengths fix a triangle: it cannot change shape. A square can.
- Two triangles are congruent when one fits exactly on the other.
- SSS, SAS and ASA are enough to prove it. Angles only, or SSA, are not.
How the lesson goes
Why are bridges built from triangles, not squares?
Both frames are rods joined by hinges. Tap the one you bet keeps its shape when pushed.
Push them.
Drag the blue dot at the top of each frame sideways.
Three rods: 6, 5 and 4. Build a triangle.
Swing the two rods until their ends meet. Then try to build a different one.
Now two sides and an angle: 7, 5 and 35°.
The 35° angle is at the left end, not between the two rods. Find every spot where the rod of 5 lands on the ray.
When are two triangles the same?
Two triangles are congruent when one fits exactly on the other, maybe after a flip.
These are enough: SSS (three sides), SAS (two sides and the angle between), ASA (two angles and the side between).
SSA is not enough: you just found two triangles. Three angles alone are not enough either: the size can change.
Game: the rule sorter.
Equal marks mean equal sides or angles. Tap the rule that proves each pair is the same, or Not enough. Three lives.
This code says two triangles are different. They are the same.
Sides [3, 4, 5] and [5, 3, 4] make the same triangle. Tap the orange code to change it, Run, and make both answers right.
By SSS, three equal sides make congruent triangles, whatever order you list them in. Sorting puts both lists in the same order first.
Make the frames rigid.
Tap a corner, then another corner, to add a brace. Use as few braces as you can.
Your triangle goes to work.
One bridge is made of squares, one of copies of your 6, 5, 4 triangle. Drive the trucks across.
Previous lesson: Why Triangles Add to 180 · Next lesson: Similar Triangles and Shadows · Part of the course: Shapes and Proof
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