Chapter 4
Congruent Triangles
A factory cuts triangular pieces for a frame. Two pieces may look alike, but will one fit exactly where the other belongs? Checking every corner and edge seems like a lot of work. What information would settle the question? Congruent triangles give us a way to investigate exact matches, even when the pieces are turned over or drawn far apart. The challenge is deciding when the measurements we have are enough.
What You'll Explore
5 lessons.
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Can three lengths determine a triangular piece, or could different shapes share those measurements? Start with what it means for two triangles to match exactly, then investigate how their parts correspond and what happens when you try to build a triangle from its sides.
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Suppose you know some angles as well as some lengths. Which combinations pin down a triangle, and does the position of the known angle or side matter? Explore different ways of assembling a triangle and explaining why a proposed match must work.
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A triangular flag may have two matching edges, but what else can you discover about it? Investigate the connections among its sides, corners, and symmetry, then consider what changes when all three edges match.
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How much confidence should you place in a list of matching measurements? Compare triangles that fit different kinds of information, including right triangles, and investigate cases where an apparently convincing match leaves more than one possibility.
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Two triangles may be hidden inside a larger drawing, sharing an edge or overlapping. How could an exact match help you establish something about the whole figure? Explore how to choose useful triangles, organize a proof, and justify a construction.