MATH 85

Until now, mathematics has mostly asked you to find answers. This year it asks a harder and more interesting question: how do you *know*? A proof is an argument so airtight that a determined skeptic — someone who refuses to take your word for anything — must still agree with your conclusion. Learning to build such arguments, and to find the cracks in arguments that only look airtight, is the real subject of this course. Geometry is simply the arena where the game is easiest to learn.

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Two figures are congruent when one is an exact copy of the other — same size, same shape, every measurement identical. You met this idea in Grade 8 through slides, flips, and turns. Now congruence becomes a *tool for proving things*: if you can show two triangles are congruent, you instantly know all six of their remaining measurements match, and that knowledge cracks open problems about bridges, bisectors, and shapes that seem to have nothing to do with triangles. The art is proving congruence from just three well-chosen pieces of information.

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Congruence demands an exact copy. Similarity relaxes exactly one requirement: the copy may be scaled. A photo and its enlargement, a building and its blueprint, a triangle and its shadow-cast twin — same shape, different size. That single relaxation makes similarity the workhorse of *indirect measurement*: it lets you measure a tree, a cliff, or a pyramid using nothing but a ruler-sized object and a proportion. It is also, as the next unit reveals, the reason trigonometry exists.

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Trigonometry has a reputation for mystery — sine, cosine, buttons on a calculator that produce nine decimal places from nowhere. The mystery dissolves the moment you see where the numbers come from: trigonometry is nothing but similarity, bottled. Because all right triangles sharing an acute angle are similar, certain side ratios depend only on the angle — not on the triangle. Name those ratios, tabulate them once, and you can solve every right triangle that will ever exist. That is the whole invention.

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A circle is the most symmetric shape there is — every point on it the same distance from the center — and that symmetry breeds theorems with almost unreasonable regularity. Angles drawn inside circles obey exact numerical laws; lines that barely graze a circle meet its radius at an exact angle, every time. This unit builds the vocabulary, then proves the two most useful laws and puts them to work.

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This unit merges the two great traditions of the course. Geometry gives you theorems about distances and right angles; algebra gives you coordinates and equations. Put them together and every geometric claim becomes something you can *compute* — and the star of the merger is an old friend. The Pythagorean Theorem, which you learned to use in Grade 8, returns for the full Meridian treatment: proved, dissected, converse-tested, and reborn as the distance formula that measures the entire coordinate plane.

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