Look down at section 8.2. Do you see the title? We'll finally prove the Pythagorean Theorem! The details are a little sticky, but the idea is really very simple: drop in the altitude to the hypotenuse, prove triangles similar, write proportions, massage algebraically.
8.1 The Altitude to the Hypotenuse
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8.2 The Pythagorean Theorem and its Converse
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8.3 The Pythagorean Inequalities
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8.4 The Special Right Triangles
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8.5 The Trigonometric Functions
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8.6 The Inverse Trigonometric Functions
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8.7Â It Ain't Right: Angles and a Side
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8.8 It Ain't Right: Two Sides and an Angle, or Three Sides
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