Chapter 1
Foundations and Reasoning
A sketch can make two lengths look equal or two lines look parallel. How much should you trust what you see, and what would convince someone who sees the drawing differently? Geometry starts by making our descriptions precise and asking what supports a claim. You already know many familiar shapes. Here, the question changes from recognizing a shape to deciding what you can justify about it, even when a sketch is tilted, stretched, or drawn badly.
What You'll Explore
6 lessons.
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Imagine drawing the edges of a room on a flat page. Which relationships belong to the room, and which come from the drawing? Explore the language geometry uses to describe positions and directions precisely.
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One mark may look halfway between the ends of a segment. How could you check that claim? Investigate what length measurements tell you about a drawing and how distances connect its different parts.
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Are the two openings between a pair of rays always the same size? Where should a protractor's zero go? Explore how to name, measure, and compare angles, and decide when their measures can be combined.
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Could you decide exactly which points belong to a circle, or copy a length without reading a ruler? Explore what circles describe and what a compass and straightedge can build from the marks and distances you are given.
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Does a claim become certain after you check many drawings? Investigate how definitions, examples, and counterexamples help you judge geometric claims, and why changing the wording of a statement can change what it says.
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Some goals ask you to show that something can never happen, but where could you begin? Explore how to plan a route through a geometric argument, choose an approach, and check that each step has a reason.