Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss similarity, students must first have a clear understanding of how dilations behave. This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria. An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.
1. Verify experimentally the properties of dilations given by a center and a scale factor:
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2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
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3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
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4. Prove theorems about triangles.
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5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
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6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
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7. Explain and use the relationship between the sine and cosine of complementary angles.
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8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.?
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1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).?
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