Similarity, Right Triangles, and Trigonometry
|Understand similarity in terms of similarity transformations.|
|1. Verify experimentally the properties of dilations given by a center and a scale factor:|
|a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through
the center unchanged.
|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.
|3. Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.|
|Prove theorems involving similarity. [Focus on validity of underlying reasoning while using variety of formats.]|
|4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally,
and conversely; the Pythagorean Theorem proved using triangle similarity.
|5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.|
|Define trigonometric ratios and solve problems involving right triangles.|
|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.
|7. Explain and use the relationship between the sine and cosine of complementary angles.|
|8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.|
|8.1 Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90°and 45°, 45°, 90°).|