# Mathematics II Similarity, Right Triangles, and Trigonometry

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#### Resources

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°).