[G.SRT.] Similarity, Right Triangles, and Trigonometry

[G.SRT.] Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations.

• • [G.SRT.1] Verify experimentally the properties of dilations given by a center and a scale factor:

  • [G.SRT.1.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.

    • [G.SRT.1.b] The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

• [G.SRT.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.

  • [G.SRT.3] Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.

Prove theorems involving similarity.

• • [G.SRT.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.

  • [G.SRT.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.

• • [G.SRT.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.

  • [G.SRT.7] Explain and use the relationship between the sine and cosine of complementary angles.

  • [G.SRT.8] (*) Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

  • [G.SRT.8.1] Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90°and 45°, 45°, 90°). (CA)

Apply trigonometry to general triangles.

• • [G.SRT.9] (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

  • [G.SRT.10] (+) Prove the Laws of Sines and Cosines and use them to solve problems.

  • [G.SRT.11] (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).