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

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.

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 AA criterion for two triangles to be similar.

G.CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
G.CO. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180∘; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

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.
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.9. (+) Derive the formula A=1/2absin(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).

F.TF.8. Prove the Pythagorean identity sin^2(θ)+cos^2(θ)=1and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

## Standards

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

- 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.
- 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 AA criterion for two triangles to be similar.

G.CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

G.CO. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180∘; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G.CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

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.

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.9. (+) Derive the formula A=1/2absin(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).

F.TF.8. Prove the Pythagorean identity sin^2(θ)+cos^2(θ)=1and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

## Examples of Learning Cycles and Tasks

Angle Pairs and their Relationships(No. Davis CA, refers to G.CO.9)Definition of similarity -- Core Academy learning cycle

Trigonometric Ratios (Murray CA, refers to G.SRT.6-7)

Pythagorean Identity (Springville CA, refers to F.TF.8)

Pythagorean Identity-The Trig Farmer's Swine and Coswine (Cache CA , refers to F.TF.8)

Define trigonometric ratios and solve problems (Orem CA, refers to G.SRT.6-7)

Define trigonometric ratios and solve problems (Orem CA, refers to G.SRT.7)

Defining Trigonometry Functions (St. George)

Trig through similar triangles (Cedar City, UT)

Geometric Proofs-A Learning Cycle (SLC-WestHS--G.CO.9, 10, 11)

Writing Formal Proofs(West HS)

How Long Is Your String (West High CA, G.SRT.8)

Relationship between sine & cosine in complementary angles (Wasatch CA, refers to G.SRT.7)

Trigonometric Ratios (Copperhills CA, refers to G.srt.6-7)

Triangle Proofs (Copperhills CA, refers to G.CO.9)

Intro to Writing Proof (Richfield CA, refers to G.CO.xx)