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### Math

## Common Core Math

# High School: Geometry: Similarity, Right Triangles, and Trigonometry

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.

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.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

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.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

- 30-60-90 triangle example problem
- Calculating angle measures to verify congruence
- Challenging similarity problem
- Corresponding angles in congruent triangles
- Determine congruent triangles
- Determine similar triangles: Angles
- Determine similar triangles: SSS
- Determining congruent triangles
- Determining similar triangles
- Find angles in congruent triangles
- Find angles in isosceles triangles
- Finding angles in isosceles triangles
- Finding angles in isosceles triangles (example 2)
- Geometry proof problem: congruent segments
- Geometry proof problem: midpoint
- Geometry proof problem: squared circle
- Geometry word problem: a perfect pool shot
- Geometry word problem: Earth & Moon radii
- Geometry word problem: the golden ratio
- Isosceles & equilateral triangles problems
- Justify constructions
- Prove parallelogram properties
- Prove triangle congruence
- Prove triangle properties
- Prove triangle similarity
- Proving triangle congruence
- Solve similar triangles (advanced)
- Solve similar triangles (basic)
- Solve triangles: angle bisector theorem
- Solving similar triangles
- Solving similar triangles: same side plays different roles
- Special right triangles
- Triangle congruence review
- Triangle similarity review
- Use ratios in right triangles
- Use similar triangles
- Using similar & congruent triangles
- Using the angle bisector theorem

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.

- Intro to the trigonometric ratios
- Side ratios in right triangles as a function of the angles
- Triangle similarity & the trigonometric ratios
- Trigonometric ratios in right triangles
- Trigonometric ratios in right triangles
- Trigonometric ratios in right triangles
- Use ratios in right triangles
- Using similarity to estimate ratio between side lengths

Explain and use the relationship between the sine and cosine of complementary angles.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

- 30-60-90 triangle example problem
- Area of a regular hexagon
- Intro to inverse trig functions
- Intro to the trigonometric ratios
- Multi-step word problem with Pythagorean theorem
- Pythagorean theorem challenge
- Pythagorean theorem in 3D
- Pythagorean theorem in 3D
- Pythagorean theorem with isosceles triangle
- Right triangle trigonometry review
- Right triangle trigonometry word problems
- Right triangle word problem
- Solve for a side in right triangles
- Solve for an angle in right triangles
- Solving for a side in right triangles with trigonometry
- Solving for a side in right triangles with trigonometry
- Special right triangles
- Special right triangles proof (part 1)
- Special right triangles proof (part 2)
- Special right triangles review
- Trigonometric ratios in right triangles
- Trigonometric ratios in right triangles
- Using right triangle ratios to approximate angle measure

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.

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Prove the Laws of Sines and Cosines and use them to solve problems.

- General triangle word problems
- Laws of sines and cosines review
- Proof of the law of cosines
- Proof of the law of sines
- Solve triangles using the law of cosines
- Solve triangles using the law of sines
- Solving for a side with the law of cosines
- Solving for a side with the law of sines
- Solving for an angle with the law of cosines
- Solving for an angle with the law of sines
- Trig word problem: stars

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

- General triangle word problems
- Laws of sines and cosines review
- Solve triangles using the law of cosines
- Solve triangles using the law of sines
- Solving for a side with the law of cosines
- Solving for a side with the law of sines
- Solving for an angle with the law of cosines
- Solving for an angle with the law of sines
- Trig word problem: stars