Standards - Mathematics

Investigate, prove, and apply theorems about triangles, including but not limited to: the \sum of the measures of the interior angles of a triangle is $180^{\circ}$; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.

Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.

Use congruence and similarity criteria for triangles to solve problems in real-world contexts.

Discover and apply relationships in similar right triangles.

Derive and apply the constant ratios of the sides in special right triangles ($45^{\circ} -45^{\circ} -90^{\circ}$ and $30^{\circ} -60^{\circ} -90^{\circ}$).

Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.

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

Demonstrate the converse of the Pythagorean Theorem.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.

Use the mathematical modeling cycle involving geometric methods to solve design problems.

Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.

Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.

Prove the Pythagorean identity $\sin^2 (\theta) + \cos^2 (\theta) = 1$ and use it to calculate trigonometric ratios.

Derive and apply the formula $A = \frac{1}{2} \cdot ab \cdot \sin(C)$ for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.

Derive and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Extend the domain of sine and cosine to include right and obtuse angles.