Learners will use visualization, spatial reasoning, geometric modeling, and measurement to investigate the characteristics of figures, perform transformations, and construct logical arguments.

MAT-10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30°-60°-90° and 45°-45°-90°).

MAT-06.AR.RP.01 Describe the concept of a ratio relationship between two quantities using ratio language and visual models.

MAT-06.AR.RP.03 Make and use tables of equivalent ratios, tape diagrams, double number line diagrams, and equations to reason about ratios, rates, and unit rates.

MAT-07.AR.RP.02 Analyze the relationships between the dependent and independent variables of a proportional relationship using graphs and tables. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, k) where k is the unit rate.

MAT-07.AR.RP.03 Identify the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by an equation of the form y = kx, where k is the constant of proportionality, and describe the meaning of each variable (y, k, x) in the context of the situation.

MAT-07.AR.RP.04 Use proportional relationships to solve multi-step problems involving ratios, percents, and scale drawings of geometric figures, including authentic problems.

MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.

MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.

MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.

MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

MAT-10.GM.16 Prove similarity theorems about triangles.

MAT-10.GM.18 Recognize how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.

MAT-10.GM.20Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30°-60°-90° and 45°-45°-90°).

MAT-10.GM.25 Explain and use the formulas for arc length and area of sectors of circles.

MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.

MAT-10.GM.29 Determine the midpoint or endpoint of a line segment using coordinates. (+) Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

MAT-10.GM.36 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; scaling a model).

MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context.

Angles/Triangles

MAT-03.GM.G.01 In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.

MAT-04.GM.G.01 Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).

MAT-04.GM.M.07 Recognize angle measures as additive and solve addition and subtraction problems to find unknown angles on a diagram.

MAT-07.GM.GF.01 Draw triangles from given conditions using appropriate tools. Defend whether a unique triangle, multiple triangles, or no triangle can be constructed when given three measures of angles or sides.

MAT-07.GM.GF.02 Describe the angle-pair relationships: supplementary angles, complementary angles, vertical angles, and adjacent angles. Solve for an unknown angle in a figure by applying facts about these angles.

MAT-08.GM.GF.04 Describe the following angle-pair relationships: interior and exterior angles of triangles and angles formed when a transversal cuts parallel lines or intersecting lines. Solve for an unknown angle in a figure by applying
facts about these angles.

MAT-08.GM.GF.05 Describe the relationship between the leg length and the hypotenuse length of a right triangle. Determine whether a triangle is a right triangle using this relationship.

MAT-08.GM.GF.06 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in two and three dimensions on and off a coordinate plane, including authentic problems.

MAT-10.GM.01 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, and plane.

MAT-10.GM.09 Prove and apply theorems about lines and angles.

MAT-10.GM.10 Prove and apply theorems about triangles.

MAT-10.GM.18 Recognize how the properties of similar right triangles allow for trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.

MAT-10.GM.19 Explain and use the relationship between the sine and cosine of complementary angles.

MAT-10.GM.20Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30º, -60º, -90º, and 45º-45º-90º).

MAT-10.GM.21 Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.

MAT-10.GM.23 Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.

MAT-10.GM.24 Construct a tangent line from a point outside a given circle to the circle.

MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.

MAT-12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π/3, π/4, and π/6.

MAT-12.AR.F.17 Use the Pythagorean Identity sin² (θ) + cos² (θ) = 1 to find sin (θ), cos (θ), or tan (θ) given sin (θ), cos (θ), or tan (θ) and the quadrant of the angle.

MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context