verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems.

Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:

verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems;

prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions;

apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles;

verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems; and

prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems. The student is expected to:

apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles; and

apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.

Similarity, proof, and trigonometry. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:

prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems; and

identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.

Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles. The student is expected to:

determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems; and

apply the relationships in special right triangles 30°-60°-90° and 45°-45°-90° and the Pythagorean theorem, including Pythagorean triples, to solve problems.

Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures. The student is expected to:

identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations of two-dimensional shapes; and

determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change.

Two-dimensional and three-dimensional figures. The student uses the process skills in the application of formulas to determine measures of two- and three-dimensional figures. The student is expected to:

apply the formula for the area of regular polygons to solve problems using appropriate units of measure;

determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure;

apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure; and

apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.

Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:

apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems;

apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems;

apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems;

describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle; and

show that the equation of a circle with center at the origin and radius \(r\) is \(x^2 + y^2 = r^2\) and determine the equation for the graph of a circle with radius r and center \((h, k)\), \((x - h)^2 + (y - k)^2 =r^2\).

Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to:

develop strategies to use permutations and combinations to solve contextual problems;

determine probabilities based on area to solve contextual problems;

identify whether two events are independent and compute the probability of the two events occurring together with or without replacement;

apply conditional probability in contextual problems; and

apply independence in contextual problems.

Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

apply mathematics to problems arising in everyday life, society, and the workplace;

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

create and use representations to organize, record, and communicate mathematical ideas;

analyze mathematical relationships to connect and communicate mathematical ideas; and

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:

use the composition of two functions to model and solve real-world problems;

demonstrate that function composition is not always commutative;

represent a given function as a composite function of two or more functions;

describe symmetry of graphs of even and odd functions;

determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations;