**10.53.516** MONTANA HIGH SCHOOL MATHEMATICS GEOMETRY CONTENT STANDARDS

(1) Mathematics geometry: congruence content standards for high school are:

(a) know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc;

(b) represent transformations in the plane using transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs; and compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch);

(c) given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself;

(d) develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments;

(e) given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software and specify a sequence of transformations that will carry a given figure onto another;

(f) use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure and given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent;

(g) use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent;

(h) explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions;

(i) prove theorems about lines and angles; theorems include: vertical angles are congruent, when a transversal crosses parallel lines, alternate interior angles are congruent, corresponding angles are congruent, and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints;

(j) 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, and the medians of a triangle meet at a point;

(k) 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;

(l) make formal geometric constructions, including those representing Montana American Indians, with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.); copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line; and

(m) construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

(2) Mathematics geometry: similarity, right triangles, and trigonometry content standards for high school are:

(a) verify experimentally the properties of dilations given by a center and a scale factor:

(i) 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; and

(ii) the dilation of a line segment is longer or shorter in the ratio given by the scale factor;

(b) given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar and 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;

(c) use the properties of similarity transformations to establish the AA criterion for two triangles to be similar;

(d) 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;

(e) use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures;

(f) 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) explain and use the relationship between the sine and cosine of complementary angles;

(h) use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems;

(i) (+) 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;

(j) (+) prove the Laws of Sines and Cosines and use them to solve problems; and

(k) (+) understand and apply the Laws of Sines and Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).

(3) Mathematics geometry: circles content standards for high school are:

(a) prove that all circles are similar;

(b) identify and describe relationships among inscribed angles, radii, and chords; include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; and the radius of a circle is perpendicular to the tangent where the radius intersects the circle;

(c) construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle;

(d) (+) construct a tangent line from a point outside a given circle to the circle; and

(e) derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius; define the radian measure of the angle as the constant of proportionality; and derive the formula for the area of a sector.

(4) Mathematics geometry: expressing geometric properties with equations content standards for high school are:

(a) derive the equation of a circle of given center and radius using the Pythagorean Theorem and complete the square to find the center and radius of a circle given by an equation;

(b) derive the equation of a parabola given a focus and directrix;

(c) (+) derive the equations of ellipses and hyperbolas given the foci and directrices;

(d) use coordinates to prove simple geometric theorems algebraically; for example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle and prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2);

(e) prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point);

(f) find the point on a directed line segment between two given points that partitions the segment in a given ratio; and

(g) use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.^{*}

(5) Mathematics geometry: geometric measurement and dimension content standards for high school are:

(a) give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone and use dissection arguments, Cavalieri's principle, and informal limit arguments;

(b) (+) give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures;

(c) use volume formulas for cylinders, pyramids, cones, and spheres to solve problems;^{*} and

(d) identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.

(6) Mathematics Geometry: modeling with geometry content standards for high school are:

(a) use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder; modeling a Montana American Indian tipi as a cone);^{*}

(b) apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot);^{* }and

(c) apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).^{*}

History: 20-2-114, MCA; __IMP__, 20-2-121, 20-3-106, 20-7-101, MCA; __NEW__, 2011 MAR p. 2522, Eff. 11/26/11.