1. (1) The transformations content standards for high school are to:

   1. (a) represent transformations in the plane using a variety of methods;

   2. (b) define the congruence of two and show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other by:

      1. (i) using 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; and
      2. (ii) verifying that two triangles are congruent if, but not only if, the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS);

   3. (c) define the similarity of two figures in terms of similarity transformations by:

      1. (i) verifying that two triangles are similar if, and only if, corresponding pairs of sides are proportional and corresponding pairs of angles are congruent; and
      2. (ii) using the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.

2. (2) The geometric arguments, reasoning, and proof content standards for high school are to:

   1. (a) investigate, conjecture, prove theorems, and communicate the proofs in a variety of ways by:

      1. (i) proving theorems about lines and angles; theorems include: vertical angles are congruent; when a transversal crosses parallel lines alternate interior angles are congruent and corresponding angles are congruent; and the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints;
      2. (ii) proving theorems about triangles; theorems include: the sum of the measures of the interior angles of a triangle is 180˚; the Pythagorean Theorem; the base angles of isosceles triangles are congruent; and a line parallel to one side of a triangle divides the other two sides proportionally;
      3. (iii) proving theorems about parallelograms and other quadrilaterals; theorems include: necessary and sufficient conditions for rectangles, parallelograms, rhombi, and kites; and
      4. (iv) proving theorems about circles; theorems 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.

3. (3) The measurement, problem solving, and geometric modeling content standards for high school are to:

   1. (a) use the Pythagorean Theorem to calculate distance in the coordinate plane;
   2. (b) derive the equation of a circle of a given center and radius using the Pythagorean Theorem;

   3. (c) use similarity to explore and define the sine ratio, cosine ratio, and tangent ratio in terms of right triangles by:

      1. (i) deriving and applying the trigonometric ratios in special right triangles; and
      2. (ii) using trigonometric ratios and the Pythagorean Theorem to solve right triangles;

   4. (d) use geometric shapes, their measures, and their properties to model objects and use those models to solve problems in context. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities by:

      1. (i) modeling and solving problems with 2D shapes by using the perimeter and area of polygons, circles, and composite shapes with portions removed;
      2. (ii) modeling and solving problems with 3D solids by using surface area and volume of solids, including composite solids and solids with portions removed; and
      3. (iii) deriving and applying the relationships between the lengths, perimeters, areas, and volumes of similar figures in relation to their scale factor.

Authorizing statute(s): Mont. Const. Art. X, sec. 9, 20-2-114, 20-7-101, MCA

Implementing statute(s): Mont. Const. Art. X, sec. 9, 20-2-121, 20-3-106, 20-7-101, MCA

History: NEW, 2025 MAR, 10-53-141, Eff. 7/1/26.