Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs.  Compare transformations that preserve distance and angle measure to those that do not.  Instructional strategies may include drawing tools, graph paper, transparencies and software programs.

Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that map the polygon onto itself.  The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.”

Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments.
• Include point reflections.
• A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector.
• A rotation requires knowing the center/point and the measure/direction of the angle of rotation.
• A line reflection requires a line and the knowledge of perpendicular bisectors.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using.  Specify a sequence of transformations that will carry a given figure onto another.
• Include point reflections.
• A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector.
• A rotation requires knowing the center/point and the measure/direction of the angle of rotation.
• A line reflection requires a line and the knowledge of perpendicular bisectors.
• Instructional strategies may include graph paper, tracing paper, and geometry software.
• Singular transformations that are equivalent to a sequence of transformations may be utilized, such as a glide reflection.  However, glide reflections are not an expectation of the course.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.  Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
• A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector.
• A rotation requires knowing the center/point and the measure/direction of the angle of rotation.
• A line reflection requires a line and the knowledge of perpendicular bisectors.

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.

Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL (Hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions.

Prove and apply theorems about lines and angles.
• Include multi-step proofs and algebraic problems built upon these concepts.
• Examples of theorems include but are not limited to:
   o Vertical angles are congruent.
   o If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
   o The points on a perpendicular bisector are equidistant from the endpoints of the line segment.

Prove and apply theorems a bout triangles.
• Include multi-step proofs and algebraic problems built upon these concepts.
• Examples of theorems include but are not limited to:
   o Angle Relationships:
      • The sum of the interior angles of a triangle is 180 degrees.
      • The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle.
   o Side Relationships:
      • The length of one side of a triangle is less than the sum of the lengths of the other two sides.
      • In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length.
   o Isosceles Triangles:
      • Base angles of an isosceles triangle are congruent.

Prove and apply theorems about parallelograms.
• Include multi-step proofs and algebraic problems built upon these concepts.
• The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.”
• Examples of theorems include but are not limited to:
   o A diagonal divides a parallelogram into two congruent triangles.
   o Opposite sides/angles of a parallelogram are congruent.
   o The diagonals of parallelogram bisect each other.
   o If the diagonals of quadrilateral bisect each other, then quadrilateral is a parallelogram.
   o If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle.
• Additional theorems covered allow for proving that a given quadrilateral is a particular parallelogram (rhombus, rectangle, square) based on given properties.

Verify experimentally the properties of dilations given by a center and a scale factor.

Verify experimentally that 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.

Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.  Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• The center and scale factor of the dilation must always be specified with dilation.
• A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector.
• A rotation requires knowing the center/point and the measure/direction of the angle of rotation.
• A line reflection requires a line and the knowledge of perpendicular bisectors.

Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar.

Prove and apply similarity theorems about triangles.
•  Include multi-step proofs and algebraic problems built upon these concepts.
• Examples of theorems include but are not limited to:
   o If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally (and conversely).
   o The length of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse.
   o The centroid of the triangle divides each median in the ratio 2:1.

Use congruence and similarity criteria for triangles to:

Solve problems algebraically and geometrically.

Prove relationships in geometric figures.

• ASA, SAS, SSS, AAS, and Hypotenuse-Leg (HL) theorems are valid criteria for triangle congruence. AA~, SAS~, and SSS~ are valid criteria for triangle similarity.
• This standard is a fluency recommendation for Geometry.  Fluency with the triangle congruence and similarity criteria will help students throughout their investigations of triangles, quadrilaterals, circles, parallelism, and trigonometric ratios.  These criteria are necessary tools in many geometric modeling tasks.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent ratios for acute angles.

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

Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems.  Special right triangles refer to the 30-60-90 and 45-45-90 triangles.

Justify and apply the formula A=(1/2)ab sin(C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Prove that all circles are similar.

Identify, describe and apply relationships between the angles and their intercepted arcs of a circle.  These relationships that pertain to the circle may be utilized to prove other relationships in geometric figures, e.g., the opposite angles in any quadrilateral inscribed in a circle are supplements of each other.

Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle.  Include algebraic problems built upon these concepts.

On the coordinate plane, algebraically prove geometric theorems and properties.
• Examples include but not limited to:
   o Given points and/or characteristics, prove or disprove a polygon is a specified quadrilateral or triangle based on its properties.
   o Given a point that lies on a circle with a given center, prove or disprove that a specified point lies on the same circle.
• This standard is a fluency recommendation for Geometry. Fluency with the use of coordinates to establish geometric results and the use of geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields.

On the coordinate plane:
a. Explore the proof for the relationship between slopes of parallel and perpendicular lines;
b. Determine if lines are parallel, perpendicular, or neither, based on their slopes; and
c. Apply properties of parallel and perpendicular lines to solve geometric problems.
This standard is a fluency recommendation for Geometry. Fluency with the use of coordinates to establish geometric results and the use of geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.  Midpoint formula is a derivative of this standard.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.  This standard is a fluency recommendation for Geometry. Fluency with the use of coordinates to establish geometric results and the use of geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields.

Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Use geometric shapes, their measures, and their properties to describe objects.

Apply concepts of density based on area and volume of geometric figures in modeling situations.

Apply geometric methods to solve design problems.  Applications may include designing an object or structure to satisfy constraints such as area, volume, mass, and cost.