Geometry (West Virginia CCR Standards)

Basics of Geometry

• M.GHS.1 Know precise definitions of angle, circle, perpendicular line, parallel line and segment, based on undefined terms.
• M.GHS.2 Build and justify logical arguments using conditional statements, symbolic form, Venn diagrams and types of reasoning.
  • a. Identify the converse, inverse and contrapositive of a conditional statement.
  • b. Translate a short verbal argument into symbolic form.
  • c. Use Venn diagrams to show relationships between sets.
  • d. Use inductive and deductive reasoning.
• M.GHS.3 Prove theorems about lines and angles, such as vertical angles and angles made by a transversal cutting parallel lines.
• M.GHS.4 Find the point on a directed line segment between two points that divides it in a given ratio.
• M.GHS.5 Make formal geometric constructions with tools such as compass and straightedge or software.
  • a. Copy a segment.
  • b. Copy an angle.
  • c. Bisect a segment.
  • d. Bisect an angle.
  • e. Construct perpendicular lines, including the perpendicular bisector of a segment.
  • f. Construct a line parallel to a given line through a point not on it.

Transformations and Congruence

• M.GHS.6 Represent transformations in the plane, describe them as functions on points and compare those that keep distance and angle with those that do not.
  • a. Represent transformations in the plane.
  • b. Describe transformations as functions that take points as inputs and give points as outputs.
  • c. Compare transformations that preserve distance and angle with those that do not.
• M.GHS.7 Describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid or regular polygon onto itself.
• M.GHS.8 Define rotations, reflections and translations in terms of angles, circles, perpendicular and parallel lines and segments.
• M.GHS.9 Draw the image of a figure under a transformation and give a sequence of transformations that maps one figure onto another.
• M.GHS.10 Use rigid motions to transform figures and predict their effect; decide whether two figures are congruent.
• M.GHS.11 Use rigid motions to show two triangles are congruent exactly when corresponding sides and angles are congruent.
• M.GHS.12 Explain how the ASA, SAS and SSS criteria follow from the definition of congruence by rigid motions.
• M.GHS.13 Use congruence criteria for triangles to solve problems and prove relationships in figures.
• M.GHS.14 Prove theorems about triangles, such as the 180° angle sum, isosceles base angles, the midsegment and medians meeting at a point.
• M.GHS.15 Prove theorems about parallelograms, such as opposite sides and angles congruent, diagonals bisecting each other and rectangle diagonals.
• M.GHS.16 Use coordinates to prove simple theorems about right triangles, quadrilaterals and circles algebraically.

Similarity and Trigonometry

• M.GHS.17 Verify by experiment the properties of dilations given by a center and scale factor.
  • a. A line not through the center goes to a parallel line; a line through the center is unchanged.
  • b. A dilation changes segment lengths by the scale factor.
• M.GHS.18 Use similarity transformations to decide whether figures are similar and explain similarity of triangles through angles and proportional sides.
• M.GHS.19 Use properties of similarity transformations to establish the AA criterion for similar triangles.
• M.GHS.20 Prove theorems about triangles using similarity, such as a line parallel to one side dividing the others proportionally and the Pythagorean theorem.
• M.GHS.21 Use similarity criteria to solve problems and derive and use the relationships in special right triangles.
• M.GHS.22 Understand that side ratios in right triangles depend on the angles, which leads to the trig ratios of acute angles.
• M.GHS.23 Explain and use the link between the sine and cosine of complementary angles.
• M.GHS.24 Use trig ratios and the Pythagorean theorem to solve right triangles in applied problems.
• M.GHS.25 Derive the formula area = ½ab sin C for a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
• M.GHS.26 Prove the Laws of Sines and Cosines, extending sine and cosine to obtuse angles.
• M.GHS.27 Use the Laws of Sines and Cosines to find unknown measures in right and non-right triangles.

Circles

• M.GHS.28 Prove that all circles are similar.
• M.GHS.29 Identify and describe relationships among inscribed angles, radii, chords and tangents, including central, inscribed and circumscribed angles.
• M.GHS.30 Use similarity to show arc length is proportional to the radius, define radian measure and derive the area of a sector.
• M.GHS.31 Construct the inscribed and circumscribed circles of a triangle and prove properties of angles of a quadrilateral inscribed in a circle.
• M.GHS.32 Construct a tangent line from a point outside a circle to the circle.
• M.GHS.33 Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle.

Extending to Three Dimensions and Modeling

• M.GHS.34 Give informal arguments for the formulas for a circle's circumference and area and the volumes of cylinders, pyramids and cones.
• M.GHS.35 Use volume formulas for cylinders, pyramids, cones and spheres, including how area and volume change for similar figures.
• M.GHS.36 Identify the shapes of cross-sections of solids and the solids made by rotating plane shapes.
• M.GHS.37 Use shapes, their measures and properties to describe and model real objects.
  • a. Use density based on area and volume in modeling situations.
  • b. Use geometric methods to solve design problems with given constraints.

Statistics and Probability

• M.GHS.38 Describe events as subsets of a sample space using characteristics of outcomes or unions, intersections and complements.
• M.GHS.39 Understand that two events are independent when P(A and B) equals P(A) times P(B), and use this to test independence.
• M.GHS.40 Understand conditional probability as P(A and B)/P(B) and interpret independence in terms of conditional probability.
• M.GHS.41 Make and interpret two-way frequency tables to decide whether events are independent and to approximate conditional probabilities.
• M.GHS.42 Recognize and explain conditional probability and independence in everyday language and situations.
• M.GHS.43 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret it.
• M.GHS.44 Apply the Addition Rule and interpret the answer in terms of the model.
• M.GHS.45 Apply the general Multiplication Rule in a uniform probability model and interpret the answer.
• M.GHS.46 Use permutations and combinations to compute probabilities of compound events and solve problems.
• M.GHS.47 Use probabilities to make fair decisions.
• M.GHS.48 Use probability concepts to analyze decisions and strategies.