Geometry (Massachusetts Curriculum Frameworks)

Quantities (GEO.N-Q)

• GEO.N-Q.A.3 Choose a level of accuracy suited to the limits of measurement when reporting quantities.
  • a. Describe how measurement error and rounding affect results, and use significant figures suited to the context and tools.

Congruence (GEO.G-CO)

• GEO.G-CO.A.1 Know precise definitions of angle, circle, perpendicular and parallel lines and line segment, from undefined terms.
• GEO.G-CO.A.2 Describe transformations as functions on points and compare those that keep distance and angle with those that do not.
• GEO.G-CO.A.3 Describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid or regular polygon onto itself.
• GEO.G-CO.A.4 Define rotations, reflections and translations using angles, circles, perpendicular and parallel lines and segments.
• GEO.G-CO.A.5 Draw the image of a shape under a transformation and give a sequence of transformations mapping one shape to another.
• GEO.G-CO.B.6 Use rigid motions to transform shapes and to decide whether two shapes are congruent.
• GEO.G-CO.B.7 Show with rigid motions that two triangles are congruent exactly when matching sides and angles are equal.
• GEO.G-CO.B.8 Explain why the ASA, SAS and SSS congruence criteria follow from congruence defined by rigid motions.
• GEO.G-CO.C.9 Prove theorems about lines and angles, e.g. vertical angles, angles with a transversal, points on a perpendicular bisector.
• GEO.G-CO.C.10 Prove theorems about triangles, e.g. angle sum 180°, isosceles base angles, the midsegment, medians meeting at a point.
• GEO.G-CO.C.11 Prove theorems about parallelograms, e.g. opposite sides and angles equal, diagonals bisect each other.
  • a. Prove theorems about polygons, including interior and exterior angle measures, and apply polygon properties to problems.
• GEO.G-CO.D.12 Make formal constructions with compass and straightedge, string, folding or software: copies, bisectors, perpendiculars, parallels.
• GEO.G-CO.D.13 Construct a regular hexagon, a square and an equilateral triangle inside a circle.

Similarity, Right Triangles, and Trigonometry (GEO.G-SRT)

• GEO.G-SRT.A.1 Check by experiment the properties of dilations with a center and scale factor.
  • a. A dilation sends a line not through its center to a parallel line; a line through the center stays put.
  • b. A dilation changes segment lengths by the scale factor.
• GEO.G-SRT.A.2 Use similarity transformations to decide if two shapes are similar and explain similarity for triangles.
• GEO.G-SRT.A.3 Prove the AA similarity criterion for triangles using properties of similarity transformations.
• GEO.G-SRT.B.4 Prove theorems about triangles, e.g. a line parallel to a side divides the other sides proportionally; Pythagoras by similarity.
• GEO.G-SRT.B.5 Solve problems and prove relationships in figures using triangle congruence and similarity criteria.
• GEO.G-SRT.C.6 Know that, because of similarity, right-triangle side ratios depend only on the angles, giving the trig ratios.
• GEO.G-SRT.C.7 Explain and use how sine and cosine of complementary angles relate.
• GEO.G-SRT.C.8 Solve right triangles in applied problems with trig ratios and the Pythagorean Theorem.
• GEO.G-SRT.D.9 (+) Derive the triangle area formula A = ½ab sin(C) by drawing an altitude.
• GEO.G-SRT.D.10 (+) Prove the Law of Sines and Law of Cosines and apply them.
• GEO.G-SRT.D.11 (+) Use the Laws of Sines and Cosines to find unknown sides and angles in right and non-right triangles.

Circles (GEO.G-C)

• GEO.G-C.A.1 Prove that all circles are similar.
• GEO.G-C.A.2 Identify and describe how inscribed angles, radii and chords relate, including angles in a semicircle and tangents.
• GEO.G-C.A.3 Construct the circles inside and around a triangle and prove angle facts for quadrilaterals inscribed in a circle.
• GEO.G-C.A.4 (+) Construct a tangent line to a circle from a point outside it.
• GEO.G-C.B.5 Use similarity to show arc length is proportional to radius, define radian measure and derive the sector area formula.

Expressing Geometric Properties with Equations (GEO.G-GPE)

• GEO.G-GPE.A.1 Derive the equation of a circle from its center and radius using Pythagoras; complete the square to find center and radius.
• GEO.G-GPE.A.2 Derive the equation of a parabola from its focus and directrix.
• GEO.G-GPE.B.4 Prove or disprove simple geometric claims with coordinates and algebra, e.g. whether four points make a rectangle.
• GEO.G-GPE.B.5 Prove the slope tests for parallel and perpendicular lines and apply them to write equations of lines.
• GEO.G-GPE.B.6 Find the point on a directed segment between two points that splits it in a given ratio.
• GEO.G-GPE.B.7 Find polygon perimeters and triangle and rectangle areas with coordinates and the distance formula.

Geometric Measurement and Dimension (GEO.G-GMD)

• GEO.G-GMD.A.1 Give informal arguments for the circumference and area of a circle and volumes of cylinders, pyramids and cones.
• GEO.G-GMD.A.2 (+) Use Cavalieri's principle informally to justify volume formulas for spheres and other solids.
• GEO.G-GMD.A.3 Solve problems with the volume formulas for cylinders, cones, pyramids and spheres.
• GEO.G-GMD.B.4 Identify cross-sections of 3D objects and 3D objects made by rotating 2D shapes.

Modeling with Geometry (GEO.G-MG)

• GEO.G-MG.A.1 Use shapes, their measures and properties to describe real objects, e.g. a tree trunk as a cylinder.
• GEO.G-MG.A.2 Apply density ideas based on area and volume in modeling, e.g. people per square mile.
• GEO.G-MG.A.3 Use geometric methods to solve design problems with physical limits or cost constraints.
• GEO.G-MG.A.4 Use dimensional analysis in unit conversions to check that expressions and equations make sense.

Conditional Probability and the Rules of Probability (GEO.S-CP)

• GEO.S-CP.A.1 Describe events as parts of a sample space using outcome features or the words "or", "and" and "not".
• GEO.S-CP.A.2 Test independence of A and B by checking if P(A and B) = P(A)·P(B).
• GEO.S-CP.A.3 Understand conditional probability P(A given B) = P(A and B)/P(B) and interpret independence with it.
• GEO.S-CP.A.4 Build and read two-way frequency tables to decide independence and estimate conditional probabilities.
• GEO.S-CP.A.5 Explain conditional probability and independence in everyday situations.
• GEO.S-CP.B.6 Work out P(A|B) as the share of outcomes in B that also lie in A, and interpret the answer in the model.
• GEO.S-CP.B.7 Use the rule P(A or B) = P(A) + P(B) − P(A and B) and interpret the result.
• GEO.S-CP.B.8 (+) Use the general multiplication rule in a model where all outcomes are equally likely and interpret the result.
• GEO.S-CP.B.9 (+) Use permutations and combinations to find probabilities of compound events and solve problems.