Geometry (New Jersey NJSLS)

Congruence (G.CO)

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

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

• 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.
• G.SRT.A.2 Use similarity transformations to decide if two shapes are similar and explain similarity for triangles.
• G.SRT.A.3 Prove the AA similarity criterion for triangles using properties of similarity transformations.
• G.SRT.B.4 Prove theorems about triangles, e.g. a line parallel to a side divides the other sides proportionally; Pythagoras by similarity.
• G.SRT.B.5 Solve problems and prove relationships in figures using triangle congruence and similarity criteria.
• G.SRT.C.6 Know that, because of similarity, right-triangle side ratios depend only on the angles, giving the trig ratios.
• G.SRT.C.7 Explain and use how sine and cosine of complementary angles relate.
• G.SRT.C.8 Solve right triangles in applied problems with trig ratios and the Pythagorean Theorem. ★

Circles (G.C)

• G.C.A.1 Prove that all circles are similar.
• G.C.A.2 Identify and describe how inscribed angles, radii and chords relate, including angles in a semicircle and tangents.
• G.C.A.3 Construct the circles inside and around a triangle and prove angle facts for quadrilaterals inscribed in a circle.
• 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 (G.GPE)

• 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. (being checked)
• G.GPE.B.4 Prove or disprove simple geometric claims with coordinates and algebra, e.g. whether four points make a rectangle.
• G.GPE.B.5 Prove the slope tests for parallel and perpendicular lines and apply them to write equations of lines.
• G.GPE.B.6 Find the point on a directed segment between two points that splits it in a given ratio. (being checked)
• G.GPE.B.7 Find polygon perimeters and triangle and rectangle areas with coordinates and the distance formula. ★

Geometric Measurement and Dimension (G.GMD)

• G.GMD.A.1 Give informal arguments for the circumference and area of a circle and volumes of cylinders, pyramids and cones.
• G.GMD.A.3 Solve problems with the volume formulas for cylinders, cones, pyramids and spheres. ★
• G.GMD.B.4 Identify cross-sections of 3D objects and 3D objects made by rotating 2D shapes.

Modeling with Geometry (G.MG) ★

• G.MG.A.1 Use shapes, their measures and properties to describe real objects, e.g. a tree trunk as a cylinder. ★
• G.MG.A.2 Apply density ideas based on area and volume in modeling, e.g. people per square mile. ★
• G.MG.A.3 Use geometric methods to solve design problems with physical limits or cost constraints. ★