# Common Core Mapping for High School: Geometry

Related Topics: Common Core for Mathematics

### Congruence

• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions
• Prove geometric theorems
• Make geometric constructions
• ### Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similarity transformations
• Prove theorems involving similarity
• Define trigonometric ratios and solve problems involving right triangles
• Apply trigonometry to general triangles
• ### Circles

• Understand and apply theorems about circles
• Find arc lengths and areas of sectors of circles
• ### Expressing Geometric Properties with Equations

• Translate between the geometric description and the equation for a conic section
• Use coordinates to prove simple geometric theorems algebraically
• ### Geometric Measurement and Dimension

• Explain volume formulas and use them to solve problems
• Visualize relationships between two-dimensional and three-dimensional objects
• ### Modeling with Geometry

• Apply geometric concepts in modeling situations

• ### Common Core Mapping for High School: Geometry

Congruence

Standard

Lessons

Worksheets/Games

HSG-CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Geometry Definitions Perpendicular and Parallel Lines

Geometric Definitions

Geometry Games

HSG-CO.A.2

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Transformations in the Plane

HSG-CO.A.3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Polygons & Symmetry

Symmetry of two-dimensional shapes

HSG-CO.A.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Define Rotations, Reflections, Translations

HSG-CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Sequence of Transformations

HSG-CO.B.6

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.

Congruence & Transformations

HSG-CO.B.7

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.

Rigid Motions & Congruent Triangles (CPCTC)

HSG-CO.B.8

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

Triangle Congruence

Congruency postulates

HSG-CO.C.9

Prove 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; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment?s endpoints.

Prove Line and Angle Theorems

HSG-CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180?; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove Triangle Theorems

Triangle Inequality Theorem

HSG-CO.C.11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Prove Parallelogram Theorems

Proof of Parallelograms

HSG-CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Geometric Constructions

Compass constructions

HSG-CO.D.13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Construct Shapes

Compass Construction 2

Similarity, Right Triangles, and Trigonometry
 Standard Lessons Worksheets/Games HSG-SRT.A.1, HSG-SRT.A.1a, HS-SRT.AG1b. Verify experimentally the properties of dilations given by a center and a scale factor: A 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. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Dilations Dilations HSG-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Similarity Transformations Defining similarity through angle-preserving transformations HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Similar Triangles (AA, SAS, SSS) HSG-SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove Triangle Theorems HSG-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. HSG-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Similarity & Trig Ratios: Sin, Cos, Tan Trigonometric functions and side ratios in right triangles HSG-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. Sin and Cos of Complementary Angles Trig Functions in right triangles HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Applying Trigonometric Ratios HSG-SRT.D.9 (+) Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Area of Triangle using Sine Area of Triangle using Sine Non-right triangle proofs HSG-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. Law of Sines and Law of Cosines HSG-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Circles
 Standard Lessons Worksheets/Games HSG-C.A.1 Prove that all circles are similar. All Circles are Similar Define similarity through angle preserving transformations HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Inscribed angles, radii, chords HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Inscribed and Circumscribed Circles Inscribing and circumscribing circles on a triangle HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle. Construct Tangent to Circle Construct tangent HSG-C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Arc Length & Sector Area

Expressing Geometric Properties with Equations
 Standard Lessons Worksheets/Games HSG-GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Equation of a Circle HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix. Equation of a Parabola HSG-GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Equations of Ellipses and Hyperbolas HSG-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ?3) lies on the circle centered at the origin and containing the point (0, 2). Coordinate Proofs Geometry problems on the coordinate plane HSG-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Parallel and Perpendicular Lines Equations of parallel and perpendicular lines HSG-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Partition Line Segment HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Area and Perimeter on Coordinate Plane Coordinate Plane word problems
Geometric Measurement and Dimension
 Standard Lessons Worksheets/Games HSG-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. Geometric Formulas HSG-GMD.A.2 (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. Geometric Formulas HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Using Volume Formulas Volume Word Problems HSG-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Cross Sections and Solids of Rotation Cross sections of 3-D objects Rotate 2D to make 3D
Modeling with Geometry
 Standard Lessons Worksheets/Games HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Geometric Shapes word problems HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Density, Mass Volume Surface and volume density word problems HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Geometric Shapes word problems

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.