Core Content Connectors by Common Core State Standards: Mathematics Geometry

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Mathematics High School—Geometry Overview

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

Congruence	HSG-CO
Experiment with transformations in the plane
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.
CCCs linked to HSG-CO.A.1	None
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).
CCCs linked to HSG-CO.A.2	None
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
CCCs linked to HSG-CO.A.3	H.GM.1c1 Construct, draw or recognize a figure after its rotation, reflection, or translation.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
CCCs linked to HSG-CO.A.4	None
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.
CCCs linked to HSG-CO.A.5	H.GM.1c1 Construct, draw or recognize a figure after its rotation, reflection, or translation.
Understand congruence in terms of rigid motions
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.
CCCs linked to HSG-CO.B.6	None
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.
CCCs linked to HSG-CO.B.7	H.GM.1b1 Use definitions to demonstrate congruency and similarity in figures.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
CCCs linked to HSG-CO.B.8	None
Prove geometric theorems
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.
CCCs linked to HSG-CO.C.9	None
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.
CCCs linked to HSG-CO.C.10	None
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.
CCCs linked to HSG-CO.C.11	None
Make geometric constructions
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.
CCCs linked to HSG-CO.D.12	H.GM.1e1 Make formal geometric constructions with a variety of tools and methods.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
CCCs linked to HSG-CO.D.13	None

Similarity, Right Triangles, and Trigonometry	HSG-SRT
Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. 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.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
CCCs linked to HSG-SRT.A.1	H.ME.2b1 Determine the dimensions of a figure after dilation
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.
CCCs linked to HSG-SRT.A.2	H.ME.2b2 Determine if 2 figures are similar
	H.ME.2b3 Describe or select why two figures are or are not similar.
	H.GM.1b1 Use definitions to demonstrate congruency and similarity in figures.
	H.GM.1d1 Use the reflections, rotations, or translations in the coordinate plane to solve problems with right angles.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
CCCs linked to HSG-SRT.A.3	None
Prove theorems involving similarity
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.
CCCs linked to HSG-SRT.B.4	None
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
CCCs linked to HSG-SRT.B.5	None
Define trigonometric ratios and solve problems involving right triangles
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.
CCCs linked to HSG-SRT.C.6	None
7. Explain and use the relationship between the sine and cosine of complementary angles.
CCCs linked to HSG-SRT.C.7	None
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
CCCs linked to HSG-SRT.C.8	None
Apply trigonometry to general triangles
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.
CCCs linked to HSG-SRT.D.9	None
10. Prove the Laws of Sines and Cosines and use them to solve problems.
CCCs linked to HSG-SRT.D.10	None
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).
CCCs linked to HSG-SRT.D.11	None

Circles	HSG-C
Understand and apply theorems about circles
1. Prove that all circles are similar.
CCCs linked to HSG-C.A.1	None
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.
CCCs linked to HSG-C.A.2	None
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
CCCs linked to HSG-C.A.3	None
4. Construct a tangent line from a point outside a given circle to the circle.
CCCs linked to HSG-C.B.4	None
Find arc lengths and areas of sectors of circles
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.
CCCs linked to HSG-C.B.5	H.ME.2b4 Apply the formula to the area of a sector (e.g., area of a slice of pie).

Expressing Geometric Properties with Equations	HSG-GPE
Translate between the geometric description and the equation for a conic section
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.
CCCs linked to HSG-GPE.A.1	None
2. Derive the equation of a parabola given a focus and directrix.
CCCs linked to HSG-GPE.A.2	None
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.
CCCs linked to HSG-GPE.A.3	None
Use coordinates to prove simple geometric theorems algebraically
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).
CCCs linked to HSF-TF.A.4	None
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).
CCCs linked to HSG-GPEB.B.5	None
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
CCCs linked to HSG-GPE.B.6	None
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
CCCs linked to HSG-GPE.B.7	None

Geometric Measurement and Dimension	HSG-GMD
Explain volume formulas and use them to solve problems
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.
CCCs linked to HSG-GMD.A.1	None
2. Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
CCCs linked to HSG-GMD.A.2	None
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
CCCs linked to HSG-GMD.A.3	None
Visualize relationships between two-dimensional and three-dimensional objects
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.
CCCs linked to HSG-GMD.B.4	None

Modeling with Geometry	HSG-MG
Apply geometric concepts in modeling situations
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).
CCCs linked to HSG-MG.A.1	H.ME.1b1 Describe the relationship between the attributes of a figure and the changes in the area or volume when 1 attribute is changed.
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
CCCs linked to HSG-MG.A.2	None
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).
CCCs linked to HSG-MG.A.3	H.ME.2b5 Apply the formula of geometric figures to solve design problems (e.g., designing an object or structure to satisfy physical restraints or minimize cost).