### Introduction

Academic standards define the expectations for knowledge and skills that students are to learn in a subject by a certain age or at the end of a school grade level. This page contains a list of standards for a specific content area, grade level, and/or course. The list of standards may be structured using categories and sub-categories.

### Congruence

CCSS.Math.Content.HSG-CO

Experiment with transformations in the plane
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line,
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points

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

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or

Understand congruence in terms of rigid motions
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
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
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a
Make geometric constructions
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding,

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

### Similarity, Right Triangles, and Trigonometry

CCSS.Math.Content.HSG-SRT

Understand similarity in terms of similarity transformations
Verify experimentally the properties of dilations given by a center and a scale factor:
There are 2 components within this standard.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
Apply trigonometry to general triangles
(+) 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.

(+) Prove the Laws of Sines and Cosines and use them to solve problems.

(+) 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

### Circles

CCSS.Math.Content.HSG-C

Understand and apply theorems about circles
Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

(+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles
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

### Expressing Geometric Properties with Equations

CCSS.Math.Content.HSG-GPE

Translate between the geometric description and the equation for a conic section
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

Derive the equation of a parabola given a focus and directrix.

(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is

Use coordinates to prove simple geometric theorems algebraically
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

### Geometric Measurement and Dimension

CCSS.Math.Content.HSG-GMD

Explain volume formulas and use them to solve problems
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

(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

Visualize relationships between two-dimensional and three-dimensional objects
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by

### Modeling with Geometry

CCSS.Math.Content.HSG-MG

Apply geometric concepts in modeling situations
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost;

Disclaimer: This website provides a reference tool for browsing academic standards and locating associated resources. We are not the originator of these academic standards. Although we strive to maintain accuracy, there may be revisions, updates, or errors within the text and structure of the information displayed. In case of any discrepancy, please respect the originator's published version (http://www.corestandards/Math) as the definitive record.