# Academic standards list

### High School: Geometry — Mathematics

### 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,…

### 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.

### Circles

CCSS.Math.Content.HSG-C

Understand and apply theorems about circles

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.

Find arc lengths and areas of sectors of circles

### 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 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.

Visualize relationships between two-dimensional and three-dimensional objects

### 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;…

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