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### HSG.CO Congruence

- HSG.CO.1 Investigate transformations in the plane
- HSG.CO.A.1 Based on the undefined notions of point, line, plane, distance along a line, and distance around a circular arc, define: angle, line segment, circle, perpendicular lines, parallel lines.
- HSG.CO.A.2 Represent transformations in the plane (e.g. using transparencies, tracing paper, geometry software, etc.). 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 dilation).
- HSG.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that carry it onto itself.
- HSG.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
- HSG.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure, (e.g., using graph paper, tracing paper, miras, geometry software, etc.). Specify a sequence of transformations that will carry a given figure onto another.
- Checkpoint opportunity

- HSG.CO.2 Understand congruence in terms of rigid motions
- 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.
- 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.
- 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. Investigate congruence in terms of rigid motion to develop the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL).
- Checkpoint opportunity

- HSG.CO.3 Apply and prove geometric theorems
- HSG.CO.C.9 Apply and prove theorems about lines and angles.
- HSG.CO.C.10 Apply and prove theorems about triangles.
- HSG.CO.C.11 Apply and prove theorems about quadrilaterals.
- Checkpoint opportunity

- HSG.CO.4 Make geometric constructions
- 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.).
- HSG.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
- Checkpoint opportunity

- HSG.CO.5 Logic and Reasoning
- HSG.CO.E.14 Apply inductive reasoning and deductive reasoning for making predictions based on real world situations using: Conditional Statements (inverse, converse, and contrapositive); Venn Diagrams.

### HSG.SRT Similarity, Right Triangles, and Trigonometry

- HSG.SRT.6 Understand similarity in terms of similarity transformations
- HSG.SRT.A.1 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.
- HSG.SRT.A.2 Given two figures: Use the definition of similarity in terms of similarity transformations to determine 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.
- HSG.SRT.A.3 Use the properties of similarity transformations to establish the AA, SAS~, SSS~ criteria for two triangles to be similar.
- Checkpoint opportunity

- HSG.SRT.7 Apply and prove theorems involving similarity
- HSG.SRT.B.4 Use triangle similarity to apply and prove theorems about triangles.
- HSG.SRT.B.5 Use congruence (SSS, SAS, ASA, AAS, and HL) and similarity (AA~, SSS~, SAS~) criteria for triangles to solve problems. Use congruence and similarity criteria to prove relationships in geometric figures.
- Checkpoint opportunity

- HSG.SRT.8 Define trigonometric ratios and solve problems involving right triangles
- 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.
- HSG.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
- HSG.SRT.C.8 Use trigonometric ratios, special right triangles, and/or the Pythagorean Theorem to find unknown measurements of right triangles in applied problems.
- Checkpoint opportunity

- HSG.SRT.9 Apply trigonometric to general triangles
- 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.
- HSG.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
- 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.
- Checkpoint opportunity

### HSG.C Circles

- HSG.C.10 Understand and apply theorems about circles
- HSG.C.A.1 Prove that all circles are similar.
- HSG.C.A.2 Identify, describe, and use relationships among angles, radii, segments, lines, arcs, and chords as related to circles.
- HSG.C.A.3 Construct the inscribed and circumscribed circles of a triangle. Prove properties of angles for a quadrilateral inscribed in a circle.
- Checkpoint opportunity

- HSG.C.11 Find arc lengths and areas of sectors of circles
- HSG.C.B.5 Derive using similarity that the length of the arc intercepted by an angle is proportional to the radius. Derive and use the formula for the area of a sector. Understand the radian measure of the angle as a unit of measure.
- Checkpoint opportunity

### HSG.GPE Expressing Geometric Properties with Equations

- HSG.GPE.12 Translate between the geometric description and the equation of a conic section
- 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.
- HSG.GPE.A.2 Derive the equation of a parabola given a focus and directrix.
- 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.
- Checkpoint opportunity

- HSG.GPE.13 Use coordinates to prove simple geometric theorems algebraically
- HSG.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically.
- HSG.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines. Use the slope criteria for parallel and perpendicular lines to solve geometric problems.
- HSG.GPE.B.6 Find the midpoint between two given points; and find the endpoint of a line segment given the midpoint and one endpoint.
- HSG.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
- Checkpoint opportunity

### HSG.GMD Geometric measurement and dimension

- HSG.GMD.14 Explain volume formulas and use them to solve problems
- 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.
- 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.
- HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, spheres, and to solve problems which may involve composite figures. Compute the effect on volume of changing one or more dimension(s).
- Checkpoint opportunity

- HSG.GMD.15 Visualize relationships between two-dimensional and three-dimensional objects
- HSG.GMD.B.4 Identify the shapes of two-dimensional cross-sections of three- dimensional objects. Identify three-dimensional objects generated by rotations of two-dimensional objects.
- Checkpoint opportunity

### HSG.MG Modeling with Geometry

- HSG.MG.16 Apply geometric concepts in modeling situations
- 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).
- 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).
- 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).