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### G.LP Logic and Proofs

- G.LP.1 Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates, methods of reasoning, and theorems). Understand the differences among supporting evidence, counterexamples, and actual proofs.
- G.LP.2 Use precise definitions for angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, and plane. Use standard geometric notation.
- G.LP.3 State, use, and examine the validity of the converse, inverse, and contrapositive of conditional (“if – then”) and bi-conditional (“if and only if”) statements.
- G.LP.4 Understand that proof is the means used to demonstrate whether a statement is true or false mathematically. Develop geometric proofs, including those involving coordinate geometry, using two-column, paragraph, and flow chart formats.
- Checkpoint opportunity

### G.PL Points, Lines, and Angles

- G.PL.1 Prove and apply theorems about lines and angles, including the following: Vertical angles are congruent; When a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and corresponding angles are congruent; When a transversal crosses parallel lines, same side interior angles are supplementary; and Points on a perpendicular bisector of a line segment are exactly those equidistant from the endpoints of the segment.
- G.PL.2 Explore the relationships of the slopes of parallel and perpendicular lines. Determine if a pair of lines are parallel, perpendicular, or neither by comparing the slopes in coordinate graphs and equations.
- G.PL.3 Use tools to explain and justify the process to construct congruent segments and angles, angle bisectors, perpendicular bisectors, altitudes, medians, and parallel and perpendicular lines.
- G.PL.4 Develop the distance formula using the Pythagorean Theorem. Find the lengths and midpoints of line segments in the two-dimensional coordinate system.
- Checkpoint opportunity

### G.T Triangles

- G.T.1 Prove and apply theorems about triangles, including the following: measures of interior angles of a triangle sum to 180°; the Isosceles Triangle Theorem and its converse; the Pythagorean Theorem; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and its converse; and the Angle Bisector Theorem.
- G.T.2 Explore and explain how the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) follow from the definition of congruence in terms of rigid motions.
- G.T.3 Use tools to explain and justify the process to construct congruent triangles.
- G.T.4 Use the definition of similarity in terms of similarity transformations, to determine if two given triangles are similar. Explore and develop the meaning of similarity for triangles.
- G.T.5 Use congruent and similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles.
- G.T.6 Prove and apply the inequality theorems, including the following: Triangle inequality; Inequality in one triangle; and the Hinge theorem and its converse.
- G.T.7 Explore the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Understand and use the geometric mean to solve for missing parts of triangles.
- G.T.8 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.
- G.T.9 Use trigonometric ratios (sine, cosine,tangent and their inverses) and the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles.
- G.T.10 Explore the relationship between the sides of special right triangles (30° – 60° and 45° – 45°) and use them to solve real-world and other mathematical problems.
- Checkpoint opportunity

### G.QP Quadrilaterals and Other Polygons

- G.QP.1 Prove and apply theorems about parallelograms, including those involving angles, diagonals, and sides.
- G.QP.2 Prove that given quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids. Include coordinate proofs of quadrilaterals in the coordinate plane.
- G.QP.3 Develop and use formulas to find measures of interior and exterior angles of polygons.
- G.QP.4 Identify types of symmetry of polygons, including line, point, rotational, and self-congruences.
- G.QP.5 Compute perimeters and areas of polygons in the coordinate plane to solve real-world and other mathematical problems.
- G.QP.6 Develop and use formulas for areas of regular polygons.
- Checkpoint opportunity

### G.CI Circles

- G.CI.1 Define, identify and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, congruent circles, and concentric circles.
- G.CI.2 Derive the fact that the length of the arc intercepted by an angle is proportional to the radius; derive the formula for the area of a sector.
- G.CI.3 Explore and use relationships among inscribed angles, radii, and chords, including the following: the relationship that exists between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; and the radius of a circle is perpendicular to a tangent where the radius intersects the circle.
- G.CI.4 Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents).
- G.CI.5 Use tools to explain and justify the process to construct a circle that passes through three given points not on a line, a tangent line to a circle through a point on the circle, and a tangent line from a point outside a given circle to the circle.
- G.CI.6 Use tools to construct the inscribed and circumscribed circles of a triangle. Prove properties of angles for a quadrilateral inscribed in a circle.
- Checkpoint opportunity

### G.TR Transformations

- G.TR.1 Use geometric descriptions of rigid motions to transform figures and to predict and describe the results of translations, reflections and rotations on a given figure. Describe a motion or series of motions that will show two shapes are congruent.
- G.TR.2 Understand 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. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
- Checkpoint opportunity

### G.TS Three-Dimensional Solids

- G.TS.1 Describe relationships between the faces, edges, and vertices of three-dimensional solids. Create a net for a given three-dimensional solid. Describe the three-dimensional solid that can be made from a given net (or pattern).
- G.TS.2 Explore and use symmetries of three-dimensional solids to solve problems.
- G.TS.3 Explore properties of congruent and similar solids, including prisms, regular pyramids, cylinders, cones, and spheres and use them to solve problems.
- G.TS.4 Solve real-world and other mathematical problems involving volume and surface area of prisms, cylinders, cones, spheres, and pyramids, including problems that involve composite solids and algebraic expression
- G.TS.5 Apply geometric methods to create and solve design problems.
- Checkpoint opportunity