Math.Content.HSG-CO: Congruence

Math.Content.HSG-CO.A: Experiment with transformations in the plane

Math.Content.HSG-CO.A.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.

Chords and Arcs
Constructing Parallel and Perpendicular Lines
Parallel, Intersecting, and Skew Lines

Math.Content.HSG-CO.A.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).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

Math.Content.HSG-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Reflections
Rotations, Reflections, and Translations

Math.Content.HSG-CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Reflections
Rotations, Reflections, and Translations
Translations

Math.Content.HSG-CO.A.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.

Reflections
Rotations, Reflections, and Translations
Translations

Math.Content.HSG-CO.B: Understand congruence in terms of rigid motions

Math.Content.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.

Reflections
Rotations, Reflections, and Translations
Translations

Math.Content.HSG-CO.C: Prove geometric theorems

Math.Content.HSG-CO.C.9: Prove theorems about lines and angles.

Constructing Parallel and Perpendicular Lines
Investigating Angle Theorems
Segment and Angle Bisectors

Math.Content.HSG-CO.C.10: Prove theorems about triangles.

Concurrent Lines, Medians, and Altitudes
Congruence in Right Triangles
Isosceles and Equilateral Triangles
Proving Triangles Congruent
Triangle Angle Sum
Triangle Inequalities

Math.Content.HSG-CO.C.11: Prove theorems about parallelograms.

Parallelogram Conditions
Special Parallelograms

Math.Content.HSG-CO.D: Make geometric constructions

Math.Content.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.).

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

Math.Content.HSG-SRT: Similarity, Right Triangles, and Trigonometry

Math.Content.HSG-SRT.A: Understand similarity in terms of similarity transformations

Math.Content.HSG-SRT.A.1: Verify experimentally the properties of dilations given by a center and a scale factor:

Math.Content.HSG-SRT.A.1.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.

Dilations

Math.Content.HSG-SRT.A.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations

Math.Content.HSG-SRT.A.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.

Dilations
Similar Figures

Math.Content.HSG-SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Proving Triangles Congruent
Similar Figures

Math.Content.HSG-SRT.B: Prove theorems involving similarity

Math.Content.HSG-SRT.B.4: Prove theorems about triangles.

Congruence in Right Triangles
Proving Triangles Congruent
Pythagorean Theorem
Similarity in Right Triangles

Math.Content.HSG-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Chords and Arcs
Perimeters and Areas of Similar Figures
Pythagorean Theorem
Similar Figures
Similarity in Right Triangles

Math.Content.HSG-SRT.C: Define trigonometric ratios and solve problems involving right triangles

Math.Content.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.

Sine, Cosine, and Tangent Ratios

Math.Content.HSG-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles.

Cosine Function
Sine Function

Math.Content.HSG-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

Math.Content.HSG-C: Circles

Math.Content.HSG-C.A: Understand and apply theorems about circles

Math.Content.HSG-C.A.2: Identify and describe relationships among inscribed angles, radii, and chords.

Chords and Arcs
Inscribed Angles

Math.Content.HSG-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes

Math.Content.HSG-C.B: Find arc lengths and areas of sectors of circles

Math.Content.HSG-C.B.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.

Radians

Math.Content.HSG-GPE: Expressing Geometric Properties with Equations

Math.Content.HSG-GPE.A: Translate between the geometric description and the equation for a conic section

Math.Content.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.

Circles

Math.Content.HSG-GPE.A.2: Derive the equation of a parabola given a focus and directrix.

Parabolas

Math.Content.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.

Ellipses
Hyperbolas

Math.Content.HSG-GPE.B: Use coordinates to prove simple geometric theorems algebraically

Math.Content.HSG-GPE.B.4: Use coordinates to prove simple geometric theorems algebraically.

Distance Formula

Math.Content.HSG-GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Distance Formula

Math.Content.HSG-GMD: Geometric Measurement and Dimension

Math.Content.HSG-GMD.A: Explain volume formulas and use them to solve problems

Math.Content.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.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

Math.Content.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.

Prisms and Cylinders
Pyramids and Cones

Math.Content.HSG-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders
Pyramids and Cones

Math.Content.HSG-GMD.B: Visualize relationships between two-dimensional and three-dimensional objects

Math.Content.HSG-GMD.B.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.

Prisms and Cylinders
Pyramids and Cones