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.

Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Inscribed Angles
Parallel, Intersecting, and Skew 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).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

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

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures

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

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

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.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

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.

Absolute Value with Linear Functions
Circles
Dilations
Holiday Snowflake Designer
Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

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).

Congruence in Right Triangles
Proving Triangles Congruent

HSG.CO.3: Apply and prove geometric theorems

HSG.CO.C.9: Apply and prove theorems about lines and angles.

Investigating Angle Theorems

HSG.CO.C.10: Apply and prove theorems about triangles.

Isosceles and Equilateral Triangles
Proving Triangles Congruent
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Triangle Angle Sum
Triangle Inequalities

HSG.CO.C.11: Apply and prove theorems about quadrilaterals.

Parallelogram Conditions
Special Parallelograms

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.).

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines

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.

Dilations
Similar Figures

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.

Circles
Dilations
Similar Figures
Similarity in Right Triangles

HSG.SRT.A.3: Use the properties of similarity transformations to establish the AA, SAS~, SSS~ criteria for two triangles to be similar.

Similar Figures

HSG.SRT.7: Apply and prove theorems involving similarity

HSG.SRT.B.4: Use triangle similarity to apply and prove theorems about triangles.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures

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.

Chords and Arcs
Congruence in Right Triangles
Constructing Congruent Segments and Angles
Dilations
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures
Similarity in Right Triangles

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.

Sine, Cosine, and Tangent Ratios

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.

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

HSG.C: Circles

HSG.C.10: Understand and apply theorems about circles

HSG.C.A.2: Identify, describe, and use relationships among angles, radii, segments, lines, arcs, and chords as related to circles.

Chords and Arcs
Inscribed Angles

HSG.C.A.3: Construct the inscribed and circumscribed circles of a triangle. Prove properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

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.

Chords and Arcs

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.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

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

Parabolas

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

HSG.GPE.13: Use coordinates to prove simple geometric theorems algebraically

HSG.GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

Distance Formula

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.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

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).

Prisms and Cylinders
Pyramids and Cones