HS.G-CO: Congruence

1.1: Experiment with transformations in the plane

HS.G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, and plane.

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

HS.G-CO.2.i: Represent transformations in the plane.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

HS.G-CO.2.ii: Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

HS.G-CO.2.iii: 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

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

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures

HS.G-CO.4: Develop or verify experimentally the characteristics of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Circles
Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

HS.G-CO.5.i: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

HS.G-CO.5.ii: Specify a sequence of transformations that will carry a given figure onto another.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

1.2: Understand congruence in terms of rigid motions

HS.G-CO.6.i: Use geometric descriptions of rigid motions to predict the effect of a given rigid motion on a given figure.

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

HS.G-CO.6.ii: Use the definition of congruence in terms of rigid motions to decide if two figures are congruent.

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

HS.G-CO.8: Prove two triangles are congruent using the congruence theorems such as ASA, SAS, and SSS.

Congruence in Right Triangles
Proving Triangles Congruent

1.3: Prove and apply geometric theorems

HS.G-CO.9: Prove and apply theorems about lines and angles.

Investigating Angle Theorems
Slope

HS.G-CO.10: Prove and apply theorems about triangle properties.

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

HS.G-CO.11: Prove and apply theorems about parallelograms.

Parallelogram Conditions
Special Parallelograms

1.4: Make geometric constructions

HS.G-CO.12: Make basic geometric constructions with a variety of tools and methods.

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

1.4.1.2: Tools may include compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

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

HS.G-CO.13: Apply basic constructions to create polygons such as equilateral triangles, squares, and regular hexagons inscribed in circles.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

HS.G-SRT: Similarity, Right Triangles, and Trigonometry

2.1: Understand similarity

HS.G-SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor.

Dilations
Similar Figures

HS.G-SRT.2.i: Given two figures, use transformations to decide if they are similar.

Circles
Dilations
Similar Figures
Similarity in Right Triangles

HS.G-SRT.2.ii: Apply 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

HS.G-SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Similar Figures

2.2: Prove theorems involving similarity

HS.G-SRT.4: Prove similarity theorems about triangles.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures

HS.G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and 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

2.3: Define trigonometric ratios and solve problems involving right triangles

HS.G-SRT.6: Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine, and tangent of an acute angle in a right triangle.

Sine, Cosine, and Tangent Ratios

HS.G-SRT.8: Use special right triangles (30°-60°-90° and 45°-45°-90°), trigonometric ratios and the Pythagorean Theorem to solve 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

HS.G-C: Circles

3.1: Understand and apply theorems about circles

HS.G-C.1: Understand and apply theorems about relationships with line segments and circles including radii, diameter, secants, tangents, and chords.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

HS.G-C.2.i: Understand and apply theorems about relationships with angles formed by radii, diameter, secants, tangents, and chords.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

HS.G-C.2.ii: Understand and apply properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

HS.G-C.3: Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.

Concurrent Lines, Medians, and Altitudes

3.2: Find arc lengths and areas of sectors of circles

HS.G-C.5: Explain and use the formulas for arc length and area of sectors of circles.

Chords and Arcs

HS.G-GPE: Expressing Geometric Properties with Equations

4.1: Understand and use conic sections

HS.G-GPE.1.i: Derive the equation of a circle of given center and radius.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

HS.G-GPE.1.ii: Derive the equation of a parabola given a focus and directrix.

Parabolas

HS.G-GPE.1.iii: Derive the equations of ellipses and hyperbolas given foci, using the fact that the sum or difference of distances from the foci is constant.

Ellipses
Hyperbolas

HS.G-GPE.3.i: Identify key features of conic sections given their equations.

Addition and Subtraction of Functions
Circles
Ellipses
Hyperbolas
Parabolas

HS.G-GPE.3.ii: Apply properties of conic sections in real world situations.

Circles

4.2: Use coordinates to verify simple geometric theorems algebraically

HS.G-GPE.5.i: Develop and verify the slope criteria for parallel and perpendicular lines.

Cat and Mouse (Modeling with Linear Systems)

HS.G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles, parallelograms, trapezoids and kites.

Distance Formula

HS.G-GMD: Geometric Measurement and Dimension

5.1: Explain surface area and volume formulas and use them to solve problems

HS.G-GMD.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

5.1.1.1: May use dissection arguments. Cavalieri’s Principle or informal limit arguments.

Pyramids and Cones

5.1.1.2: Cavalieri’s Principle: 2D: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.

Pyramids and Cones

HS.G-GMD.2: Calculate the surface area for prisms, cylinders, pyramids, cones, and spheres to solve problems.

Surface and Lateral Areas of Prisms and Cylinders
Surface and Lateral Areas of Pyramids and Cones

HS.G-GMD.3: Know and apply volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders
Pyramids and Cones