Congruence  eTAP Lesson 

Experiment with transformations in the plane.  
Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane.
NY.GEOG.CO.1 
Definitions  Geometry Definition  Circles Arcs and Angles 
Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.
NY.GEOG.CO.2 
Transformations Definitions of Figures Line and Angle Relationships 
Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that carry the polygon onto itself.
NY.GEOG.CO.3 
Symmetry Properties of Isometrics 
Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments.
NY.GEOG.CO.4 
Tessellations Using Translations, Rotations, and Glide Reflections Tessellations with Polygons 
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.
NY.GEOG.CO.5 

Understand congruence in terms of rigid motions.  
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.
NY.GEOG.CO.6 
SSS and SAS Congruencies ASA Congruencies 
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.
NY.GEOG.CO.7 
Triangles Sum Conjecture Properties of Isosceles Triangles 
Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL(Hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions.
NY.GEOG.CO.8 
Triangle Inequalities AAS and HL Congruencies 
Prove geometric theorems.  
Prove and apply theorems about lines and angles.
NY.GEOG.CO.9 

Prove and apply theorems about triangles.
NY.GEOG.CO.10 
Triangles 
Prove and apply theorems about parallelograms.
NY.GEOG.CO.11 

Make geometric constructions.  
Make, justify,and apply formal geometric constructions.
NY.GEOG.CO.12 
Duplicating Segments and Angles Constructing Perpendicular Bisectors 
Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle
NY.GEOG.CO.13 
Constructing Perpendiculars Constructing Angle Bisectors 
Similarity, Right Triangles, and Trigonometry  eTAP Lesson 
Understand similarity in terms of similarity transformations.  
Verify experimentally the properties of dilations given by a center and a scale factor.
NY.GEOG.SRT.1 
Constructing Parallel Lines Constructing Points of Concurrency The Centroid Postulates of Geometry 
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
NY.GEOG.SRT.2 
Corresponding Parts 
Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar.
NY.GEOG.SRT.3 
Proportions with Area and Volume 
Prove theorems involving similarity.  
Prove and apply similarity theorems about triangles.
NY.GEOG.SRT.4 
Similar Triangles 
Use congruence and similarity criteria for triangles to:
NY.GEOG.SRT.5 
Law of Contraposition 
Define trigonometric ratios and solve problems involving right triangles.  
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent ratios for acute angles.
NY.GEOG.SRT.6 
Special Right Triangles Indirect Measurement with Similar Triangles 
Explain and use the relationship between the sine and cosine of complementary angles.
NY.GEOG.SRT.7 
Converse 
Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems.
NY.GEOG.SRT.8 
The Theorem of Pythagoras 
Apply Trigonometry to general triangles.  
Justify and apply the formula A= 1/2ab sin (C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
NY.GEOG.SRT.9 
Areas of Triangles, Trapezoids and Kites Areas of Rectangles and Parallelograms 
Circles  eTAP Lesson 
Understand and apply theorems about circles.  
Prove that all circles are similar.
NY.GEOG.C.1 

Identify, describe and apply relationships between the angles and their intercepted arcs of a circle.
NY.GEOG.C.2a 
Circumference/Diameter Ratio Arc Length 
Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle.
NY.GEOG.C.2b 
Chord Properties Tangent Properties 
Find arc lengths and area of sectors of circles.  
Using proportionality, find one of the following given two others; the central angle, arc length, radius or area of sector.
NY.GEOG.C.5 
Areas of Circles 
Expressing Geometric Properties with Equations  eTAP Lesson 
Translate between the geometric description and the equation of a conic section.  
Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Find the center and radius of a circle, given the equation of the circle.
NY.GEOG.GPE.1a 
Geometric Proofs 
Graph circles given their equation.
NY.GEOG.GPE.1b 

Use coordinates to prove simple geometric theorems algebraically.  
On the coordinate plane, algebraically prove geometric theorems and properties.
NY.GEOG.GPE.4 

On the coordinate plane:
NY.GEOG.GPE.5 
Parallelograms and Parallel Lines Related Proofs Proving Angle Conjectures 
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
NY.GEOG.GPE.6 
Symbols of Logic 
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
NY.GEOG.GPE.7 
Direct Proofs 
Geometric Measurement and Dimension  eTAP Lesson 
Explain volume formulas and use them to solve problems.  
Provideinformal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
NY.GEOG.GMD.1 
Solids with Curved Surfaces Areas of Regular Polygons Surface Area 
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
NY.GEOG.GMD.3 
Polyhedrons, Prisms and Pyramids Cylinders Cones 
Visualize relationships between twodimensional and threedimensional objects.  
Identify the shapes of plane sections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.
NY.GEOG.GMD.4 
Conditional Proof 
Modeling with Geometry  eTAP Lesson 
Apply geometric concepts in modeling situations.  
Use geometric shapes, their measures, and their properties to describe objects.
NY.GEOG.MG.1 
Definition 
Apply concepts of density based on area and volume of geometric figures in modeling situations.
NY.GEOG.MG.2 
Number and Picture Patterns Mathematical Modeling 
Apply geometric methods to solve design problems.
NY.GEOG.MG.3 
Indirect Proof 