Standardized Test Preparation

New York Regents

Geometry - Grade 10

Assessment Exam - NY Regents Geometry
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.
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.

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.

Properties of Isometrics
Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments.
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.
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.
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.
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.
Triangle Inequalities

AAS and HL Congruencies
Prove geometric theorems.
Prove and apply theorems about lines and angles.
Prove and apply theorems about triangles.
Prove and apply theorems about parallelograms.
Make geometric constructions.
Make, justify,and apply formal geometric constructions.
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
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.
  1. Verify experimentally that 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.
  2. Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

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.
Corresponding Parts
Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar.
Proportions with Area and Volume
Prove theorems involving similarity.
Prove and apply similarity theorems about triangles.
Similar Triangles
Use congruence and similarity criteria for triangles to:
  1. Solve problems algebraically and geometrically.
  2. Prove relationships in geometric figures.

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.
Special Right Triangles

Indirect Measurement with Similar Triangles
Explain and use the relationship between the sine and cosine of complementary angles.
Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems.
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.
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.
Identify, describe and apply relationships between the angles and their intercepted arcs of a circle.
Circumference/Diameter Ratio

Arc Length
Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle.
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.
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.
Geometric Proofs
Graph circles given their equation.
Use coordinates to prove simple geometric theorems algebraically.
On the coordinate plane, algebraically prove geometric theorems and properties.
On the coordinate plane:
  1. Explore the proof for the relationship between slopes of parallel and perpendicular lines
  2. Determine if lines are parallel, perpendicular, or neither, based on their slopes
  3. Apply properties of parallel and perpendicular lines to solve geometric problems.

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.
Symbols of Logic
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
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.
Solids with Curved Surfaces

Areas of Regular Polygons

Surface Area
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Polyhedrons, Prisms and Pyramids


Visualize relationships between two-dimensional and three-dimensional objects.
Identify the shapes of plane sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Conditional Proof
Modeling with Geometry eTAP Lesson
Apply geometric concepts in modeling situations.
Use geometric shapes, their measures, and their properties to describe objects.
Apply concepts of density based on area and volume of geometric figures in modeling situations.
Number and Picture Patterns

Mathematical Modeling
Apply geometric methods to solve design problems.
Indirect Proof