New York State Regents Examinations Geometry (Regents Examination) Syllabus

Geometry (Regents Examination) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Geometry (Regents Examination) in New York State Regents Examinations, not a summary of it.

1. 1

   Tools, Constructions, and Foundations

   3 topics

   • Geometric Definitions and Postulates
     • Points, lines, planes, segments, and angles
     • Angle pair relationships
   • Compass-and-Straightedge Constructions
     • Bisecting segments and angles
     • Perpendicular and parallel lines
     • Constructing equilateral triangles, squares, and inscribed figures
   • Reasoning and Logic in Geometry
     • Conditional statements, converse, and contrapositive
     • Inductive and deductive reasoning
2. 2

   Transformations and Congruence

   3 topics

   • Rigid Motions
     • Translations, reflections, and rotations
     • Compositions of transformations
   • Triangle Congruence
     • SSS, SAS, ASA, AAS, and HL criteria
     • Proving triangles congruent
   • Symmetry
     • Line and rotational symmetry of figures
3. 3

   Similarity, Right Triangles, and Trigonometry

   3 topics

   • Similarity Transformations
     • Dilations and scale factor
     • AA, SAS, and SSS similarity criteria
   • Right Triangle Relationships
     • Pythagorean theorem and its converse
     • Geometric mean and altitude-on-hypotenuse
   • Right Triangle Trigonometry
     • Sine, cosine, and tangent ratios
     • Angles of elevation and depression
4. 4

   Circles

   3 topics

   • Angles and Segments in Circles
     • Central, inscribed, and tangent-chord angles
     • Chord, secant, and tangent segment relationships
   • Arc Length and Sector Area
     • Radian and degree measure of arcs
   • Equations of Circles
     • Deriving and using the standard form
     • Completing the square to find center and radius
5. 5

   Coordinate Geometry and Proof

   3 topics

   • Coordinate Methods
     • Distance, midpoint, and slope formulas
     • Partitioning a segment in a given ratio
   • Proving Properties Algebraically
     • Classifying quadrilaterals on the coordinate plane
     • Parallel and perpendicular lines
   • Equations of Lines in Proofs
     • Writing equations to verify geometric relationships
6. 6

   Measurement: Area, Surface Area, and Volume

   3 topics

   • Two-Dimensional Measurement
     • Area of polygons and composite figures
   • Three-Dimensional Solids
     • Volume of prisms, cylinders, pyramids, cones, and spheres
     • Cavalieri's principle
   • Modeling with Geometry
     • Density and design problems
     • Cross-sections and solids of revolution

Geometry (Regents Examination) flashcards for New York State Regents Examinations

18 of 50 cards from the Geometry (Regents Examination) deck — real questions with worked answers.

1. What is a postulate (axiom) in geometry?

   A statement accepted as true without proof, serving as a basic building block for proving theorems.

2. State the definition of congruent figures in terms of rigid motions.

   Two figures are congruent if there exists a sequence of rigid motions (translations, reflections, rotations) that maps one exactly onto the other.

3. How is the midpoint of a segment defined, and what does it create?

   The midpoint is the point that divides a segment into two congruent segments; it bisects the segment into two equal halves.

4. What does it mean for two lines to be perpendicular?

   They intersect to form four right angles (90 degrees each).

5. Describe the compass-and-straightedge construction of the perpendicular bisector of a segment.

   Open the compass more than half the segment's length; draw arcs from each endpoint above and below the segment; connect the two arc intersections with a straightedge.

6. How do you construct an angle bisector with compass and straightedge?

   From the vertex, draw an arc crossing both sides; from those two intersection points draw equal arcs that cross inside the angle; draw a ray from the vertex through that crossing point.

7. How do you copy an angle using compass and straightedge?

   Draw an arc through both sides of the original angle; replicate that arc on a new ray; measure the arc's width between the sides and mark it on the new arc; draw a ray from the new vertex through that mark.

8. How is an equilateral triangle constructed on a given segment?

   Use the segment as radius; draw an arc centered at each endpoint; the arcs intersect at the third vertex; connect it to both endpoints.

9. What is the difference between inductive and deductive reasoning?

   Inductive reasoning draws a general conclusion from specific observations (a conjecture); deductive reasoning uses accepted facts and logic to reach a guaranteed conclusion (a proof).

10. What is the converse of a conditional statement 'If p, then q'?

    'If q, then p' — the hypothesis and conclusion are switched.

11. What is a counterexample and what does it prove?

    An example that satisfies the hypothesis but not the conclusion; it proves a statement is false.

12. List the three basic rigid motions and state what each preserves.

    Translation (slide), reflection (flip), and rotation (turn); all three preserve distance (length) and angle measure.

13. How does a translation by vector (a, b) transform a point (x, y)?

    (x, y) maps to (x + a, y + b).

14. What are the coordinate rules for reflecting (x, y) over the x-axis, the y-axis, and the line y = x?

    Over x-axis: (x, -y); over y-axis: (-x, y); over y = x: (y, x).

15. What are the coordinate rules for rotating (x, y) about the origin by 90 degrees, 180 degrees, and 270 degrees counterclockwise?

    90 degrees: (-y, x); 180 degrees: (-x, -y); 270 degrees: (y, -x).

16. List the five triangle congruence criteria.

    SSS, SAS, ASA, AAS, and HL (Hypotenuse-Leg, for right triangles).

17. Why is SSA (or AAA) not a valid triangle congruence criterion?

    SSA can produce two different triangles (the ambiguous case), and AAA only guarantees similarity, not congruence (same shape, different size).

18. What does CPCTC stand for and when is it used?

    Corresponding Parts of Congruent Triangles are Congruent; used after proving two triangles congruent to conclude their corresponding sides/angles are congruent.

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