Geometry (Common Core) Regents Review Flashcards

Study Guide Overview and Dedication

• Resource Identity: This document is the fourth edition of the "Geometry (Common Core) Facts You Must Know Cold for the Regents Exam," published in the spring of 2018. It was developed by Trevor Clark for NYS Mathematics Regents Preparation.

• Purpose: The guide is designed as an exhaustive resource containing essential formulas, theorems, and concepts required to pass or master the Geometry Regents exam. It is intended for both students (to memorize and understand) and teachers (to ensure curriculum coverage).

• Study Strategy:

  • Step 1: Memorize the material "cold."

  • Step 2: Take a practice Geometry Regents exam with this guide as a reference.

  • Step 3: If stuck, consult the guide to identify the necessary theorem; this process aids retention.

  • Step 4: Use supplemental videos at www.nysmathregentsprep.com for full explanations.

• New to This Edition (4th Ed):

  • Speed and average speed formulas (from Algebra 1).

  • Coordinate geometry proof properties for quadrilaterals.

  • Updated formatting, improved diagrams, and typo corrections.

• Organization and Support: NYS Mathematics Regents Preparation is a non-for-profit organization. It offers resources for Algebra 1, Algebra 2, and eventually AP Calculus, AP Statistics, and SAT/ACT prep. Support is sought via Patreon.

• Dedication: The author, Trevor Clark, dedicates this guide to his former teachers at Farmingdale High School: Mrs. Mary-Elena D’Ambrosio, Mrs. Laura Angelo-Provenza, Mrs. Louise Corcoran, Mrs. Efstratia Vouvoudakis, Mr. Scott Drucker, and Mr. Ed Papo. Additional inspiration came from Mrs. Jacquelyn Passante-Merlo and Mrs. Mary Ann DeRosa (W. E. Howitt Middle School) and Ms. Elizabeth Bove (Massapequa High School).

Polygons: Interior and Exterior Angles

• Sum of Interior Angles: The total degree measure of all interior angles in any polygon with $n$ sides is given by:

  • $\text{Sum} = 180(n - 2)$

• Each Interior Angle (Regular Polygon): For a regular polygon (where all sides and angles are congruent), the measure of a single interior angle is:

  • $\text{Angle} = \frac{180(n - 2)}{n}$

• Sum of Exterior Angles: For any polygon, the sum of the exterior angles (one at each vertex) is always constant:

  • $\text{Sum} = 360^{\circ}$

• Each Exterior Angle (Regular Polygon): The measure of a single exterior angle of a regular polygon is:

  • $\text{Angle} = \frac{360}{n}$

Triangle Classification and Fundamental Properties

• Classifying by Sides:

  • Scalene: No congruent sides.

  • Isosceles: Exactly 2 congruent sides.

  • Equilateral: All 3 sides are congruent.

• Classifying by Angles:

  • Acute: All internal angles are less than $90^{\circ}$.

  • Right: Exactly one angle measures $90^{\circ}$.

  • Obtuse: Exactly one angle is greater than $90^{\circ}$.

  • Equiangular: All 3 angles are congruent (each is exactly $60^{\circ}$, making it equilateral as well).

• Angle Sum Theorem: All triangles contain interior angles that sum to $180^{\circ}$.

• Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent (remote) interior angles.

• Triangle Midsegment: A segment joining the midpoints of two sides of a triangle.

  • It is always parallel to the third side.

  • Its length is exactly $\frac{1}{2}$ the length of the third side.

  • It splits the triangle into two similar triangles.

• Isosceles Triangle Specifics:

  • Found to have 2 congruent sides and 2 congruent base angles.

  • The altitude drawn from the vertex angle to the base also serves as the median (bisecting the base) and the angle bisector of the vertex.

  • Theorem: If two sides of a triangle are congruent, the angles opposite those sides are congruent.

• Triangle Inequality Theorems:

  • The sum of any two sides must be strictly greater than the third side.

  • The difference of any two sides must be strictly less than the third side.

  • The longest side is always opposite the largest angle; the shortest side is always opposite the smallest angle.

Coordinate Geometry and Line Relationships

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a specific point on the line.

• Slope Formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel and Perpendicular Lines:

  • Parallel: Lines have the same slope.

  • Perpendicular: Lines have negative reciprocal slopes (invert the fraction and negate the sign).

• Collinear Points: Points that lie on the exact same line.

• Midpoint Formula: $M = \begin{pmatrix} \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \end{pmatrix}$

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Partitioning Line Segments: To find a point that partitions a segment into a specific ratio, use linear proportions:

  • $\frac{x - x_1}{x_2 - x} = \frac{\text{part}_1}{\text{part}_2}$

  • $\frac{y - y_1}{y_2 - y} = \frac{\text{part}_1}{\text{part}_2}$

Parallel Lines and Transversals

• When two parallel lines are cut by a transversal:

  • Alternate Interior Angles: Congruent.

  • Alternate Exterior Angles: Congruent.

  • Corresponding Angles: Congruent.

  • Same-Side Interior Angles: Supplementary (sum to $180^{\circ}$).

• Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.

Trigonometry and Right Triangles

• SOHCAHTOA:

  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

  • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

• Calculator Use: Use standard functions (sin, cos, tan) to find sides; use inverse functions ($\sin^{-1}$, $\cos^{-1}$, Spirit $\tan^{-1}$) to find angle measures.

• Algebra 1 Retrieval:

  • $\text{Average Speed} = \frac{\Delta \text{Distance}}{\Delta \text{Time}}$

  • $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Cofunctions: Sine and Cosine are cofunctions and are complementary.

  • $\sin(A) = \cos(90^{\circ} - A)$

  • $\cos(A) = \sin(90^{\circ} - A)$

  • If $A$ and $B$ are acute angles of a right triangle, then $\sin(A) = \cos(B)$.

• Pythagorean Theorem: In a right triangle with legs $a$ and $b$ and hypotenuse $c$:

  • $a^2 + b^2 = c^2$

• Mean Proportional Theorems:

  • Altitude Theorem (SAAS/Heartbeat Method): The altitude ($h$) is the geometric mean between the two segments of the hypotenuse ($p_1$ and $p_2$).

    • $\frac{p_1}{h} = \frac{h}{p_2}$

  • Leg Theorem (HYLLS/PSSW): The leg ($L$) is the geometric mean between the segment of the hypotenuse it touches ($s$) and the whole hypotenuse ($H$).

    • $\frac{H}{L} = \frac{L}{s}$

Congruence and Similarity

• Triangle Congruence Theorems: SSS, SAS, ASA, AAS, and HL (Hypotenuse-Leg for right triangles only).

• CPCTC: Corresponding Parts of Congruent Triangles are Congruent. Use this after proving triangles are congruent.

• Similar Triangle Theorems: AA, SAS ($\sim$), and SSS ($\sim$).

• Similarity Properties:

  • Figures have congruent angles and proportional sides.

  • CSSTP: Corresponding Sides of Similar Triangles are in Proportion.

  • In a proportion, the product of the means equals the product of the extremes.

Transformational Geometry

• Rigid Motion: A transformation that preserves distance, congruency, angle measure, size, and shape.

• Reflections (FLIP):

  • $r_{x\text{-axis}}(x, y) = (x, -y)$

  • $r_{y\text{-axis}}(x, y) = (-x, y)$

  • $r_{y=x}(x, y) = (y, x)$

  • $r_{y=-x}(x, y) = (-y, -x)$

  • $r_{\text{origin}}(x, y) = (-x, -y)$

• Rotations (TURN):

  • $R_{90^{\circ}}(x, y) = (-y, x)$

  • $R_{180^{\circ}}(x, y) = (-x, -y)$

  • $R_{270^{\circ}}(x, y) = (y, -x)$

• Translations (SHIFT/MOVE):

  • $T_{a,b}(x, y) = (x + a, y + b)$

• Dilations (ENLARGEMENT/REDUCTION):

  • $D_k(x, y) = (kx, ky)$

  • Dilations create similar figures (proportional sides, congruent angles).

  • Dilations are not always rigid motions because they do not preserve distance/congruency (unless $k=1$).

• Composition of Transformations:

  • Notation: $R_{90^{\circ}} \circ T_{3,-4}$

  • Rule: Work from right to left (Perform the translation first, then the rotation).

  • Two reflections over parallel lines = one translation.

  • Two reflections over intersecting lines = one rotation.

• Rotational Symmetry: A regular polygon with $n$ sides maps onto itself in increments of its central angle: $\frac{360}{n}$.

Circles: Equations and Graphing

• Circle Definition: A 2D shape where all points on the curve are equidistant from a fixed center.

• Equations:

  • General/Standard Form: $x^2 + y^2 + Dx + Ey + F = 0$

  • Center-Radius Form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• Completing the Square (to find center-radius form):

  1. Ensure squared terms have a coefficient of 1.

  2. Move the constant to the right side of the equation.

  3. Group $x$ terms and $y$ terms; insert blank spaces for perfect-square trinomials.

  4. Add $(\frac{b}{2})^2$ for both linear terms to both sides.

  5. Factor as $(x - h)^2$ and $(y - k)^2$.

• Review of Factoring:

  • GCF: $ab + ac = a(b + c)$

  • DOTS (Difference of Two Perfect Squares): $x^2 - y^2 = (x + y)(x - y)$

  • Trinomial (AM Method): $x^2 - x + 6 = (x + 2)(x - 3)$

• Graphing Steps:

  1. Identify center and radius.

  2. Plot the center.

  3. Plot four loci points equidistant (the radius length) from the center (up, down, left, right).

  4. Connect points using a compass or freehand.

Angle and Segment Relationships in Circles

• Angle Types:

  • Central Angle: Angle = Intercepted Arc.

  • Inscribed Angle: Angle = $\frac{1}{2}$ Intercepted Arc.

  • Tangent-Chord Angle: Angle = $\frac{1}{2}$ Intercepted Arc.

  • Two Chord Angle (Internal): Angle = $\frac{\text{Arc}_1 + \text{Arc}_2}{2}$

  • Outside Angle (Two Secants, Two Tangents, or Secant-Tangent): Angle = $\frac{\text{Big Arc} - \text{Little Arc}}{2}$

• Special Circle Theorems:

  • A tangent is perpendicular to the radius at the point of tangency ($90^{\circ}$ angle).

  • An angle inscribed in a semicircle is a right angle ($90^{\circ}$).

  • If a quadrilateral is inscribed in a circle, opposite angles are supplementary ($180^{\circ}$).

  • Parallel chords intercept congruent arcs.

  • If chords are congruent, their intercepted arcs are congruent.

  • A diameter perpendicular to a chord bisects the chord and its arc.

• Segment Lengths:

  • Chords: (Part)(Part) = (Part)(Part) or $(a)(b) = (c)(d)$

  • Secants: (Whole)(External) = (Whole)(External) or $(W_1)(E_1) = (W_2)(E_2)$

  • Secant-Tangent: (Whole)(External) = $\text{Tangent}^2$ or $(W)(E) = T^2$

Sector Area, Arc Length, and 3-D Figures

• Area of a Sector:

  • Using Radians: $A = \frac{1}{2}r^2 \theta$

  • Using Degrees: $A = \frac{n}{360} \pi r^2$

• Arc/Sector Length:

  • Using Radians: $s = r\theta$

• Cavalieri’s Principle: If two solids have the same height and the same cross-sectional area at every level, they have the same volume.

• Volume Formulas:

  • Total Volume = (Area of Base) $\times$ (Height).

  • Includes Prism, Cone, Pyramid, Cylinder, and Sphere.

• Density Formulas:

  • $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Cross Sections: A 2D surface exposed by a straight cut through a 3D object.

Quadrilateral Properties and Proofs

• The Quadrilateral Family Tree (Properties inherit downward):

  • Quadrilateral: 4-sided polygon.

  • Trapezoid: At least one pair of parallel sides. Median length = $\frac{1}{2}(\text{base}_1 + \text{base}_2)$.

  • Isosceles Trapezoid: Congruent base angles, congruent diagonals, congruent legs (non-parallel sides).

  • Parallelogram: Opposite sides parallel and congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other.

  • Rectangle: Parallelogram properties plus all right angles and congruent diagonals.

  • Rhombus: Parallelogram properties plus all sides congruent, perpendicular diagonals, diagonals bisect opposite angles, and diagonals form four congruent right triangles.

  • Square: Inherits all properties of rectangles and rhombuses; diagonals form four congruent isosceles right triangles.

• Proving Quadrilaterals:

  • To prove a Parallelogram: Show both pairs of opposite sides are parallel OR congruent; OR show one pair of opposite sides is both parallel and congruent; OR show diagonals bisect each other.

  • To prove a Rectangle: Show it is a parallelogram and then show one right angle OR congruent diagonals.

  • To prove a Rhombus: Show it is a parallelogram and then show one pair of congruent consecutive sides OR perpendicular diagonals OR one diagonal is an angle bisector.

• Coordinate Proofs: Use the distance, midpoint, and slope formulas to verify specific geometric properties as part of a formal plan.

Geometric Constructions

• Essential constructions for the Regents include:

  • Copying a line segment.

  • Creating a perpendicular bisector.

  • Creating a perpendicular line through a point (on or off the line).

  • Constructing an equilateral triangle.

  • Inscribing a square or a regular hexagon in a circle.

  • Constructing parallel lines.

  • Drawing a median (to the midpoint).

  • Bisecting an angle.

  • Finding the Centroid (intersection of 3 medians).