Geometry (Common Core) Regents Review Flashcards

Vocabulary and formula flashcards covering essential concepts for the Geometry Common Core Regents Exam, including polygons, trigonometry, transformations, circles, and 3D geometry.

Sum of Interior Angles

$$180(n - 2)$$

Each Interior Angle of a Regular Polygon

$$\frac{180(n - 2)}{n}$$

Exterior Angle Theorem

The exterior angle is equal to the sum of the two non-adjacent interior angles.

Midsegment

A segment joining two midpoints that is parallel to the third side and $\frac{1}{2}$ its length.

Slope-Intercept Form

$y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

Slope Formula

$$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$

Perpendicular Lines Slopes

Negative reciprocal slopes (flip the fraction and change the sign).

Collinear Points

Points that lie on the same line.

Midpoint Formula

M = \begin{pmatrix} \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Distance Formula

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

Same-side Interior Angles

Angles formed by parallel lines and a transversal that are supplementary ($180^\circ$).

Side-Splitter Theorem

If a line is parallel to a side of a triangle and intersects the other two sides, it divides those sides proportionally.

Triangle Inequality Theorems

The sum of 2 sides must be greater than the third; the difference of 2 sides must be less than the third.

Average Speed Formula

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

Cofunctions

Sine and Cosine are cofunctions: $\sin(A) = \cos(90^\circ - A)$. In a right triangle, if $\angle A$ and $\angle B$ are acute, $\sin(A) = \cos(B)$.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

Pythagorean Theorem

$a^2 + b^2 = c^2$ where $a$ and $b$ are legs and $c$ is the hypotenuse.

Mean Proportional Altitude Theorem (SAAS)

$\frac{part_1}{altitude} = \frac{altitude}{part_2}$ (The altitude is the geometric mean between 2 segments of the hypotenuse).

Mean Proportional Leg Theorem (HYLLS)

$$\frac{\text{hypotenuse}}{\text{leg}} = \frac{\text{leg}}{\text{segment touched}}$$

Rigid Motion

A transformation that preserves distance, congruency, angle measure, size, and shape.

Dilation

$D_k(x, y) = (k \cdot x, k \cdot y)$. It creates similar figures with proportional sides and congruent angles.

Rotational Symmetry Theorem

A regular polygon with $n$ sides maps onto itself in increments of $\frac{360^\circ}{n}$.

Difference of Two Perfect Squares (DOTS)

$$x^2 - y^2 = (x + y)(x - y)$$

Center-Radius Equation of a Circle

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

Inscribed Angle

$$\angle = \frac{1}{2} \text{arc}$$

Tangent-Chord Angle

$$\angle = \frac{1}{2} \text{arc}$$

Exterior Circle Angle (2 Secants/Tangents)

$$\angle = \frac{\text{Big Arc} - \text{Little Arc}}{2}$$

Segment Relationships (Chords)

$$(\text{Part})(\text{Part}) = (\text{Part})(\text{Part})$$

Segment Relationships (Secants)

$$(\text{Whole})(\text{External}) = (\text{Whole})(\text{External})$$

Area of a Sector (Radians)

$$A = \frac{1}{2} r^2 \theta$$

Sector Length (Radians)

$$s = r \cdot \theta$$

Cavalieri’s Principle

If two solids have the same height and the same cross-sectional area at every level, they have the same volume.

Density (Mass)

$$Mass = (Density) \cdot (Volume)$$

Trapezoid Median

$$\text{median} = \frac{1}{2}(\text{base}_1 + \text{base}_2)$$

Parallelogram

A quadrilateral with opposite sides parallel and congruent, opposite angles congruent, consecutive angles supplementary, and diagonals that bisect each other.

Rhombus

A parallelogram where all sides are congruent, diagonals are perpendicular, and diagonals bisect opposite angles.

Centroid

The intersection point of 3 medians in a triangle.