Geometry (Common Core) Facts for the Regents Exam

Flashcards of key vocabulary and theorems from the Geometry (Common Core) Facts You Must Know Cold for the Regents Exam.

Sum of Interior Angles (Polygon)

180(n-2)

Each Interior Angle of a Regular Polygon

180(n-2) / n

Sum of Exterior Angles (Polygon)

360°

Each Exterior Angle of a Regular Polygon

360 / n

Scalene Triangle

No congruent sides

Isosceles Triangle

2 congruent sides

Equilateral Triangle

3 congruent sides

Acute Triangle

All angles are < 90°

Right Triangle

One right angle that is 90°

Obtuse Triangle

One angle that is > 90°

Equiangular Triangle

3 congruent angles (60°)

Exterior Angle Theorem

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

Midsegment (Triangle)

A segment that joins two midpoints; parallel to the third side and half its length.

Slope-Intercept Form

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

Point-Slope Form

y - y1 = m(x - x1) where m is the slope, and (x1, y1) are the values of a given point on the line.

Slope Formula

m = (y2 - y1) / (x2 - x1)

Parallel Lines Slopes

Have the same slope

Perpendicular Lines Slopes

Have negative reciprocal slopes (flip the fraction & change the sign)

Collinear Points

Points that lie on the same line

Midpoint Formula

((x1 + x2)/2, (y1 + y2)/2)

Distance Formula

√((x2 − x1)² + (y2 − y1)²)

Segment Ratios to Partition Line Segments

(x > x1 / x2 > x) = (part / whole); (y > y1 / y2 > y) = (part / whole)

Isosceles Triangle Theorem

2 ≅ sides and 2 ≅ base angles; The altitude drawn from the vertex is also the median and angle bisector

Parallel Lines and Angles

Alternate interior angles are congruent; Alternate exterior angles are congruent; Corresponding angles are congruent; Same-side interior angles are supplementary

Side-Splitter Theorem

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

Triangle Inequality Theorems

The sum of 2 sides must be greater than the third side; The difference of 2 sides must be less than the third side; The longest side of the triangle is opposite the largest angle; The shortest side of the triangle is opposite the smallest angle

SOHCAHTOA

Recall from Algebra 1: sin = opposite / hypotenuse and cos = adjacent / hypotenuse

Cofunctions

Sine and Cosine are cofunctions, which are complementary sine = cos(90° − θ) cose = sin(90° − θ)

Triangle Congruence Theorems

Side-Side-Side (SSS); Side-Angle-Side (SAS); Angle-Side-Angle (ASA); Angle-Angle-Side (AAS); Hypotenuse-Leg (HL)

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Similar Triangle Theorems

Angle-Angle (AA); Side-Angle-Side (SAS); Side-Side-Side (SSS)

CSSTP

Corresponding Sides of Similar Triangles are in Proportion

Pythagorean Theorem

a² + b² = c² where a and b are the legs, and c is the hypotenuse

Mean Proportional Altitude Theorem

altitude /leg1 = leg2/ segment1

Leg Theorem

whole hypotenuse/ leg 1 = leg 1/ adjacent hypotenuse segment 1

Reflection over x-axis

(x, y) = (x, −y)

Reflection over y-axis

(x, y) = (−x, y)

Reflection over y=x

(x, y) = (y, x)

Reflection over y=-x

(x, y) = (−y, −x)

Reflection over the origin

(x, y) = (−x, −y)

Rotation 90 degrees

(x, y) = (−y, x)

Rotation 180 degrees

(x, y) = (−x, −y)

Rotation 270 degrees

(x, y) = (y, −x)

Translation

(x, y) = (x + a, y + b)

Dilation

(x, y) = (k ⋅ x, k ⋅ y)

Rigid Motion

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

Rotational Symmetry Theorem

A regular polygon with & sides always has rotational symmetry, with rotations in increments equal to its central angle of 360°/n.

General/Standard Equation of a Circle

x² + y² + Ax + By + C = 0 where A, B, and C are constants.

Center – Radius Equation of a Circle

(x − h)² + (y − k)² = r² where (h, k) is the center and r is the radius.

Greatest Common Factor (GCF)

ax + ay = a(x + y)

Difference of Two Perfect Squares (DOTS)

x² − y² = (x + y)(x − y)

Central Angle (Circle)

∠x = arc

Inscribed Angle (Circle)

∠x = 1/2 arc

Tangent-Chord Angle (Circle)

∠x = 1/2 arc

Two Chord Angles (Circle)

∠x = (arc1 + arc2) / 2

Area of a Sector

A = (n / 360) πr² where A is the area of the sector, n is the amount of degrees in the central angle, and r is the radius

Sector Length

= r ⋅ θ where s is the sector length, r is the radius, and θ is an angle in radians.

Cavalieri’s Principle

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

Prism Volume

(Area of Base) ⋅ (Height)

Cylinder Volume

(Area of Base) ⋅ (Height)

Density

Density = (Mass) / (Volume)

Trapezoid

At least one pair of parallel sides

Isosceles Trapezoid

Each pair of base angles are congruent, diagonals are congruent, one pair of congruent sides (legs)

Parallelogram

Opposite sides are parallel, opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other

Rectangle

All angles at its vertices are right angles, diagonals are congruent

Rhombus

All sides are congruent, diagonals are perpendicular, diagonals bisect opposite angles, diagonals form four congruent right triangles, diagonals form two pairs of two congruent isosceles triangles

Square

Diagonals form four congruent isosceles right triangles

Centroid

The intersection of 3 medians

Median

A median is drawn to its midpoint