Regents Examination in Geometry: Summary of Geometric Relationships and Formulas

Angles and Lines
The sum of the measures of adjacent angles around a point equals 360°.

m∠1 + m∠2 + m∠3 + m∠4 = 360°

Vertical angles are congruent.

∠1 ≅ ∠3 and ∠2 ≅ ∠4
A linear pair adds to 180°.

m∠1 + m∠2 = 180°

Parallel Lines

If line m || line n:
Alternate interior angles are congruent: ∠4 ≅ ∠6 and ∠3 ≅ ∠5.
Corresponding angles are congruent: ∠2 ≅ ∠6, ∠3 ≅ ∠7, ∠1 ≅ ∠5, and ∠4 ≅ ∠8.
Same side interior angles are supplementary: m∠3 + m∠6 = 180° and m∠4 + m∠5 = 180°.

Polygons
The sum of the interior angles of a polygon with n sides equals 180(n − 2).
The sum of the exterior angles of a polygon with n sides equals 360°.

Special Segments in Triangles



Angle and Segment Relationships in Triangles
 

Angle sum theorem

m∠A + m∠B + m∠C = 180°
 

Exterior angle theorem

m∠1 = m∠3 + m∠4
 

Isosceles triangle


 

Pythagorean theorem


Transformations
 

Transformation	Figure	Relationships
Reflection		is the ⊥ bisector of and
Rotation		∠ ACA′ ≅ ∠BCB′
Translation
Dilation

Transformations on the Coordinate Plane

Triangle Congruence Postulates
SSS
SAS
ASA
AAS
HL

CPCTC
Corresponding parts of congruent triangles are congruent.

Coordinate Geometry Formulas
Given two points (x1, y1) and (x2, y2):

The distance between the points is

The midpoint of the segment joining the points is

 

The slope of the segment joining the points is

Find the points that divide a segment proportionally using



Equations of Curves

Triangle Similarity Postulates
AA
SSS
SAS

Similarity Relationships in Triangles
Segment parallel to a side forms two similar triangles

Side splitter theorem




 

Centroid divides a median in a 1 : 2 ratio

 and 

Midsegment theorem—A segment joining the midpoints of two sides of a triangle is parallel to the opposite side and its length is
 the opposite side.

and

Altitude to the hypotenuse of a right triangle theorem—An altitude to the hypotenuse of a right triangle divides the triangle into two similar triangles, each of which is also similar to the original right triangle.

ΔBDC ~ ΔCDA ~ ΔBCA

Scaling Length, Area, and Volume
Length is proportional to the scale factor.
Area is proportional to the (scale factor)2.
Volume is proportional to the (scale factor)3.

Trigonometric Relationships






Cofunction Relationships:
sin(A) = cos(90 − A)
cos(A) = sin(90 − A)

Parallelograms
A parallelogram is a quadrilateral whose opposite sides are parallel. All parallelograms have the following properties:
Opposite sides are congruent.
Opposite angles are congruent.
Adjacent angles are supplementary.
The diagonals bisect each other.
The two diagonals each divide the parallelogram into two congruent triangles.

Special Parallelograms


Additional Properties of Special Parallelograms

Trapezoids





Isosceles Trapezoids



Circle Relationships
Central and inscribed angles

Congruent chords


Parallel chords

The perpendicular bisector of any chord passes through the center of the circle.
A diameter or radius that is perpendicular to a chord bisects the chord.
A diameter or radius that bisects a chord is perpendicular to the chord.

A tangent is perpendicular to a radius.

Tangents from the same point are congruent.

Angles formed by chords, secants, and tangents:

Vertex inside the circle	Angle measure the sum of the intercepted arcs
Vertex on the circle	Angle measure of the intercepted arc
Vertex outside the circle	Angle measure  the difference of the intercepted arcs

Segments formed by chords, secants, and tangents:

Intersecting chords	Products of the parts are equal a · b = c · d
Intersecting secants	outside · whole = outside · whole PA · PC = PB · PD
Intersecting tangent and secant	outside · whole = outside · whole PA2 = PB · PC

Radian Measure
radians =  degrees
degrees =  radians

Arc Length and Sector Area

Arc Length	Sector Area
r · θ

where θ is the measure of the central angle in radians and r is the radius.

Volume
Prism: V = Bh
Cylinder: V = πr2h
Cone and pyramids: 

Sphere: 
where B is the area of the base, h is the height, r is the radius.

Solids of revolution

Figure Rotated	Solid
Right triangle, rotate 360° about leg	Cone
Rectangle ABCD, rotate 360° around side	Cylinder
Circle P, rotate 180° about diameter	Sphere

Cavalieri’s Principle
If two solids are contained between two parallel planes and every parallel plane between these two planes intercepts regions of equal area, then the solids have equal volume. Also any two parallel planes intercept two solids of equal volume.


Modeling with Density
Mass = volume × density