Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
How To Pass The
Geometry Regents…
(a student- made, condensed survival guide version)
180 days of lessons… how are you supposed to study THAT?
Especially since its…sigh…geometry… (EW. I know.)
But you have to pass it somehow, right?
Nope, cheating is NOT THE ANSWER.
So, READ This 14 Page Study Guide (like 10 times*) and you will
almost guaranteed pass with an 95 or above!*
 Good luck!
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By Julia “Ruby Rasberry” (on Facebook)
For any suggestions or corrections, email me at:
Vinagolb@bxscience.edu
*dependent on personal intelligence, laziness, procrastination level, motivation and interest
Note: I used class work, home work, previous regents tests, regentsprep.org, and the “Ultimate Bronx Science
Geometry Review Sheet” our teachers made for some information.
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
LOGIC
Remember this, too!
Inverse- the negation of a statement. X ~X (you negate symbols, too: ^v)
Converse- the sequence is switched. (ae becomes ea)
Contrapositive- when you do both the inverse and converse: you flip the sequence and negate it.
IMPORTANT: this is logically equivalent to the original statement. (a~b) becomes (b~a)
Tautology- a statement that’s always true, no matter the truth value of its constituents.
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Points/Lines/Segments
PARALLEL LINES
-If 2 lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
-If 2 lines are cut by a transversal so alternate interior angles are congruent, then the lines are parallel.
-If 2 lines are cut by a transversal so alternate exterior angles are congruent, then the lines are parallel.
-If 2 lines are cut by a transversal so same side interior angles are supplementary, the lines are parallel.
-If 2 lines are perpendicular to the same line, then they are parallel.
-Parallel Postulate Through a point not on a line, there is 1 and only 1line parallel to the given line.
-Coplanar lines are parallel if and only if they have no points in common, (or if the lines coincide)
-If 2 parallel lines are cut by a transversal, then corresponding angles are congruent.
-If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.
-If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent.
-If 2 parallel lines are cut by a transversal, then same side interior angles are supplementary.
PERPENDICULAR LINES
-If 2 lines intersect to form congruent adjacent angles, then the lines are perpendicular.
-2 lines are perpendicular if and only if they meet to form right angles.
OTHER LINES
-Coplanar lines are parallel if and only if they have no points in common, or if the lines coincide, and
therefore have no points in common.
-Skew lines lines that do not lie on the same plane.
-Transversal a line that intersects two other lines at two different points.
SEGMENTS
-Segment addition postulate ab+bc=ac
-Addition postulate if a=c, b=d, then a+b=c+d
-Subtraction postulate if a=c, b=d, then a-b=c-d
-Multiplication postulate if a=c, b=d, then ab=cd
-Division postulate if a=c, b=d, then a/b=c/d
-Substitution postulate if a=c, and a=b, then b=c (ONLY FOR EQUALITIES)
-2 segments are congruent if and only if they have equal lengths.
-Partition Postulatea whole is equal to the sum of its parts.
--Between points: AB + BC = AC
--Angle Addition Postulate: m<ABC + m<CBD = m<ABD
PROPERTIES OF EQUALITY
-Reflexive Property A quantity is related to itself. AB=AB.
-Symmetric Property A relation might be expressed in either order. A=B, B=A
-Transitive Property If quantities are related to the same quantity, then they are related to each other;
(i.e., if A=B, B=C, then A=C…same for parallels…etc.)
-Equivalence Property a relation satisfying the reflexive, symmetric and transitive properties
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Triangles
TYPES OF TRIANGLES
Acute Triangles
-A triangle is acute if and only if it has 3 acute angles.
-Acute angles of a right triangle are complementary (4 Corollaries)
Equilateral + Equiangular Triangles
-A triangle is equilateral if and only if it has 3 congruent sides
-A triangle is equiangular if and only if it has 3 congruent angles
-Each angle of an equiangular triangle measures 60 degrees. (4 Corollaries)
-Equilateral triangles are equiangular.
Scalene Triangles
-A triangle is scalene if and only if it has 3 sides of different lengths
Obtuse Triangles
-A triangle is obtuse if and only if it has an obtuse angle
-A triangle can have at most one obtuse or one right angle (4 Corollaries)
Right Triangles
-A triangle is right if and only if it has a right angle.
-An angle is right if and only if it is 90 degrees.
-Hypotenuse Leg Theorem 2 right triangles are congruent if the hypotenuse and a leg of 1 triangle are
congruent to the corresponding parts of the other (SSA postulate for right triangles)
-If 2 angles are right, then they are congruent.
Isosceles Triangles
-A triangle is isosceles if and only if it has 2 congruent sides.
-Isosceles Triangle Theorem If 2 sides of a triangle are congruent, then the angles opposite to these
sides are congruent.
-Converse Isosceles Triangle Theorem If 2 angles are congruent, then the sides opposite to these
angles are congruent.
TRIANGLE ANGLES
-
Triangle Exterior Angle Theorem exterior angle measure = the sum of remote interior angles.
Triangle Exterior Angle Inequality Theorem exterior angle > both remote interior angles.
If 2 angles of a triangle are congruent to 2 angles of another triangle, their third sides are congruent.
-In a triangle, the largest side is opposite to the largest angle.
-In a triangle, the sum of any 2 sides must be greater than the third.
Point of Concurrency
Circumcenter
Incenter
Centroid
Orthocenter
Intersection of the…
Perpendicular
bisectors
Angle bisectors
Medians
Altitudes
Special properties
Center of the circumscribed circle
Inscribed circle
Divides median into a 2:1 radio
Right-vertex of right angle Obtuse-outside Acute-inside
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
HOW TO PROVE TRIANGLES CONGRUENT
-SSS Postulate
-SAS Postulate
-ASA Postulate
-AAS Theorem
-THESE DO NOT EXIST: SSA (we all know why), AAA
- Hypotenuse Leg Theorem 2 right triangles are congruent if the hypotenuse and a leg of 1 triangle are
congruent to the corresponding parts of the other (like the SSA postulate for right triangles)
***corresponding parts of congruent triangles are congruent***
HOW TO PROVE TRIANGLES SIMILAR
-AA postulate
-SSS theorem
-SAS similarity theorem
***corresponding angles of similar triangles are congruent***
***the lengths of corresponding sides are in proportion***
ANGLES/BISECTORS
ANGLES
-Angle Addition Postulate If point s lies in the interior of <PQR, then <PQS+<SQR=<PQR
-Angle Subtraction Postulate If point s lies in the interior of <PQR, then <PQR-<PQS=<SQR
-Complements and supplements of congruent angles are congruent.
-2 angles are congruent if and only if they are equal in measure.
-If 2 angles are right, then they are congruent.
-If 2 angles form a linear pair, then they are supplementary.
-If 2 angles are supplements of the same angle, then they are congruent.
-If 2 angles are vertical, then they are congruent.
-An angle is straight if and only if it is 180 degrees.
-An angle is right if and only if it is 90 degrees.
-Alternate angles are on opposite sides of a transversal.
BISECTORS
-Angle Bisector TheoremIf a point is on the angle bisector of an angle, then it is equidistant from the
two sides of the angle.
-Converse Angle Bisector TheoremIf a point is equidistant from 2 sides of the angle, then it is on the
angle bisector.
-Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is
equidistant from the endpoints of the segment.
- Converse Perpendicular Bisector TheoremIf a point is equidistant from the endpoints of a segment,
then it is on the perpendicular bisector of a segment.
-Angle bisector a bisector of an angle is a ray whose endpoint is the vertex of the angle, and that
divides the angle into 2 congruent angles.
-Segment bisector the bisector of a segment is a line or subset of a line that intersects the segment at
its midpoint.
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Quadrilaterals/Parallelograms
* If a quadrilateral is a parallelogram, the opposite
sides are parallel.
About Sides
* If a quadrilateral is a parallelogram, the opposite
Parallelograms
sides are congruent.
* If a quadrilateral is a parallelogram, the opposite
angles are congruent.
About Angles
* If a quadrilateral is a parallelogram, the
consecutive angles are supplementary.
* If a quadrilateral is a parallelogram, the diagonals
bisect each other.
About Diagonals
* If a quadrilateral is a parallelogram, the diagonals
form two congruent triangles.
* If both pairs of opposite sides of a quadrilateral
are parallel, the quadrilateral is a parallelogram.
Parallelogram Converses
About Sides * If both pairs of opposite sides of a quadrilateral
are congruent, the quadrilateral is a parallelogram.
* If both pairs of opposite angles of a quadrilateral
are congruent, the quadrilateral is a parallelogram.
About Angles
* If the consecutive angles of a quadrilateral are
supplementary, the quadrilateral is a parallelogram.
* If the diagonals of a quadrilateral bisect each
About Diagonals
other, the quadrilateral is a parallelogram.
* If the diagonals of a quadrilateral form two congruent
triangles, the quadrilateral is parallelogram.
If one pair of sides of a quadrilateral is BOTH parallel and congruent, the
Parallelogram
quadrilateral is a parallelogram.
If a parallelogram has one right angle it is a rectangle
A parallelogram is a rectangle if and only if its diagonals are congruent.
Rectangle
A rectangle is a parallelogram with four right angles.
A rhombus is a parallelogram with four congruent sides.
If a parallelogram has two consecutive sides congruent, it is a rhombus.
Rhombus
A parallelogram is a rhombus if and only if each diagonal bisects a pair of
opposite angles.
A parallelogram is a rhombus if and only if the diagonals are perpendicular.
A square is a parallelogram with four congruent sides and four right angles.
Square
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Trapezoid
An isosceles trapezoid is a trapezoid with congruent legs.
A trapezoid is isosceles if and only if the base angles are congruent
Isosceles Trapezoid
A trapezoid is isosceles if and only if the diagonals are congruent
If a trapezoid is isosceles, the opposite angles are supplementary.
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Circles
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Finding out LENGTHS of a circle
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Finding out ANGLES/ARCS of a circle
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
3Dimensional Solids
FORMULA SHEET FOR GEOMETRIC SOLIDS
BA=Base area
P=Perimeter
S=Slant height
Name
Prism
Definition
Polyhedron
Volume
BA(H)
Pyramid
Polyhedron
1/3 (BH)
Lateral Area
Sum of all
lateral faces
Surface Area
(For a right prism,
it is (Pbasex
Heightprism)
Sum of all bases/faces
1/2 (PS)
BA + (LA)
(perimeter of base
x slant height) /2
(base area + lateral area)
Cylinder
Non-Polyhedron
πR2(h)
2 πR(h)
2 πR(h) + 2(πR2)
Cone
Sphere
Non-Polyhedron
Non-Polyhedron
1/3 πR2 (h)
πSR
πR2+LA
4/3 ( πR3)
n/a 
4 πR2
 Euler’s Law: (for any polyhedron)
(Faces+vertices)-edges=2
 Regular polyhedra (platonic solids)
A regular polyhedron is a polyhedron whose faces are congruent, regular
polygons, and that has the same number of faces intersecting at each vertex.
 Trig ratios
SOH-CAH-TOA
 Area of regular polygons
ASN (apothem x side length x number of sides)
2
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Coordinate Geometry
FORMULAS FOR LINE SEGMENTS
Midpoint=
M=
_____________________________________________________________________________________
Distance=
_____________________________________________________________________________________
Slope=
SLOPES
Standard (general) form
Ax+By=C
Slope intercept form
Y=mx+b
Point-Slope form
y-y1=m(x-x1)
Locus and Loci
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
Transformations
o
o
o
o
Transformation: A transformation occurs when a figure is altered from its original position or size.
Reflection: (Flip) A reflection is a mirror image of an object.
Translation: (Slide) A translation occurs when a figure is moved up, down, right, left, or a combination of
these directions.
Rotation: (Turn) A rotation occurs when a figure is turned in a circular motion.
ISOMETRIES
o
o
o
o
o
o
o
o
The product of two isometries is an isometry: For all transformations F and G, if F and G are isometries, then
GF is an isometry. (Product means composition of functions: (GF)(X) = G(F(X)).)
The inverse of an isometry is an isometry: For all transformations F, if F is an isometry and G is its inverse, then
G is an isometry. (G is the inverse of F if GF is the identity, i.e. G(F(X)) = X for all X.)
The product of isometries is associative: For all isometries F, G, H, (HG)F = H(GF).
The product of isometries is not commutative: There exist isometries F and G such that GF is not equal to FG.
(For example, suppose F1 is a reflection with mirror m1 and F2 is a reflection with mirror m2, and suppose that
m1 and m2are not parallel. Let O be the intersection point of m1 and m2, and let a be the measure of the angle
from m1 to m2.
A transformation is just some change to the plane—it can even be zero change!
You should know the different classes of transformations:
o Line reflections
o Point reflections
o Rotations (remember positive is clockwise and negative is counterclockwise!)
o Translations
o Glide Reflections
**********Compositions: Remember that these are to be followed right to left! No exceptions!**********
Symmetry: An object is symmetrical if it has its own image after a transformation. As such:
Line Symmetry
Point Symmetry (180o Rotational)
Rotational Symmetry
(The triangle has 120o symmetry, since 1/3 of a turn will yield the identical image, and the pentagon has 72 o
symmetry, since 1/5 of a turn yields the same image as the original.)
Geometry Regents Study Guide
Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)
vinagolb@bxscience.edu
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