web viewnote: i used class work, home work, previous regents tests, regentsprep.org, and the...

Geometry Regents Study Guide Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)[email protected]

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!

By Julia “Ruby Rasberry” (on Facebook)For any suggestions or corrections, email me at: [email protected]

*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.

LOGIC

mailto:[email protected]



Remember this, too!

Inverse- the negation of a statement. X ~X (you negate symbols, too: ^v)Converse- the sequence is switched. (ae becomes ea)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.

Points/Lines/Segments


Geometry Regents Study Guide Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)[email protected] 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 Postulatea 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

TrianglesTYPES OF TRIANGLES


Geometry Regents Study Guide Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)[email protected] 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.Point of Concurrency Intersection of the… Special propertiesCircumcenter Perpendicular

bisectorsCenter of the circumscribed circle

Incenter Angle bisectors Inscribed circleCentroid Medians Divides median into a 2:1 radioOrthocenter Altitudes Right-vertex of right angle Obtuse-outside Acute-inside

- -In a triangle, the sum of any 2 sides must be greater than the third.HOW TO PROVE TRIANGLES CONGRUENT

-SSS Postulate-SAS Postulate-ASA Postulate-AAS Theorem


Geometry Regents Study Guide Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)[email protected] 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/BISECTORSANGLES

-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 TheoremIf a point is on the angle bisector of an angle, then it is equidistant from the two sides of the angle.-Converse Angle Bisector TheoremIf 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 TheoremIf 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.

Quadrilaterals/ParallelogramsParallelograms

About Sides* If a quadrilateral is a parallelogram, the opposite sides are parallel.



* If a quadrilateral is a parallelogram, the opposite sides are congruent.

About Angles

* If a quadrilateral is a parallelogram, the opposite angles are congruent.* If a quadrilateral is a parallelogram, the consecutive angles are supplementary.

About Diagonals

* If a quadrilateral is a parallelogram, the diagonals bisect each other.* If a quadrilateral is a parallelogram, the diagonals form two congruent triangles.

Parallelogram Converses

About Sides

* If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram.* If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

About Angles

* If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.* If the consecutive angles of a quadrilateral are supplementary, the quadrilateral is a parallelogram.

About Diagonals

* If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.* If the diagonals of a quadrilateral form two congruent triangles, the quadrilateral is parallelogram.

Parallelogram If one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.

RectangleIf a parallelogram has one right angle it is a rectangleA parallelogram is a rectangle if and only if its diagonals are congruent.A rectangle is a parallelogram with four right angles.

Rhombus

A rhombus is a parallelogram with four congruent sides.If a parallelogram has two consecutive sides congruent, it is a 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.

Square A square is a parallelogram with four congruent sides and four right angles.A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Isosceles Trapezoid

An isosceles trapezoid is a trapezoid with congruent legs.A trapezoid is isosceles if and only if the base angles are congruentA trapezoid is isosceles if and only if the diagonals are congruentIf a trapezoid is isosceles, the opposite angles are supplementary.



Circles


Geometry Regents Study Guide Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)[email protected] out LENGTHS of a circle



Finding out ANGLES/ARCS of a circle



3Dimensional SolidsFORMULA SHEET FOR GEOMETRIC SOLIDSBA=Base areaP=PerimeterS=Slant heightName Definition Volume Lateral Area Surface AreaPrism Polyhedron

BA(H)Sum of all lateral faces(For a right prism, it is (Pbasex Heightprism)

Sum of all bases/faces

Pyramid Polyhedron 1/3 (BH) 1/2 (PS)(perimeter of base x slant height) /2

BA + (LA)(base area + lateral area)

Cylinder Non-Polyhedron πR2(h) 2 πR(h) 2 πR(h) + 2(πR2)

Cone Non-Polyhedron 1/3 πR2 (h) πSR πR2+LASphere Non-Polyhedron

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 ratiosSOH-CAH-TOA

Area of regular polygonsASN ( a pothem x s ide length x n umber of sides)

2



Coordinate GeometryFORMULAS FOR LINE SEGMENTSMidpoint=

M= _____________________________________________________________________________________ Distance=

_____________________________________________________________________________________Slope=

SLOPES

Locus and Loci

Standard (general) form Ax+By=C

Slope intercept form Y=mx+b

Point-Slope form y-y1=m(x-x1)



Transformationso Transformation : A transformation occurs when a figure is altered from its original position or size.o Reflection : (Flip) A reflection is a mirror image of an object.o Translation : (Slide) A translation occurs when a figure is moved up, down, right, left, or a combination of

these directions.o Rotation : (Turn) A rotation occurs when a figure is turned in a circular motion.

ISOMETRIESo 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)).)

o 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.)

o The product of isometries is associative: For all isometries F, G, H, (HG)F = H(GF).

o 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.

o A transformation is just some change to the plane—it can even be zero change!o You should know the different classes of transformations:

o Line reflectionso Point reflectionso Rotations (remember positive is clockwise and negative is counterclockwise!)o Translationso Glide Reflections

o **********Compositions: Remember that these are to be followed right to left! No exceptions!**********o Symmetry: An object is symmetrical if it has its own image after a transformation. As such:

Line Symmetry

Point Symmetry (180o Rotational)


Geometry Regents Study Guide Julia Vinagolu-Baur (Facebook: Julia Ruby-Rasberry)[email protected] 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.)


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