Reviewer Geometry

7/21/2019 Reviewer Geometry

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CSI Review for UP College Admission Test

1

Reviewer for Geometry

I. Undefined Terms in Geometry: Point, Line and Plane

*Why undefined?- Their meanings are accepted without definition;

- They are not defined by words but can be defined in a way that their meaning can be accepted at all

A. Point simply represented by a dot and usually named using a capital letter (e.g.P for point P) dimensionless; has no length, width or thickness indicates place or position

B. Line a set points that can extend indefinitely in both directions can be determined by at least two points can either be straight or curved lines

denoted by AB or has one dimension

Subsets of a line1. Line segment

set of points consisting of two points on the line called endpoints and thepoints on the line between these endpoints

denoted by AB

2. Half-line Set of points on one side of the point of division

3. Ray set of points consisting of one point on a line (endpoint) and all points on

one side of the endpoint

denoted by ; the first letter is always the endpoint of the ray

C. Plane Set of points that form a flat surface extending indefinitely

in all directions Can be named either by using a small letter (e.g. plane p),

the letters representing the vertices (or corners) of a plane(e.g. plane ABCD) or by using three letters representingthree points in the plane that are not collinear (e.g planeJKL)

Collinear points - points that do not lie on the same straight line

Congruent segments - lines having the same measure

Midpoint of a segment - a point on the segment that divides itinto 2 equal parts

Bisector of a line segment - a line (or a part of a line) thatintersects a line segment at its midpoint

A

B

k

A

B

A

B

A B

p

C D

J

L

K

D E F

m


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Reviewer for Geometry 2

II. Rays and Angles

Opposite rays - rays of the same line having their endpointas their only common point

Angle - a set of points formed by uniting two rays in onecommon endpoint

A. Parts of Angles1. Sides2. Vertex

In naming angles, the symbol is used.

B. Ways of naming angles1. By a capital letter that names its vertex: _________

2. By a lowercase letter/number placed inside the angle: ________

3. By three capital letters (the middle letter is the vertex; the first and third letters are from twopoints on different rays/sides: ________

Degree () - a common unit used in measuring angles

Protractor - simple device used in measuring angles

C. Types of Angles

1.

Acute angle - greater than 0 but less than90 2.

Right angle - measures 90

3. Obtuse angle - greater than 90 but lessthan less than 180

4. Straight angle - measures 180

Notation for measure of angles: mfollowed by the name of the angle

A

BC

E F

D

G

H I

J K L

D E F

ray ray

G

HJ

y


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Congruent angles -angles with equal measures

Angle bisector - a ray whose endpoint is that of the angle, anddivides the angle into two congruent angles

Perpendicular lines - two lines that intersect to form right angleSymbol used to denote perpendicular lines:

Adjacent angles - two angles in the same plane sharing thesame vertex and one common side but do not have any interiorpoint in common

Complementary angles - two angles whose sum is 90

Supplementary angles - two angles whose sum is 180

Linear angle pair - pair of adjacent angles that aresupplementary of one another; their sum forms astraight angle

Vertical angles - two angles in which the sides of one angle are opposite rays to the side of the otherangle

VZW and XZY are vertical angles. VZX and WZY are vertical angles.

III. Parallel Lines

Coplanar points/linespoints/lines that lie on the same plane

Parallel Lines coplanar lines having no points in common (or all points incommon)Symbol used to denote parallel lines:

Transversala line that intersects two other coplanar lines in twodifferent points

Interior anglesangles found between two lines intersected by thetransversal

Exterior anglesangles not found between two lines intersected bythe transversal

Alternate interior anglesinterior angles that are on opposite sides of the transversal and do not have a

common vertex

V

W

YX

Z

Lq

km

n12

4356

87

P

Q

R

S

M N P

QR


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Alternate exterior anglesexterior angles that are on opposite sides of the transversal and do not havea common vertex

Corresponding anglesone exterior and one interior angle that are on the same side of the transversaland do not have a common vertex

Properties of Parallel Lines Through a given point not on the given line, there exists one and only one line parallel to the

given line and passes through the given point. If, in a plane, a line intersects one of two parallel lines, then it intersects the other line. Transversal forms congruent alternate interior angles. Transversal forms congruent corresponding angles. Transversal forms supplementary interior angles lie on the same side of the transversal If a transversal is perpendicular to one of the two parallel lines, then it is also perpendicular to

the other. A third line in the same plane parallel to one of the two parallel lines is parallel to the other.

IV.

Triangles

Polygon - a closed figure formed by union of line segments such that segments intersect only at theirendpoints and no segments sharing an endpoint are collinear

Triangle - a polygon that exactly has three (3) sides and three (3) corners

A. Classification:

Parts of an Isosceles Triangle

Triangles

Sides Angles

Scalene Isosceles

Equilateral

Acute Obtuse

Equiangular

Right

Isoscelesright

Can be classified according to


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B. Centers of a TriangleAltitude a line segment drawn from one vertexperpendicular to its opposite side. CF is the altitude to AB EB isthe altitude to AC DA isthe altitude to CB

Orthocenter point where altitudes of a triangleintersect Point G is the orthocenter

Median a line segment that connects one vertex and

the midpoint of its opposite side In LJK, the medians are , and .

Centroidpoint where three medians intersect; divideseach median in the ratio of 2:1

Point O is the centroid of LJK.

Perpendicular bisector of the sides of a trianglea line(or line segment) perpendicular to one side of thetriangle at its midpoint

Circumcenter point where the three perpendicularbisectors of the sides of a triangle intersect; the centerof the circle that can be circumscribed about thetriangle

Point V is the centroid of STU.

Angle bisector a line segment thatbisects one angle and ends at any pointon the opposite side GM is the angle bisector of JGH. JQ is the angle bisector of GJH. HL is the angle bisector of JHG.

IncenterPoint where angle bisectorsof a triangle intersect; the center of the

circle that can be inscribed in the triangle Point R is the incenter ofJGH.

C. Triangle Congruence Theorems

Congruent triangles

A B

C

E

D

F

G

G H

J

L

M

Q

R

J K

L

M N

P

O

S U

T

V


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- Triangles having the size and shape- Triangles wherein every side of one triangle is congruent to the other triangle; every angle of onetriangle is congruent to the other

Symbol used forcongruenttriangles (or polygons):

Included side - side between two angles of a triangle

Included angle - angle between two sides of a triangle

Congruence Theorems:1. Side-Angle-Side (SAS) Congruence - two sides and included angle of one triangle are congruent

to those of the other triangle2. Angle-Side-Angle (ASA) Congruence - two angle and included side of one triangle are congruent

to those of the other triangle3. Side-Side-Side (SSS) Congruence - three sides of one triangle are congruent to those of the other4. Angle-Angle-Side (AAS) Congruence - two angles and one side opposite to either angle of one

triangle are congruent to those of the other

ExercisesIdentify the theorem that proves the congruence of the following pairs of triangles. If the givensides/angles are not sufficient to prove their congruence, write X.

1.__________

2.

KLM and NOPwhere mK = mN, mL = mO and mM = mP

__________

3.__________

4.__________

S T

V U

B

CD

E

F

R

S

T

U


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5.(FIG and FHG)

__________6.

__________

D. Triangle Similarity Theorems

Two triangles are similar if:1. All pairs of corresponding angles are congruent; and2. The ratios of the lengths of all pairs of corresponding sides are equal

Symbol used for similar triangles (or polygons): tilde (~)

Similarity Theorems:1. AA Similarity Theorem - two angles of one triangle are congruent to two angles of the other

triangle2. SSS Similarity Theorem - ratios of corresponding sides of two triangles are equal

3.

SAS Similarity Theorem - Ratios of two pairs of corresponding sides are equal and thecorresponding included angles are congruent

ExercisesGive the theorem that proves the similarity of the given triangle pairs. If none, write X.

1.__________

2.__________

F G

I H

J

Q

R

ST

A

C H

F K

U

S T

P

L M

12 20

28

3 5

7


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3. UPD and QTHGiven: mU = mQ; UP = 100; UD = 80;QH = 20; QT = 16

__________

4.__________

E. Triangle Inequality Theorem

-The length of one side of a triangle is always less than the sum of the lengths of the other two sides.

V. Quadrilaterals- A polygon with four sides

Consecutive/Adjacent Vertices - vertices at endpoints of the same side

Consecutive/Adjacent Sides - sides having a common endpoint

Consecutive angles - angles whose vertices are consecutive vertices

Opposite sides - sides having no common endpoint

Opposite angles - nonconsecutive/nonadjacent angles

Diagonal - a line segment whose endpoints are nonconsecutive vertices

*In a quadrilateral, sum of measures of all angles = 360

A. Classification

U

P

L B

O2

16


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1. Parallelogram- Two pairs of parallel sides- Opposite sides are parallel.- Opposite angles are congruent.- Two consecutive angles are supplementary.- Diagonals bisect each other.- Each diagonal divides it into two congruent triangles.

2.

Rectangle- A parallelogram whose angles are right- Diagonals are congruent

3. Rhombus- A parallelogram with all sides congruent- Diagonals are perpendicular- Each diagonal bisects opposite angles

4. Square- A rectangle with all sides congruent

- Possesses properties of rhombus

5. Trapezoid- Two and only two sides are parallelParts of a trapezoid:Bases - parallel sidesLegs - nonparallel sides

*Isosceles trapezoid- A trapezoid whose non-parallel sides are congruent- Diagonals are congruent

Median of a trapezoid- Line segment whose endpoints are midpoints of the nonparallel sides

TrapeziumTrapezoid

IsoscelesTrapezoid

Parallelogram

Rectangle

Square

Rhombus

Kite

Quadrilaterals


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- Parallel to the bases- Length is one-half the sum of lengths of the parallel sides

6. Kite- A special quadrilateral that consists of two pairs of consecutive sides that are parallel to eachother

- With two perpendicular diagonals wherein one bisects the other diagonal while the latter doesnot bisect the former

7. Trapeziuma quadrilateral with no parallel sides

Cyclic quadrilateral- Quadrilateral that can be inscribed in a circle (circumcircle)- Only quadrilaterals whose perpendicular bisectors of all four sides areconcurrent (those four bisectors intersect at only one pointcircumcenter)can be inscribed in a circle.- For a cyclic quadrilateral, opposite angles are supplementary.

ExercisesIdentify whether each of the following statements about quadrilaterals is TRUE or FALSE.1. A square is a rhombus.2. In a rhombus, all angles are right angles.3. A parallelogram is a square.4. In an isosceles trapezoid, the nonparallel sides are congruent.5. The base angles of a trapezoid are always congruent.6. In a trapezoid, there are always two pairs of supplementary angles.7. Any isosceles trapezoid can be inscribed in a circle.8. The diagonals of a kite bisects each other.

9.

The sum of measures of opposite angles in a cyclic quadrilateral is 360.10.Opposite sides of any parallelogram are congruent.

B. Perimeter (P)-sum of length of all sides of the quadrilateral

For square/rhombus: P = 4sWhere s = length of one side

For rectangle: P = 2(l + w)Where l = length and w = width

C. Area (A)- Number of non-overlapping square units contained in a closed figure

Area of Quadrilaterals

Area Notes

Square A = sA = d

s = length of side;

d = length of diagonal

Rectangle

A = lw l = length

w =width

Parallelogram A = bh b = baseh = height


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Rhombus A = bhA =12 dd

d1and d2= length of two

diagonals

Trapezoid A =(b+ b)(h)2 b1and b2= length of the two

bases;

h = height

Kite

A =12 dd

d1and d2= length of the two

diagonals

Exercises1. Find the area.

A = ________A = ________

A = ________

Polygon BEAUTY, given that BY ET AU A = ________

2.

What is the area of a square whose perimeter is 24 cm?3. The width of a rectangle is 5 less than twice its length. If the area of the rectangle is 12 dm2, whatis its perimeter?

VI.Polygons- classified as:

Convex polygon - each interior angle measures less than 180Concave polygon - at least one interior angle measures greater than 180


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Convex Polygon Concave Polygon

Consecutive anglesangles whose vertices are endpoints of a common sides

Consecutive/adjacent verticesvertices of consecutive angles

Diagonalsline segments whose endpoints are two non-adjacent vertices

Internal anglesangles formed by 2 adjacent sides of the polygonSum of measures of internal angles for a polygon with n sides = 180(n-2)For equiangular polygon:

Measure of internal angle on each vertex of the polygon =

External anglesangles that form a linear pair with the internal angles; internal and external angles aresupplementary.

Sum of measures of external angles for a polygon with n sides = 360For equiangular polygon:

Measure of external angle on each vertex of the polygon =

Regular Polygonpolygon that is both equilateral and equiangular Centerpoint inside a regular polygon equidistant from each vertex Apothem line segment from the center of the polygon to the midpoint of one side of the

polygon

VII. Surface Area, Lateral Area and Volume of Solids

Dihedral angleunion of two half-planes with a common edgeMeasure of a dihedral anglemeasure of the plane angle formed by two rays each in a different half-plane of the angle and each perpendicular to the common edge at the same point on the edge.

Perpendicular Planestwo intersecting planes forming a right dihedral angleParallel Planesplanes having no point in common

A. Polyhedronthree-dimensional figure formed by the union of the surfaces enclosed by plane figures facesportions of planes enclosed by a plane figure edgesintersection of the faces verticesintersection of the edges

*Prisma polyhedron in which two of the faces (bases) are congruent polygons in parallel planes Lateral sidessurfaces between the corresponding sides of the bases


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Lateral edgescommon edges of the lateral sides Altitude line segment perpendicular to the two bases that

corresponds to the height of the prism (perpendicular distancebetween the two bases)

*Rectangular Solid (or Rectangular Parallelepiped) solid having

rectangular bases and lateral edges perpendicular to the bases

*Cubea special rectangular solid with all its sides equal

B. Pyramid solid figure with a base that is a polygon and lateral faces that aretriangles Vertexpoint where all lateral edges meet Altitude perpendicular line segment from the vertex to the base

corresponding to the height of the pyramid Slant heightlength of the altitude of each triangular lateral face

*Regular pyramida pyramid whose base is a regular polygon hence its lateral faces are congruentisosceles triangles

C. CylinderSolid figure formed by congruent parallel closed curves and the surfacethat joins them Basesclosed curves Lateral surfacesurface that joins the bases Altitude line segment perpendicular to the bases with endpoints on the

bases

*Right Circular Cylinderbases are congruent circles with segments joining the

centers of which are perpendicular to the bases

D. (Right Circular) Conea solid figure similar to a pyramid but has a circular cross-section. Vertex (or apex)the tip of the cone Basea circle instead of a polygon Altitudeline segment perpendicular to the base extending from the vertex

to the center of the base Slant heightdistance from the vertex to any point on the circular base.

E. Spheresset of points in space equidistant to a fixed point (center) Radius length of the line segment connecting the center and any

point on the sphere.

F. Lateral area, surface area and volume

Lateral areasum of the areas of the lateral faces

(Total) surface arealateral area + sum of areas of the bases

Volumespace that is occupied by a solid or figures of three dimensions.

Lateral Area, Surface Area and Volume of some Solids

Lateral Area (LA) Surface Area (SA) Volume (V) Notes


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Points outside the circle (exterior of the circle)

A. Other parts of the circle

1. Radius (r)distance between the center and any point on the circle2. Secanta line that intersects a circle at two points

3.

Tangenta line in the plane of the circle that intersects a circle in one and only one point

Point of tangencypoint on the circle where the tangent line passes through

Common tangenta line tangent to eachof the two circleso Common external tangent atangent line that does not cross the linesegments joining the centers of the twocircleso Common internal tangent atangent line that crosses the line segmentsjoining the centers of the two circles

Two circles tangent to each other they aretangent to the same line at the same point

o Externally tangent Every point of onecircle, except the point of tangency, is anexterior point of the other circle

o Internally tangent Every point of onecircle, except the point of tangency, is aninterior of the other circle

4. Chorda line segments whose endpoints are points on the circle5. Diameter (d)a chord that passes through the center of the circle; length is twice that of the

radius (d = 2r)6. Apothemline segment perpendicular to the chord extending from the center of the circle to

the midpoint of the chord7. Central anglean angle whose vertex is the center of the circle8. Arcpart/portion of a circle between two points on the circle

*Intercepted arcarc intercepted by an angle whose endpoints are on different rays of theangle and other points on the arc are in the interior of the angle

*Degree measurement is equal to the central angle that intercepts the arc

Types of arc: Minor arcdegree measure of the arc is less than 180; usually denoted by two letters Major arcdegree measure of the arc is greater than 180; denoted by three letters Semicircledegree measure = 180; denoted by three letters


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B. Inscribed Angles and Angles formed by Tangents, Chords and Secants

1.

Inscribed anglean angle whose sides are chords of the circleand whose vertex is a point on the circle.

*Measure of inscribed angle = degree measure of theintercepted arc

mY =12 (mV)=12 (mWX)

2. Angle formed by a Tangent and a Chord measure is equal to one-half the degree measure of theintercepted arc

mEDF =12 (mED)

3. Angle formed by Two Chords Intersecting within a Circlemeasure is equal to one-half thesum of the degree measures of the arcs intercepted by the angle and its vertical angle

4. Angles formed by a Tangent and a Secant, two Secants, and two tangents (or circumscribedangles)measure is equal to one-half the difference of degree measures of the interceptedarcs*For circumscribed angle:Circumscribed angle = 180 - degree measure of the intercepted minor arc

C. Measures of Chords, Tangent Segments and Secant Segments

1. Segments of two Intersecting chordsthe product of the lengths of segments of one chordis equal to the product of the measures of segments of the other chord.

Parts of a Circle

Center

Radius

Secant

Tangent

Chord

Diameter

Apothem

Central angle

Minor arc

Major arc

Semicircle


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2. Segments formed by a tangent segment and a secant segment from an external point

Tangent segment segment of a tangent line with one of the endpoints at the point oftangency

Secant segmentsegment of a line extending from an exterior point of the circle, passing

through two points on the circle and ends on the point of intersection that is far from thestarting point

o External segment of the secantpart of the secant segment that is outside the circle

The square of the length of the tangent segment is equal to the product of the lengths ofthe entire secant segment and its external segment.

3. Two secants formed from an external point The product of the lengths of one secantsegment and its external segment is equal to the product of the lengths on the other secantsegment and its external segment.

Angle Measure LengthmZVW = mYVX =

(mZW + mXY)

mZVY = mWVX =(mWX + mYZ)

(ZV)(VX) = (YV)(VW)

mP =

(mUTW -

mVW)

(PW)2= (PU)(PV)

mP =(mUT - mVW)

(PU)(PV) = (PT)(PW)


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5. If mST = 100 and mQR = 40, find mP

6. If PQ = 8, PS = 25 and PR = 10, find PT.

7. If mQT = 170 and mQR = 70, find mP.

8. If mQR = 70 and mRT = 120, find mP.

9.

If PQ = 6 and PT = 9, find RT.

10.If mRQ = 100, find mP.

D. Circles in Cartesian Coordinate Plane1. Equations of a circle

a. General equation:

+ + + + = 0Where A = B

b. Standard (or center-radius) equation:For a circle with radius rand center at (h,k)

( )+ ( )=

*Transformation from general equation to standard equation: Completing the square*Transformation from standard equation to general equation: Binomial expansion, thenrearrangement

11.Tangents and Secants in the Coordinate plane

a.

Tangents in the coordinate plane: Equation of the line tangent to the circle at point(x1,y1)


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i. Get the slope of the line passing through the center (h,k) and point point (x1,y1)ii. Get the negative reciprocal of this slope (to obtain the slope of the tangent line)iii. Find the equation of this line given the point of tangency and the slope of the

tangent line using the point-slope form: = ( )

b.

Secants in the coordinate plane: Points of intersection of a circle and a secant linei. Get the equations of the circle and the secant lineii. Solve the pair of equation algebraically.

Exercises1. Write the standard and general equations for the following circles that has the given point as center

and r as the length of the radius.a. (1,3); r = 5b. (-2,0); r = 6c. (0,-6); r = 9

d.

(4,-2); r = 10e. (-3,-4); r =2

2. Transform the equation x2+ y2+ 6x - 4y - 12 = 0 to its center-radius form.

3. Write the equation of a line tangent to a circle with equation x2+ y2= 25 at (4,-3).

4. Find the point/s of intersection ofa. The circle x2+ (y+2)2= 4 and the line x - y = 4

b. The circle x2 + y2 = 100 and x + y = 14

E. Circumference and Arc Length1. Circumference (C)distance around a circle

C = d = 2r

Where (Greek letter pi) = 3.1415; d = diameter and r = radius

2. Arc length (s)for a given measure of arc (), in degrees, of a circle with radius r:

s = 180 r

F. Area

A = r=d

4

Exercises1. What is the circumference and area of a circle with radius of 2.5 in?

2.

What is the area of a circle if its circumference is 20cm?


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C. Right Triangle Congruence TheoremHypotenuse-Leg Triangle Congruence Theorem the hypotenuse and one leg of one triangle iscongruent to those of the other triangle

Exercises

WriteYESif HL congruence theorem can be applied to prove the congruence of the following triangle pairs;otherwise, write NO.

1.

________

2.(WRE and TER)________

3.

________

D. Proportions in the Right Triangle

Projection of a point on the linethe foot of the perpendicular line drawn from the point to the line

Projection of a segment on the line when the segment is not perpendicular to the line, is thesegment whose endpoints are the projections of the endpoints of the given segment on the line

1. The altitude to the hypotenuse of a right triangledivides it into two similar triangles that are alsosimilar to the original triangle.

BDC ~CDA~BCA

2. The length of each leg of a right triangle is thegeometric mean between the length of the projectionof that leg and the length of the hypotenuse.

CA =(DA)(BA)BC =(BD)(BA)

A

C

BD


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Reviewer Geometry