Download Geometry 3rd GP Notes 032212 Pointers 1st and 2nd Term

Geometry
3rd GP Notes
032212
Pointers
1st and 2nd Term
 Undefined and basic terms
 Angles and angle pairs
 Parallel lines and theorems
3rd Term
 Areas of polygonal regions
 Circles
 Similarities
 Solids
I. Undefined Terms
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 Half Line
 set of points which is the union of all the points of
a line on one side of a given point excluding the
given point
Plane
flat surface
usually represented by four sided figures
has infinite length and width but no thickness
named by 3 non collinear points or a capital letter
on the corner
Point
Line
Line Segment
Ray
Opposite Rays
Half Lines
no formal definition
can only be represented or described
can also be dealt with through their properties
which are taken up as postulates
Point
represented by a dot
named by a capital letter
 points contained on the same line
Non collinear Points
 points not contained on the same line
Coplanar Points
 points contained on the same plane
Non coplanar Points
 points not contained on the same plane
Line
set of points extending infinitely in both directions
has length but no width nor thickness
arrowhead
named using 2 points on each end or a small letter
in the middle
Line Segment
 set of points in between two points
 named by its endpoints
 definite
Ray
 part of a line with one endpoint and extends
infinitely in the other direction
 named by two points, name starting at end points
 Opposite Rays
 two rays with a common endpoint contained on
the same line but going in opposite directions
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Plane
A
II. Basic Terms
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Postulates
assumed to be true
also known as an Axiom
accepted without proof
Theorems
further need to be proved
Space
set of all points
Collinear
points are located on the same line
Non Collinear
points are not located on the same line
Coplanar
points are located on the same plane
Non Coplanar
points are not located on the same plane
Congruent Segments
segments with the same length
Midpoint of a Segment
divides the segment into two congruent parts
also the bisector of a segment
Betweenness
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not all betweenness are midpoints but all
midpoints are betweenness
 Bisector
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midpoint of a segment and may be a line, half line,
ray, or segment passing through the midpoint
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all midpoints are bisectors but not all bisectors
are midpoints
III. Angles and Angle Pairs
Angle
 union of two non collinear rays with a common
endpoint
 Angle Bisector- a ray bisects an angle if it divides the
angle into two congruent angles
 Congruent Angles- angles whose measurements are
equal
 Perpendicular lines- forms 4 90 degree angles
 Perpendicular bisector- cuts into 2 congruent 90
degree parts
 Angle bisector postulate- one angle bisector in every
angle
 Angle addition postulate- add two angles to make a
bigger angle
Types of Angles
 Acute Angle : measures less than 90
 Right Angle: measure is equal to 90
 Obtuse: measure is more than 90 but less than 180
Angle Pairs
 Complementary Angles
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two angles whose measures add up to 90
 Supplementary Angles
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two angles whose measures add up to 180
 Adjacent Angles
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two angles with a common side but do not have
common interior points
 Vertical Angles
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two nonadjacent angles formed by two
intersecting lines
 Linear Pair
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two adjacent angles whose non common sides also
form a pair of opposite rays
IV. Parallel Lines and Planes (not sure if we need the
other theorems but I added everything)
Parallel Lines
 lines which are coplanar and do not intersect
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Skew Lines
 lines which are non coplanar and do not intersect.
Parallel Lines Theorem
 “Two parallel lines lie in exactly one plane.”
Transversal
 a line which intersects two or more coplanar lines at
two or more distinct points.
Same-Side Interior Angles
 a pair of nonadjacent interior angles on the same
side of the transversal
Alternate Interior Angles
 a pair of nonadjacent interior angles on opposite
sides of the transversal
Alternate Exterior Angles
 a pair of nonadjacent exterior angles on opposite
sides of the transversal
Corresponding Angles
 a pair of nonadjacent angles, one interior, one
exterior, on the same side of the transversal
PCA Postulate
 “If two parallel lines are cut by a transversal, then
the corresponding angles are congruent.”
PAI Theorem
 “If two parallel lines are cut by a transversal, then
the alternate-interior angles are congruent.”
PAE Theorem
 “If two parallel lines are cut by a transversal, then
the alternate-exterior angles are congruent.”
PIS Theorem
 “If two parallel lines are cut by a transversal, then
the same-side interior angles are supplementary.”
CAP Postulate
 “If two lines are cut by a transversal and a pair of
corresponding angles are congruent, then the lines
are parallel.”
AIP Theorem
 “If two lines are cut by a transversal and a pair of
alternate-interior angles are congruent, then the
lines are parallel.”
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AEP Theorem
 “If two lines are cut by a transversal and a pair of
alternate-exterior angles are congruent, then the
lines are parallel.”
ISP Theorem
 “If two lines are cut by a transversal and a pair of
same-side interior angles are supplementary, then
the lines are parallel.”
Parallel Postulate
 “Through an external point, there is exactly one line
parallel to the given line.”
Transitivity of Parallelism Theorem (TPT) or Transitivity
of Relation Parallel (TRP)
 “In a plane, if two lines are parallel to the same line,
they are parallel to each other.”
Theorem 9.9
 “In a plane, two lines perpendicular to the same line
are parallel.”
 it is convex if each line that contains a side of the
polygon contains not a single point in the interior of
the polygon
 it is regular if it is convex, equilateral and
equiangular
 it is inscribed in a circle if the vertices of the polygon
lie on the circle
 it is circumscribed about a circle if each side of the
polygon intersects the circle at only one point
Quadrilateral
 four sided polygon
Parallelogram
 quadrilateral in which both pairs of opposite sides
are parallel
Trapezoid
 quadrilateral with exactly one pair of opposite sides
parallel
Kite
 quadrilateral in which one diagonal is
perpendicular bisector of the other diagonal
Theorem 9.10
 “In a plane, if a line is perpendicular to one of two
parallel lines, then it is perpendicular to the other.”
Rectangle
 parallelogram with four right angles
Parallel Planes
 planes that do not intersect
Rhombus
 parallelogram with four congruent sides
Theorem 9.11
 “If two planes are perpendicular to the same line,
then the planes are parallel.”
Square
 rectangle with four congruent sides
Theorem 9.12
 “If two parallel lines are cut by a third plane, then
the lines of intersection are parallel.”
V. Polygons and Polygonal Regions (not sure if we still
need the definitions but I added everything)
Polygon
 union of three or more coplanar segments such that
each segment intersects exactly two other segments
(one at each endpoint) and no two intersecting
segments are collinear.
 parts: sides, vertices, angle; consecutive sides,
opposite sides, diagonals
 three sets of points: polygon, interior, exterior
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the
Properties of a Parallelogram
 opposites are congruent
 consecutive angles are supplementary
 each diagonal separates the parallelogram into 2
congruent triangles
 diagonals bisect each other
Midsegment
 segment joining midpoints of each side of a triangle
 the length of the midsegment is always half of the
base
Properties of a special parallelogram
 diagonals of a rectangle are congruent
 if a parallelogram has one right angle then it is a
rectangle
 diagonals of a rhombus bisect opposite angles
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 Alternation
1/3=2/6
 Inversion
6/3=2/1
 Addition
1+2/2=3+6/6
 Subtraction
1-2/2=3-6/6
 Summation
1/2=3/6=1+2/3+6
Properties of an Isosceles Trapezoid
 base angles are congruent
 diagonals are congruent
 trapezoid median= b1+b2/2
Sum of Interior and Exterior angles of Polygons
 Sn (Sum of Interior angles)= (n-2)180
 Se (Measurement of each Exterior angle)= 360/n
 Diagonal= n(n-3)/2
Area of Polygons
 Area of a Square: A = s2
 Area of a Rectangle: A = lw
 Area of a Parallelogram: A = bh
 Area of a Triangle: A = ½bh
Area of a Right Triangle: A = ½l1l2
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 Area of a Equilateral Triangle: A = (s2√3)/4
 Area of a Rhombus: A = ½d1d2
 Area of a Trapezoid: A = ½h(b1 + b2)
 Area of a Regular Polygon: A = ½ap
VIII. Similar Polygons and Triangles
Definition
 Similar: have congruent angles and proportional
sides
 Congruent: have congruent angles and sides, exactly
the same size and shape
Postulates to determine Similarity
 AA
 SAS
 SSS
VI. Special Right Triangles
45
60
IX. Basic Proportionality
x sqrt 2
2x
x
x
30
x sqrt 3
30 – 60 – 90 Triangle
Definition
2
4
45
Definition
 Ratio
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1 : 2 or 1 / 2
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Relationship of 2 numbers
 Proportion
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1/2=1/4
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Equality of 2 numbers
Properties
1/ 2 = 3/ 6
 Means and Extremes
1x6=2x3
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x
Theorems
A
45 – 45 – 90 Triangle
B
Pythagorean Theorem- a2 + b2 = c2
VII. Similarity and Proportion
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C
D
E
 Basic Proportionality Theorem (BPT)
If segment BE // segment CD then
AB / BC = AE / ED ; AC / AB = AD / AE
 Converse of the Basic Proportionality Theorem
(CBPT)
If AB / BC – AE / ED then segment BC // segment CD
Geometric Mean
 x, y, and z are all real numbers
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x/y=y/z
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y2 = xz
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y = sqrt xz
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Theorem 13. 7
 Altitude- line where 2 lines intersect
 Tangent circles: two coplanar circles that are
tangent to the same line at the same point
 Internally tangent circles: two coplanar circles
whose centers are on the same side of their common
tangent
 Externally tangent circles: coplanar circles whose
centers are on opposite sides of their common
tangent
Area of Similar Triangles
5
2
1
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2
2 / 5 = (2 / 5)2 = 4 / 25
X. Parts and Theorems of Circles
Refer to Chapter 15 for the figures
Basic Terms
 Circle: set of all points in a plane at a certain
distance from a center point
 Radius: segment whose endpoints are the center of
the circle and a point on the circle
 Diameter: segment containing the center and whose
endpoints are points on the circle
 Interior: set of points whose distance is less than the
radius
 Exterior: set of points whose distance is greater than
the radius
 Chord: segment whose endpoints are points on the
circle
 Secant: line that intersects a circle in exactly two
points
 Tangent: line in the plane of the circle that intersects
the circle at exactly one point
 Ray: segment that is a subset of a tangent line and
contains the point of tangency
Relations of Circles
 Congruent circles: circles with congruent radii
 Concentric circles: two or more circles with the
same center
 Common tangent: line that is tangent to each of two
coplanar circles
 Common external tangent: does not intersect the
segment joining the centers of the circle
 Common internal tangent: intersects the segment
joining the centers of the circle
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Tangent, Secants, and Chords
 Theorem 15.1: “A line perpendicular to a radius at
its outer end is tangent to the circle.”
 Theorem 15.2: “Every tangent to a circle is
perpendicular to the radius drawn to the point of
tangency.”
 Theorem 15.3: The Two-Tangent Theorem: “The
two tangent segments to a circle from an external
point are congruent and determine congruent angles
with the segment from the external point to the
center.”
 Theorem 15.4: “The line from the center of a circle
perpendicular to a chord, bisects the chord.”
 Theorem 15.5: “The line containing the center of a
circle and the midpoint of a chord that is not a
diameter, is perpendicular to the chord.”
 Theorem 15.6: “In the plane of the circle, the
perpendicular bisector of a chord passes through
the center.”
 Corollary 15.6-1: “No three points of a circle are
collinear.”
 Theorem 15.7: “In the same or in congruent circles,
chords equidistant from the center are congruent.”
 Theorem 15.8: “In the same or in congruent circles,
any two congruent chords are equidistant from the
center.”
 Theorem 15.9: “The graph of the equation (x-h)2 +
(y-k)2 = r2 is the circle with center (h, k) and radius
r.”
 Theorem 15.10: “Every circle is the graph of the
equation of the form: x2 + y2 + Ax + By + C = 0.”
 Theorem 15.11: “The graph of the equation x2 + y2 +
Ax + By + C = 0 is a circle (+), a point (0), or an
empty set (-).”
Arcs and Angles Related to a Circle
 Arc: subset of a circle with two endpoints
 Central angle: angle in the plane of the circle whose
vertex is the center of the circle
 Minor arc: set of points on the circle that lie on a
central angle or in the interior of the central angle
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 Major arc: set of points on the circle that lie on a
central angle or in the exterior of the central angle
 Semicircle: union of the endpoints of a diameter and
the points of the circle on one side of the diameter
 Degree measurement of a circle: 360
 Degree measurement of a semi circle: 180
 Degree measurement of a minor arc: measure of its
central angle
 Degree measurement of a major arc: 360 – minor arc
Power Theorems
 Postulate 33: Arc Addition Postulate: “If Q is on arc
RS then measurement of arc SQ + m arc QR = m arc
SQR.”
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Congruent arcs: arcs that have the same measure
and lie on the same or on congruent circles
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Angle inscribed: angle whose sides contain the
endpoints of an arc and whose vertex is a point on
the arc other than the endpoints
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Intercepted arc: arc whose endpoints lie on
difference sides of an angle and whose other
points lie in the interior of the angle
 Theorem 15.12: Inscribed Angle Theorem: “the
measure of an inscribed angle is one-half the
measure of its intercepted arc.”
 Corollary 15.12-1: “Two inscribed angles that
intercept the same arc or congruent arcs are
congruent.”
 Corollary 15.12-2: “An angle inscribed in a
semicircle is a right angle.”
 Corollary 15.12-3: “The opposite angles of an
inscribed quadrilateral are supplementary.”
 Theorem 15.13: “The measure of an angle formed by
two secants which intersect in the interior of a circle
is one –half the sum of the measures of the arcs
intercepted by the angle and its vertical angle.”
 Theorem 15.14: “The measure of an angle formed by
two secants that intersect in the exterior of a circle
is one-half the difference of the measures of the
intercepted arcs.”
 Theorem 15.15: “The measure of an angle formed by
a tangent and a secant that intersect at the point of
tangency is one-half the measure of the intercepted
arc.”
 Theorem 15.16: “The measure of an angle formed by
a secant and a tangent that intersect in the exterior
of a circle is one-half the difference of the measure s
of the intercepted arcs.”
 Theorem 15.17: “The measure of an angle formed by
two intersecting tangents is one-half the difference
of the measures of the intercepted arcs.
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 Theorem 15.18: The Two-Chord Power Theorem: “If
two chords intersect in a circle, then the product of
the lengths of the segments on one chord is equal to
the product of the lengths of the segments on the
other.”
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If a segment intersects a circle in two points, and
exactly one of these is an endpoint of the segment,
then the segment is called a secant segment to the
circle.
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The part of the secant segment in the exterior of
the circle is called its external secant segment.
 Theorem 15.19: The Two-Secant Power Theorem:
“If two secants intersect at a point in the exterior of
the circle, the product of the lengths of one secant
segment and its external secant segment is equal to
the product of the lengths of the other secant
segment and its external secant segment.”
 Theorem 15.20: The Tangent-Secant Power
Theorem: “If a tangent and secant intersect at a
point in the exterior of the circle, the square of the
length of the tangent segment is equal to the product
of the lengths of the secant segment and its external
secant segment.”
XI. Area and Circumference of Circles and Arcs
Area and Circumference of a Circle
 Limit: fixed number which a sequence of numbers
approaches
 Circumference: limit of the perimeters of the
inscribed regular polygons as the number of sides
increases
 Area: limit of the ares of the inscribed polygons as
the number of sides increases
 Ratio of the circumference of a circle to the diameter
is denoted by , = c / d
Inscribed and Circumscribed
 “A circle is inscribed in a polygon when the sides of
the polygon are tangent to the circle.”
 “A circle is circumscribed about a polygon when the
vertices of the polygon lie on the circle.”
Sectors and Segments of a Circle
 Sector: region bounded by two radii and the
intercepted minor or major arc
 Theorem 15.21: “In a circle of radius r, the ratio of
the length L of an arc to the circumference C of the
circle is the same as the ratio of the arc measure m
to 360.” L / C = m / 360 or L = m / 180 r
 Asector= m / 360 r2
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 Segment: Region bounded by a chord and the
intercepted minor or major arc
 Sphere: set of points in space that are at a given
distance, called the radius, from a given point, called
the center
XII. Prisms
Solids and Polyhedrons
 Solid: three-dimensional figure bounded by flat
surfaces, curved surfaces, or both flat and curved
surfaces
 Polyhedron: union of four or more noncoplanar
polygonal regions which encloses a part of space
 Enclose space: interior of the polyhedron
 Solid Polyhedron: union of a polyhedron and its
interior
Area and Volume of Prisms
 Prism: polyhedron with two congruent faces
contained in parallel planes and its other faces are
parallelogram regions
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The bases are congruent and lie on parallel
planes.
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The lateral faces are parallelogram regions.
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The lateral edges are parallel and congruent.
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Regular Prism: right prism whose bases are
regular polygonal regions
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Parallelepiped: prism whose bases are
parallelogram regions
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Rectangular parallelepiped: right rectangular
prism
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Cube: rectangular parallelepiped all of whose
edges are congruent
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Postulate 34: Volume Postulate: “To every solid
region there corresponds a unique positive real
number called the volume of the solid region.”
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Postulate 35: Congruent Solids Postulate: “If two
solids are congruent, then they have exactly the
same volume.”
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Postulate 36: Volume Addition Postulate: “If a
solid region is separated into nonoverlapping
regions, then the sum of the volumes of these
regions equals the volume of the given region.”
 Cylinder: three-dimensional figure bounded by a
curved surface with two congruent circular bases
contained in parallel planes
 Pyramid: polyhedron formed by a polygonal region
in a plane, a point not on the plane, and the
triangular regions formed by joining the vertices of
the polygonal region by the point
 Cone: three-dimensional figure bounded by a curved
surface with a vertex not on the plane and a circular
region base
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Formulas for Prisms
 Prism
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d= sqrt l2 + w2 + h2
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sum of all lateral faces
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TA = 2B + LA
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V= bh
 Cylinder
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LA = 2rh
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TA = 2r(h + r)
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V= r2h
 Pyramid
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LA= 1 / 2 (ps)
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TA= LA + B
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V= 1 / 3 (bh)
 Cone
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LA = rs
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TA = LA + B
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V = 1 / 3 bh
 Sphere
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SA = 4r2
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VA = 4 / 3 r3
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