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MATHEMATICS 8UNIT 8
TERMS IN GEOMETRYMEASURING AND CLASSIFYING ANGLES
ANGLE RELATIONSHIPSPARALLEL AND PERPINDICULAR LINES
LINES OF SYMMETRY
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TERMS IN GEOMETRYSOME IMPORTANT GEOMETRIC TERMS
POINT: A pointrepresents or identifiesan exact position in
space. A capital letteris used to name a point.
LINE: A line is astraight line pf pointsextending infinitelyin both directions.
Identified by usingany two points alongits length or a smallletter.A
BLINESEGMENT: Apartof a line having twoendpoints
RAY: A part of a line that has oneendpoint and extends infinitely in one
direction.
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ANGLESAngles are formed when two rays or line segments have
a common endpoint. This endpoint is called avertex.
A
B Cy
ABC
CBA
y
Sides, or arms of the angle
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CLASSIFYING ANGLES
ACUTELess than 90 degrees
RIGHT90 degrees
OBTUSEBetween 90 and
180 degrees
STRAIGHT180 degrees
REFLEXBetween 180 and 360 degrees
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ANGLE RELATIONSHIPS
1) OPPOSITE ANGLES HAVE THE SAMEMEASURE
2) SUPPLEMENTARY ANGLES ARE TWOANGLES WHOSE SUM MEASUREMENT = 180
DEGREES
3) COMPLEMENTARY ANGLES ARE TWOANGLES WHOSE SUM MEASUREMENT = 90
DEGREES
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a
bc
d
e
f
g
h
i
j
OPPOSITE ANGLESa and cd and bh and fe and g
SUPPLEMENTARY ANGLESa and b, a and d,d and c, c and bh and g, g and f,f and e, e and h
i and j
COMPLEMENTARYANGLES
none
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If we know the measure of one angle, we can find
the measure of the other unknown angles.A
B
C
D
E67
F
AFE = 113 SATBFA = 67 OAT or SATDFC = 90 (given)AFC = 90 SAT
BFC = 23 CAT
OAT = Opposite Angle TheoremSAT = Supplementary Angle TheoremCAT = Complimentary Angle Theorem
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A transversal is a line or line segment that crosses
two or more lines. When a transversal crossestwo lines, eight angles are formed. If the two linesare parallel, certain angle relationships are
formed.
a b
c d
e f
g h
Alternate AnglesZ PATTERN
Co-interior AnglesC PATTERN
Corresponding AnglesF PATTERN
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a b
c d
e f
g h
Alternate AnglesZ PATTERNc and fd and e
Alternate angles have equal measures
Co-interior AnglesC PATTERN
c and ed and f
Co-interior angles total 180 degrees
Corresponding AnglesF PATTERN
a and e
c and gb and fd and h
Corresponding angles have equal measures
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a b
c d
e f
g h
65
ANGLE MEASUREMENTSd = 65 (given)
e = 65 Alternate Angle to dc = 115 Co-interior Angle to ef =115 Co-interior to d, alternate to cb = 115 Corresponding to fa = 65 Corresponding to e, OAT
g = 115 Corresponding to ch = 65 Corresponding to d
USINGTHESEANGLE
PROPERTIES
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MATHEMATICS 8UNIT 9
TRIANGLES AND ANGLESPOLYGONS
USING ANGLE RELATIONSHIPS
CONGRUENT POLYGONSPYTHAGOREAN THEOREM
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TRIANGLES AND ANGLESTRIANGLES ARE CLASSIFIED BY THELENGTHS OF THEIR SIDES AND THE
MEASURES OF THEIR ANGLES
Acute: 3 acute angles Right: 1 right angle Obtuse: 1 obtuse angle
Scalene: no equal sides Isosceles: 2 equal sides Equilateral: 3 equal sides
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A
B
C
D
E
61
37
71
p q
The interior angles of a triangle have a combined
measure of 180 degrees.
r
p = 82 Triangle 180q = 82 OATr = 27 Triangle 180
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x xx
x
x
IsoscelesTriangle2 sides equal in
length2 angles equal indegrees
EquilateralTriangle
All sides equalAll angles equal
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POLYGONS
A POLYGON IS A CLOSED PLANE FIGUREMADE UP OF THREE OR MORE LINE
SEGMENTS
IN A REGULAR POLYGON, ALL SIDES HAVETHE SAME LENGTH
A DIAGONAL JOINS TWO VERTICES IN APOLYGON AND IS NOT A SIDE
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FAMOUS POLYGONS
Triangle 3Quadrilateral 4
Pentagon 5
Hexagon 6Heptagon 7Octogon 8Nonagon 9
Decagon 10Dodecagon 12
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Polygons and Interior Angles
We know that the sum of the interior angles in atriangle is 180 degrees
Lets look at a quadrilateral
A
B
CD
DB is a diagonal of ABCD. Itdivides this quadrilateral into twotriangles, each having an interiormeasure of 180 degrees.
2 x 180 = 360Therefore, the sum of the interiorangles in a quadrilateral is 360degrees.
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How about a pentagon?
A B
C
D
E
EB and EC arediagonals of ABCDE
The two diagonals create three triangles within the pentagon,each having an interior measure of 180 degrees.3 x 180 = 540 degreesTherefore, a pentagons interior angles measure 540 degrees
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Formula For Interior Angles ofa Polygon
THE SUM OF THE INTERIOR ANGLES OF A
POLYGON WITH n SIDES IS
n represents the number of sides
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USING ANGLERELATIONSHIPS
A
E
C
D
B
>
>72 51
w
x y
z
x = 72, alternate to ABCy = 51, alternate to ACBw = 57, Triangle 180z = 129, SAT
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CONGRUENT POLYGONS
CONGRUENT FIGURES HAVE THE SAME SIZEAND SHAPE
A
B
C
D
E
F
GH
1
2
3
4
5
6
7
8
ANGLES1 = 52 = 63 = 7
4 = 8
SIDES
AB = EFBC = FGCD = GHDA = HE
These two figures are said to becongruent. Their correspondingparts are both congruent and equal.
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PYTHAGOREAN THEOREMFOR RIGHT ANGLE TRIANGLES
HypotenuseLeg
Leg
The hypotenuse is the sidedirectly across from the rightangle in a right angle triangle
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ca
b
The Pythagorean Theorem:
22
2
a = 3b = 4
a + b = c3 + 4 = c
9 + 16 = c25 = c
5 = c
2 2 2
2 2 2
2
2