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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