Math 8 And 9

MATHEMATICS 8 UNIT 8 TERMS IN GEOMETRY MEASURING AND CLASSIFYING ANGLES ANGLE RELATIONSHIPS PARALLEL AND PERPINDICULAR LINES LINES OF SYMMETRY

TERMS IN GEOMETRY SOME IMPORTANT GEOMETRIC TERMS

POINT: A point

represents or identifies an exact position in space. A capital letter is used to name a point.

A B LINE SEGMENT: Apart of a line having two endpoints

LINE: A line is a

straight line pf points extending infinitely in both directions. Identified by using any two points along its length or a small letter.

RAY: A part of a line that has one endpoint and extends infinitely in one direction.

ANGLES Angles are formed when two rays or line segments have a common endpoint. This endpoint is called a vertex.

ABC

A

CBA y B

y

C

Sides, or arms of the angle

CLASSIFYING ANGLES

ACUTE

OBTUSE

Less than 90 degrees

RIGHT

Between 90 and 180 degrees

90 degrees

STRAIGHT 180 degrees

REFLEX

Between 180 and 360 degrees

ANGLE RELATIONSHIPS 1) OPPOSITE ANGLES HAVE THE SAME MEASURE 2) SUPPLEMENTARY ANGLES ARE TWO ANGLES WHOSE SUM MEASUREMENT = 180 DEGREES 3) COMPLEMENTARY ANGLES ARE TWO ANGLES WHOSE SUM MEASUREMENT = 90 DEGREES

OPPOSITE ANGLES a and c d and b h and f e and g

d

a

c b j i COMPLEMENTARY ANGLES none

SUPPLEMENTARY ANGLES a and b, a and d, d and c, c and b h and g, g and f, f and e, e and h i and j

h

e

g f

If we know the measure of one angle, we can find the measure of the other unknown angles. A OAT = Opposite Angle Theorem SAT = Supplementary Angle Theorem CAT = Complimentary Angle Theorem

F

B

E

67

C AFE = 113 SAT BFA = 67 OAT or SAT DFC = 90 (given) AFC = 90 SAT BFC = 23 CAT

D

PARALLEL AND PERPENDICULAR LINES PARALLEL LINES ARE LINES IN THE SAME PLANE THAT NEVER INTERSECT PERPENDICULAR LINES ARE LINES THAT INTERSECT TO FORM RIGHT ANGLES

A transversal is a line or line segment that crosses two or more lines. When a transversal crosses two lines, eight angles are formed. If the two lines are parallel, certain angle relationships are formed. Alternate Angles Z PATTERN

a

b c

d e

Co-interior Angles C PATTERN

f g

h

Corresponding Angles F PATTERN

Alternate Angles Z PATTERN

Corresponding Angles

c and f d and e Alternate angles have equal measures

a

F PATTERN

a and e c and g b and f d and h Corresponding angles have equal measures

b c

d e

Co-interior Angles f

g

h

C PATTERN c and e d and f Co-interior angles total 180 degrees

USING THESE ANGLE PROPERTIES a

b c

d 65 e

f g

h

ANGLE MEASUREMENTS d = 65 (given) e = 65 Alternate Angle to d c = 115 Co-interior Angle to e f =115 Co-interior to d, alternate to c b = 115 Corresponding to f a = 65 Corresponding to e, OAT g = 115 Corresponding to c h = 65 Corresponding to d

MATHEMATICS 8 UNIT 9 TRIANGLES AND ANGLES POLYGONS USING ANGLE RELATIONSHIPS CONGRUENT POLYGONS PYTHAGOREAN THEOREM

TRIANGLES AND ANGLES TRIANGLES ARE CLASSIFIED BY THE LENGTHS OF THEIR SIDES AND THE MEASURES OF THEIR ANGLES

Acute: 3 acute angles

Right: 1 right angle

Scalene: no equal sides

Isosceles: 2 equal sides

Obtuse: 1 obtuse angle

Equilateral: 3 equal sides

The interior angles of a triangle have a combined measure of 180 degrees. A

D

61

71

p

C

q

37

B

r

p = 82 Triangle 180 q = 82 OAT r = 27 Triangle 180

E

x

x

x

Isosceles Triangle

2 sides equal in length 2 angles equal in degrees

x

x

Equilateral Triangle

All sides equal All angles equal

POLYGONS A POLYGON IS A CLOSED PLANE FIGURE MADE UP OF THREE OR MORE LINE SEGMENTS IN A REGULAR POLYGON, ALL SIDES HAVE THE SAME LENGTH A DIAGONAL JOINS TWO VERTICES IN A POLYGON AND IS NOT A SIDE

FAMOUS POLYGONS Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octogon 8 Nonagon 9 Decagon 10 Dodecagon 12

Polygons and Interior Angles We know that the sum of the interior angles in a triangle is 180 degrees

Let’s look at a quadrilateral A B D

C

DB is a diagonal of ABCD. It divides this quadrilateral into two triangles, each having an interior measure of 180 degrees. 2 x 180 = 360 Therefore, the sum of the interior angles in a quadrilateral is 360 degrees.

How about a pentagon? A

EB and EC are diagonals of ABCDE

B C

E D

The two diagonals create three triangles within the pentagon, each having an interior measure of 180 degrees. 3 x 180 = 540 degrees Therefore, a pentagon’s interior angles measure 540 degrees

Formula For Interior Angles of a Polygon THE SUM OF THE INTERIOR ANGLES OF A POLYGON WITH n SIDES IS

n represents the number of sides

USING ANGLE RELATIONSHIPS A w

D B

72

x

> >

y z

x = 72, alternate to ABC y = 51, alternate to ACB w = 57, Triangle 180 z = 129, SAT

E 51

C

CONGRUENT POLYGONS CONGRUENT FIGURES HAVE THE SAME SIZE AND SHAPE

A

E

1 2

B

6 7

3 4

D

C

5

8

F G

H

These two figures are said to be congruent. Their corresponding parts are both congruent and equal.

ANGLES 1=5 2=6 3=7 4=8 SIDES AB = EF BC = FG CD = GH DA = HE

PYTHAGOREAN THEOREM FOR RIGHT ANGLE TRIANGLES

The hypotenuse is the side directly across from the right angle in a right angle triangle

Hypotenuse

Leg

Leg

a=3 b=4

2

c

a b

The Pythagorean Theorem: 2

2

2

a2 + b2 = c 2 3+4=c 2 9 + 16 = c 2 25 = c 5=c 2

2