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CHAPTER 5CHAPTER 5CHAPTER 5CHAPTER 5

Logic and GeometryLogic and Geometry

SECTION 5SECTION 5--11SECTION 5SECTION 5--11Elements of Elements of GeometryGeometry

GEOMETRY– is the study of points in space

POINT– indicates a specific location and is represented by a dot and a letter

• R • S• T

LINE – is a set of points that extends without end in two opposite directions

R Sline RS

PLANE – is a set of points that extends in all directions along a flat surface

•• WY

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COLLINEAR POINTS –points that lie on the

same line

• • •C D E

F •

NONCOLLINEAR POINTS – points that do not lie on the same

line

• • •C D E

F •

COPLANAR POINTS – are points that lie in

the same plane

••

•A B C

•D

• E

NONCOPLANAR POINTS – are points that do not lie in the same plane

••

•A B C

•D

• E

INTERSECTION – set of all points common to two geometric figures

•P

LINE SEGMENT - part of a line that begins at one endpoint and ends at

another

• •F G

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•RAY – part of a line that begins at one endpoint and continues in the other direction, without ending

A B

CONGRUENT LINE SEGMENTS - have the

same measure

≅ means congruent to

MIDPOINT - the point that divides the segment

into two congruent segments.

BISECTOR - is any line, segment, ray or plane that intersects the

segment at its midpoint

POSTULATE - 1

Through any two points, there is exactly one

line.

POSTULATE - 2

Through any three noncollinear points, there

is exactly one plane.

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POSTULATE - 3

If two points lie in a plane, then the line

joining them lies in that plane

POSTULATE - 4

If two planes intersect, then their intersection

is a line

SECTION 5SECTION 5--22SECTION 5SECTION 5--22ANGLES AND ANGLES AND

PERPENDICULAR LINESPERPENDICULAR LINES

ANGLE– the union of two rays with a common

endpoint.VERTEX – endpoint of an

angleA

C

B• ••

Right Anglemeasures exactly 90º

Acute Anglemeasure is greater than 0 º and less than 90º

•

•

•

5

Obtuse Anglemeasure is greater than 90º and less than 180º

Straight Anglemeasures exactly 180º

COMPLEMENTARY

angles whose sum

measures 90º

J

39º

K51º

SUPPLEMENTARY angles whose sum measures 180º

J

K121º

59°

ADJACENT ANGLEStwo angles in the same plane that share a common side and a common vertex O

•

•

•

A

C

B

AOB and BOC

ADJACENTANGLES

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CONGRUENT ANGLESangles having the same

measure

PERPENDICULAR LINES - two lines that intersect to form right

angles

VERTICAL ANGLES two angles whose sides

form two pairs of opposite rays. Vertical angles are congruent.

ANGLE BISECTOR -ray that divides the

angle into two congruent adjacent angles O

•

•

•

A

C

B

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EXAMPLE 1

Find the measure of angle JXF

SOLUTION

Read the Protractor

Angle JXF = 90ºright

EXAMPLE 2

Find the measure of angle HXL

SOLUTION

Read the Protractor

Angle HXL = 120ºobtuse

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EXAMPLE 3

Find the measure of angle KXG

SOLUTION

Read the Protractor

Angle KXG = 120ºobtuse

EXAMPLE 4

Find the measure of angle GXJ

SOLUTION

Read the Protractor

Angle GXJ = 65ºacute

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SECTION 5SECTION 5--33

PARALLEL LINES and PARALLEL LINES and TRANSVERSALSTRANSVERSALS

SECTION 5SECTION 5--33

PARALLEL LINES and PARALLEL LINES and TRANSVERSALSTRANSVERSALS

PARALLEL LINES -coplanar lines that do

not intersect

•

Parallel Lines

m

n

PARALLEL PLANES -planes that do not

intersect

•

Parallel Planes

mn

Skew Lines -noncoplanar lines that do not intersect and are

not parallel

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Skew Lines

m nn

Transversal -is a line that intersects

each of two other coplanar lines in

different points to produce interior and

exterior angles

Transversal

65

21

4

3

87l

ALTERNATE INTERIOR ANGLES -two nonadjacent interior angles on opposite sides

of a transversal

Alternate Interior Angles

21

4

3

SAME SIDE INTERIOR ANGLES -interior angles on the

same side of a transversal

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Same Side Interior Angles

21

4

3

ALTERNATE EXTERIOR ANGLES -

two nonadjacent exterior angles on

opposite sides of the transversal

Alternate Exterior Angles

65

87

Corresponding Angles -two angles in

corresponding positions relative to two lines cut

by a transversal

Corresponding Angles

65

21

4

3

87

PARALLEL LINE PARALLEL LINE POSTULATESPOSTULATESPARALLEL LINE PARALLEL LINE POSTULATESPOSTULATES

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POSTULATE - 5

If two parallel lines are cut by a transversal, then corresponding angles are congruent

∠2 ≅ ∠8, ∠6 ≅ ∠4,∠5 ≅ ∠3, ∠1 ≅ ∠7

65

21

43

87

STATEMENT - 5A

If two parallel lines are cut by a

transversal, then alternate interior

angles are congruent

∠2 ≅ ∠3, ∠1 ≅ ∠4

21

43

STATEMENT - 5B

If two parallel lines are cut by a

transversal, then alternate exterior angles are congruent

∠6 ≅ ∠7, ∠5 ≅ ∠8

65

87

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SECTION 5SECTION 5--44SECTION 5SECTION 5--44

Properties of Properties of TrianglesTriangles

Triangle – is a figure formed by the

segments that join three noncollinear

points

Vertex – point of a triangle

Side – segment of a triangle

Congruent Segments –segments with the

same length

Congruent Angles –angles with the same

measure

Scalene Triangle – is a triangle with all three sides of

different length.

Isosceles Triangle – is a triangle with two sides (legs) of equal length and a third side called the base and

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Angles at the base are called base angles and the third angle is the

vertex angle.

Equilateral Triangle –is a triangle with

three sides of equal length

Acute Triangle – is a triangle with three acute angle (<90°)

Obtuse Triangle – is a triangle with one obtuse angle (>90°)

Right Triangle – is a triangle with one right

angle (90°)

Equiangular Triangle –is a triangle with

three angles of equal measure.

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Interior angles – angles determined by the sides

of a triangle

Exterior angle – an angle that is both adjacent

and supplementary to an interior angle

Base angle – angles opposite congruent

sides

PROPERTIES of TRIANGLES The sum of the

measures of the angles of a triangle is

180°

The sum of the lengths of any two sides is greater than the length of the third

side.

The longest side is opposite the largest

angle, and the shortest side is opposite the smallest angle.

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If one side of a triangle is extended, then the

exterior angle formed is equal to the sum of the two remote interior angles of the triangle.

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

SECTION 5SECTION 5--55SECTION 5SECTION 5--55CONGRUENT CONGRUENT TRIANGLESTRIANGLES

Congruent Triangles - the vertices can be matched so that corresponding parts fit exactly over

each other, and

Corresponding angles lie opposite corresponding sides, and vice versa.

Corresponding Sides -corresponding sides of congruent triangles are

congruent.

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Corresponding Angles -corresponding angles of congruent triangles are

congruent.

POSTULATES

If three sides of one triangle are congruent

to three sides of another triangle, then

the triangles are congruent (SSS)

If two sides and the included angle of one

triangle are congruent to two sides and the

included angle of another triangle, then the

triangles are congruent (SAS)

If two angles and the included side of one

triangle are congruent to two angles and the

included side of another triangle, then the

triangles are congruent (ASA)

SECTION 5SECTION 5--66SECTION 5SECTION 5--66QUADRILATERALS and QUADRILATERALS and PARALLELOGRAMSPARALLELOGRAMS

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Quadrilateral - a closed plane figure that has four sides

Parallelogram - is a quadrilateral with

both pairs of opposite sides parallel.

Trapezoid - is a quadrilateral with exactly one pair of

sides parallel.

Rectangle - is a quadrilateral with four right angles.

Rhombus - is a quadrilateral with four sides of equal

length.

Square - is a quadrilateral with

four right angles andfour sides of equal

length.

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Opposite angles - two angles that do not share a common side

Consecutive angles -two angles that share

a common side

Opposite sides - two sides that do not share a common

endpoint

Consecutive sides -two sides that share a common endpoint.

PROPERTIES of PARALLELOGRAMS

The opposite sides of a parallelogram are

congruent.

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The opposite angles of a parallelogram are congruent.

The consecutive angles of a

parallelogram are supplementary.

The sum of the angle measures of a

parallelogram is 360°.

The diagonals of a parallelogram bisect

each other.

The diagonals of a rectangle are congruent.

The diagonals of a rhombus are perpendicular.

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SECTION 5SECTION 5--77SECTION 5SECTION 5--77DIAGONALS and DIAGONALS and

ANGLES of POLYGONSANGLES of POLYGONS

Polygon – is a closed plane figure that is formed by joining three or more coplanar segments at their endpoints, and

Each segment of the polygon is called a side, and the point

where two sides meet is called a vertex

A polygon is Convex if each line containing a

side contains no points in the interior of the

polygon.

Convex

A polygon is Concave if a line containing a side contains a point in the interior of the polygon.

ConcaveRegular Polygon - a polygon that has all sides congruent and all angles congruent.

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Diagonal - a segment of a polygon that

joins two vertices but is not a side.

THEOREMS

The sum of the measures of the

angles of a polygon with n sides is

(n-2)180°

The measure of each interior angle of a

regular polygon with nsides is (n-2)180°

n

SECTION 5SECTION 5--88SECTION 5SECTION 5--88PROPERTIES of PROPERTIES of

CIRCLESCIRCLES

Circle - the set of all points in a plane that are a given distance from a fixed point in

the plane, and

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The fixed point is called the CENTER. The given distance is the

RADIUS.

Radius - is a segment that has one endpoint at the center and one

on the circle.

Chord - is a segment with both endpoints on the

circle.

Diameter - is a chord that passes through the center of the

circle.

Circumference - is the distance around a

circle.

Arc - is a section of the circumference of a

circle.

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Semicircle - is a arc with endpoints that are

the endpoints of a diameter.

Minor Arc - is an arc that is smaller than a

semicircle.

Major Arc - is an arc that is larger than a

semicircle.

Central Angle - is an angle with its vertex at the center of a circle.

*The measure of a central angle is equal to the measure of the arc it

intercepts.

Inscribed Angle - is an angle whose vertex lies on the circle and whose sides contain chords of

the circle, and

The measure of an inscribed angle is ½ the measure of the arc it

intercepts.

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SECTION 5SECTION 5--99SECTION 5SECTION 5--99 CIRCLE GRAPHSCIRCLE GRAPHSCIRCLE GRAPHSCIRCLE GRAPHS

Circle Graph - is a way to display data to make

comparisons.

THE END!THE END!THE END!THE END!