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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
2
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
3
•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.
4
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
6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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.
15
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.
16
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.
17
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
18
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.
19
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.
20
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.
21
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.
22
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
23
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.
24
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.