slide 9 - 1 copyright © 2009 pearson education, inc. 6.1 points, lines, planes, and angles
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Slide 9 - 1Copyright © 2009 Pearson Education, Inc.
6.1
Points, Lines, Planes, and Angles
Slide 9 - 2Copyright © 2009 Pearson Education, Inc.
Basic Terms
A point, line, and plane are three basic terms in geometry that are NOT given a formal definition, yet we recognize them when we see them.
A line is a set of points. Any two distinct points determine a unique line. Any point on a line separates the line into three
parts: the point and two half lines. A ray is a half line including the endpoint. A line segment is part of a line between two
points, including the endpoints.
Slide 9 - 3Copyright © 2009 Pearson Education, Inc.
Basic Terms
Line segment AB
Ray BA
Ray AB
Line AB
SymbolDiagramDescription
AB�������������� �
AB��������������
BA��������������
AB
A B
A
A
A
B
B
B
Slide 9 - 4Copyright © 2009 Pearson Education, Inc.
Plane
We can think of a plane as a two-dimensional surface that extends infinitely in both directions.
Any three points that are not on the same line (noncollinear points) determine a unique plane.
A line in a plane divides the plane into three parts, the line and two half planes.
Any line and a point not on the line determine a unique plane.
The intersection of two planes is a line.
Slide 9 - 5Copyright © 2009 Pearson Education, Inc.
Angles
An angle is the union of two rays with a common endpoint; denoted
The vertex is the point common to both rays. The sides are the rays that make the angle. There are several ways to name an angle:
R.
RABC, RCBA, RB
Slide 9 - 6Copyright © 2009 Pearson Education, Inc.
Angles
The measure of an angle is the amount of rotation from its initial to its terminal side.
Angles can be measured in degrees, radians, or, gradients.
Angles are classified by their degree measurement. Right Angle is 90° Acute Angle is less than 90° Obtuse Angle is greater than 90° but less
than 180° Straight Angle is 180°
Slide 9 - 7Copyright © 2009 Pearson Education, Inc.
Types of Angles
Adjacent Angles-angles that have a common vertex and a common side but no common interior points.
Complementary Angles-two angles whose sum of their measures is 90 degrees.
Supplementary Angles-two angles whose sum of their measures is 180 degrees.
Slide 9 - 8Copyright © 2009 Pearson Education, Inc.
Example
If are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle.
RABC and RCBD
A B
C
D180
6 180
7 180
25.7
m ABC m CBD
x x
x
x
mRABC 154.2o
mRCBD 25.7o
Slide 9 - 9Copyright © 2009 Pearson Education, Inc.
More definitions
Vertical angles are the nonadjacent angles formed by two intersecting straight lines.
Vertical angles have the same measure. A line that intersects two different lines, at two
different points is called a transversal.
Special angles are given to the angles formed by a transversal crossing two parallel lines.
Slide 9 - 10Copyright © 2009 Pearson Education, Inc.
Special Names
5 6
1 2
4
87
3
One interior and one exterior angle on the same side of the transversal–have the same measure
Corresponding angles
Exterior angles on the opposite sides of the transversal–have the same measure
Alternate exterior angles
Interior angles on the opposite side of the transversal–have the same measure
Alternate interior angles
5 6
1 2
4
87
3
5 6
1 2
4
87
3
Slide 9 - 11Copyright © 2009 Pearson Education, Inc.
6.2
Polygons
Slide 9 - 12Copyright © 2009 Pearson Education, Inc.
Polygons
Polygons are named according to their number of sides.
Icosagon20Heptagon7
Dodecagon12Hexagon6
Decagon10Pentagon5
Nonagon9Quadrilateral4
Octagon8Triangle3
NameNumber of Sides
NameNumber of Sides
Slide 9 - 13Copyright © 2009 Pearson Education, Inc.
Triangles
The sum of the measures of the interior angles of an n-sided polygon is (n - 2)180°.
Example: A certain brick paver is in the shape of a regular octagon. Determine the measure of an interior angle and the measure of one exterior angle.
Slide 9 - 14Copyright © 2009 Pearson Education, Inc.
Triangles continued
Determine the sum of the interior angles.
The measure of one interior angle is
The exterior angle is supplementary to the interior angle, so
180° - 135° = 45°
( 2)180
(8 2)(180 )
6(180 )
1080
S n
1080135
8
Slide 9 - 15Copyright © 2009 Pearson Education, Inc.
Types of Triangles
Acute Triangle
All angles are acute.
Obtuse Triangle
One angle is obtuse.
Slide 9 - 16Copyright © 2009 Pearson Education, Inc.
Types of Triangles continued
Right Triangle
One angle is a right angle.
Isosceles Triangle
Two equal sides.
Two equal angles.
Slide 9 - 17Copyright © 2009 Pearson Education, Inc.
Types of Triangles continued
Equilateral Triangle
Three equal sides. Three equal angles (60º) each.
Scalene Triangle
No two sides are equal in length.
Slide 9 - 18Copyright © 2009 Pearson Education, Inc.
Similar Figures
Two polygons are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion.
4
3
4
6
6 6
9
4.5
Slide 9 - 19Copyright © 2009 Pearson Education, Inc.
Example
Catherine Johnson wants to measure the height of a lighthouse. Catherine is 5 feet tall and determines that when her shadow is 12 feet long, the shadow of the lighthouse is 75 feet long. How tall is the lighthouse?
x
7512
5
Slide 9 - 20Copyright © 2009 Pearson Education, Inc.
Example continued
x
7512
5
Therefore, the lighthouse is 31.25 feet tall.
ht. lighthouse
ht. Catherine=
lighthouse's shadow
Catherine's shadowx
5
75
1212x 375
x 31.25
Slide 9 - 21Copyright © 2009 Pearson Education, Inc.
Congruent Figures
If corresponding sides of two similar figures are the same length, the figures are congruent.
Corresponding angles of congruent figures have the same measure.
Slide 9 - 22Copyright © 2009 Pearson Education, Inc.
Quadrilaterals
Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360°.
Quadrilaterals may be classified according to their characteristics.
Slide 9 - 23Copyright © 2009 Pearson Education, Inc.
Classifications
Trapezoid
Two sides are parallel.
Parallelogram
Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.
Slide 9 - 24Copyright © 2009 Pearson Education, Inc.
Classifications continued
Rhombus
Both pairs of opposite sides are parallel. The four sides are equal in length.
Rectangle
Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.
Slide 9 - 25Copyright © 2009 Pearson Education, Inc.
Classifications continued
Square
Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.
Slide 9 - 26Copyright © 2009 Pearson Education, Inc.
6.3
Perimeter and Area
Slide 9 - 27Copyright © 2009 Pearson Education, Inc.
Formulas
P = s1 + s2 + b1 + b2
P = s1 + s2 + s3
P = 2b + 2w
P = 4s
P = 2l + 2w
Perimeter
Trapezoid
Triangle
A = bhParallelogram
A = s2Square
A = lwRectangle
AreaFigure
12A bh
11 22 ( )A h b b
Slide 9 - 28Copyright © 2009 Pearson Education, Inc.
Example
Marcus Sanderson needs to put a new roof on his barn. One square of roofing covers 100 ft2
and costs $32.00 per square. If one side of the barn roof measures 50 feet by 30 feet, determine
a) the area of the entire roof.
b) how many squares of roofing he needs.
c) the cost of putting on the roof.
Slide 9 - 29Copyright © 2009 Pearson Education, Inc.
Example continued
a) The area of the roof is
A = lw
A = 30(50)
A = 1500 ft2
1500 x 2 (both sides of the roof) = 3000 ft2
b) Determine the number of squares
area of roof
area of one square
3000
10030
Slide 9 - 30Copyright © 2009 Pearson Education, Inc.
Example continued
c) Determine the cost
30 squares x $32 per square
$960
It will cost a total of $960 to roof the barn.
Slide 9 - 31Copyright © 2009 Pearson Education, Inc.
Pythagorean Theorem
The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse.
leg2 + leg2 = hypotenuse2
Symbolically, if a and b represent the lengths of the legs and c represents the length of the hypotenuse (the side opposite the right angle), then
a2 + b2 = c2 a
b
c
Slide 9 - 32Copyright © 2009 Pearson Education, Inc.
Example
Tomas is bringing his boat into a dock that is 12 feet above the water level. If a 38 foot rope is attached to the dock on one side and to the boat on the other side, determine the horizontal distance from the dock to the boat.
12 ft
38 ft rope
Slide 9 - 33Copyright © 2009 Pearson Education, Inc.
Example continued
The distance is approximately 36.06 feet.
a2 b2 c2
122 b2 382
144 b2 1444
b2 1300
b2 1300
b 36.06
1238
b
Slide 9 - 34Copyright © 2009 Pearson Education, Inc.
Circles
A circle is a set of points equidistant from a fixed point called the center.
A radius, r, of a circle is a line segment from the center of the circle to any point on the circle.
A diameter, d, of a circle is a line segment through the center of the circle with both end points on the circle.
The circumference is the length of the simple closed curve that forms the circle.
Slide 9 - 35Copyright © 2009 Pearson Education, Inc.
Example
Terri is installing a new circular swimming pool in her backyard. The pool has a diameter of 27 feet. How much area will the pool take up in her yard? (Use π = 3.14.)
A r 2
A (13.5)2
A 572.265
The radius of the pool is 13.5 ft.
The pool will take up about 572 square feet.
Slide 9 - 36Copyright © 2009 Pearson Education, Inc.
6.4
Volume and Surface Area
Slide 9 - 37Copyright © 2009 Pearson Education, Inc.
Volume
Volume is the measure of the capacity of a figure.
It is the amount of material you can put inside a three-dimensional figure.
Surface area is sum of the areas of the surfaces of a three-dimensional figure.
It refers to the total area that is on the outside surface of the figure.
Slide 9 - 38Copyright © 2009 Pearson Education, Inc.
Volume Formulas
Sphere
Cone
V = πr2hCylinder
V = s3Cube
V = lwhRectangular Solid
DiagramFormulaFigure
213V r h
343V r
Slide 9 - 39Copyright © 2009 Pearson Education, Inc.
Surface Area Formulas
Sphere
Cone
SA = 2πrh + 2πr2Cylinder
SA= 6s2Cube
SA = 2lw + 2wh +2lhRectangular Solid
DiagramFormulaFigure
SA r2
r r2
h2
SA 4r 2
Slide 9 - 40Copyright © 2009 Pearson Education, Inc.
Example
Mr. Stoller needs to order potting soil for his horticulture class. The class is going to plant seeds in rectangular planters that are 12 inches long, 8 inches wide and 3 inches deep. If the class is going to fill 500 planters, how many cubic inches of soil are needed? How many cubic feet is this?
Slide 9 - 41Copyright © 2009 Pearson Education, Inc.
Example continued
We need to find the volume of one planter.
Soil for 500 planters would be
500(288) = 144,000 cubic inches
V lwh
V 12(8)(3)
V 288 in.3
144,000
172883.33 ft3
Slide 9 - 42Copyright © 2009 Pearson Education, Inc.
Polyhedron
A polyhedron is a closed surface formed by the union of polygonal regions.
Slide 9 - 43Copyright © 2009 Pearson Education, Inc.
Euler’s Polyhedron Formula
Number of vertices - number of edges + number of faces = 2
Example: A certain polyhedron has 12 edges and 6 faces. Determine the number of vertices on this polyhedron.
# of vertices - # of edges + # of faces = 2
There are 8 vertices.
12 6 2
6 2
8
x
x
x
Slide 9 - 44Copyright © 2009 Pearson Education, Inc.
Volume of a Prism
V = Bh, where B is the area of the base and h is the height.
Example: Find the volume of the figure.Area of one triangle. Find the volume.
8 m
6 m
4 m
12
12
2
(6)(4)
12 m
A bh
A
A
3
12(8)
96 m
V Bh
V
V
Slide 9 - 45Copyright © 2009 Pearson Education, Inc.
Volume of a Pyramid
where B is the area of the base and h is the height.
Example: Find the volume of the pyramid.Base area = 122 = 144
13V Bh
12 m
12 m
18 m
13
13
3
(144)(18)
864 m
V Bh
V
V