6.1

Points, Lines, Planes, and Angles

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.

Basic Terms

Line segment AB

Ray BA

Ray AB

Line AB

SymbolDiagramDescription

AB�������������� �

AB��������������

BA��������������

AB

A B

A

A

A

B

B

B

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.

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:

BCBAABC

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°

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.

Example

If are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle.

A B

C

D

CBDABCand

Example

If are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle.

A B

C

D

CBDABCand

2.154

7.25

7.25

1807

1806

180

ABCm

CBDm

x

x

xx

CBDmABCm

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.

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

6.2

Polygons

Polygons

Polygons are named according to their number of sides.

Icosagon20Heptagon7

Dodecagon12Hexagon6

Decagon10Pentagon5

Nonagon9Quadrilateral4

Octagon8Triangle3

NameNumber of Sides

NameNumber of Sides

Types of Triangles

Acute Triangle

All angles are acute.

Obtuse Triangle

One angle is obtuse.

Types of Triangles continued

Right Triangle

One angle is a right angle.

Isosceles Triangle

Two equal sides.

Two equal angles.

Types of Triangles continued

Equilateral Triangle

Three equal sides. Three equal angles (60º) each.

Scalene Triangle

No two sides are equal in length.

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

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

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

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.

Quadrilaterals

Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360°.

Quadrilaterals may be classified according to their characteristics.

Classifications

Trapezoid

Two sides are parallel.

Parallelogram

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.

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.

Classifications continued

Square

Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.

6.3

Perimeter and Area

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

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.

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

Example continued

c) Determine the cost

30 squares x $32 per square

$960

It will cost a total of $960 to roof the barn.

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

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

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

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.

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.

6.4

Volume and Surface Area

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 the 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.

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?

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

Polyhedron

A polyhedron is a closed surface formed by the union of polygonal regions.

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

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