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Chapter 2

Geometry

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Section 2.1 Lines and Angles

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Point: Usually named using a capital letter

Line: (1-dimensional) Usually named using two points included on the line

Plane: (2-dimensional)

Ray: (half-line) Has one endpoint and extends infinitely in one direction

The Foundation

AB

AB

A

An angle is formed by two ___________ with a common

_____________, called the ____________ of the angle.

The amount of ________________ of the terminal ray is called the angle.

A

CB

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Angles

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Types of Angles

1. Right Angle ________________

2. Straight Angle ________________

3. Acute Angle ________________

4. Obtuse Angle _________________

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AB CD

Types of Lines

1. Lines in the same plane that do not intersect are called ____________________ lines.

2. Lines that intersect at right angles are called _______________________ lines.

A

C

D

B

1 2

1

2

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More on Angles

1. ________________________ angles are two angles whose sum is 90°.

2. ________________________ angles are two angles whose sum is 180°.

3. ________________________ angles share a common vertex and a common side.

E F G

H

8

When two lines intersect, the angles that are formed on opposite sides of the point of intersection are called ___________________ angles.

_______________ angles are equal in measure (congruent).

1

23

4

9

If a line intersects two or more lines in a plane, it is called a

_____________________________.

When two parallel lines are cut by a transversal, the

corresponding angles are equal (congruent)

alternate interior angles are equal (congruent)

alternate exterior angles are equal (congruent)

M N

P Q

MN PQ���������������������������������������� ���

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1. Pairs of corresponding angles:

________&_________; ________&_________; ________&_________; ________&_________

2. Pairs of alternate interior angles:

________&_________; ________&_________

3. Pairs of alternate exterior angles:________&_________; ________&_________

M N

P Q

MN PQ���������������������������������������� ���1

3

2

4

5 6

7 8

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If 4 = 41°, find the measures of the remaining angles.

1 =______ 2 = ______ 3 = ______

5 =______ 6 = ______ 7 = ______

8 =______

Example

M N

P Q

MN PQ���������������������������������������� ���1

3

2

4

5 6

7 8

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When more than two parallel lines are cut by two transversals, the segments of the transversals between the same two parallel lines are called corresponding segments.

AB CD EF������������������������������������������������������������������ �����

A B

C D

E F

W X

Y Z

W Y W XOR

X Z Y Z

Ratios of corresponding segments of the transversal are equal.

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Example p 52 # 23-28

Find the measures of the angles.

1) BDF =

2) ABE =

3) DEB =

4) DBE =

5) DFE =

6) ADE =

44°A

B C

D E

F

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Section 2.2 Triangles

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PolygonsA polygon is a closed figure with straight sides.

Some common polygons:

3 sides: __________________ 4 sides: __________________ 5 sides: __________________ 6 sides: __________________ 7 sides: __________________ 8 sides: __________________ 9 sides: __________________10 sides: __________________12 sides: __________________

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PolygonsPerimeter:

_______________________________________________________________________________________________________________________________________

Units used for perimeter:

Area: __________________________________________________________________________________________

Units used for area:

Triangles

Triangles are polygons with three sides and three interior angles.

The sum of the measures of the three angles of a triangle is __________.

Triangles are often classified according to the types of sides or the types of angles they contain.

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18

Types of Triangles (classified by sides)

1. Scalene:

2. Isosceles:

3. Equilateral:

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Right Triangles

Pythagorean Theorem:

In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides (legs).

leg

leg

hypotenuse

ac

b

Example: If a = 5 and c = 11, find the exact value of b.

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Area of a TriangleThe area of a triangle is one-half the product of a

base and its height.

b

h

Area of a triangle:

1

2A bh

The height (or altitude) of a triangle is the line segment drawn from a vertex perpendicular to the opposite side (base).

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Find the perimeter and area of the triangle shown below (all numbers are approximate).

2.2 cm4.8 cm

6.5 cm

1.3 cm

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Find the perimeter and area (to 2 s.d.) of an isosceles triangle whose two equal sides measure 35 mm and height from the vertex angle to the base measures 28 mm.

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A Super Hero!

If we know the three lengths of the sides of a triangle, but not the height, we can use Hero’s Formula (Alternate: Heron’s Formula) to calculate the area.

For a triangle with sides of length , , & :

1,

2A s s a s b

a b c

where ss bc a c

Hero’s Formula

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ExampleA triangular-shaped park is bounded by three streets. The lengths of the three sides of the park are found to be 358 ft, 437 ft, and 509 ft. What is the area of the park (to 3 sig dig)?

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Congruent TrianglesTwo or more triangles are congruent if the measures of

each of the ___________ and the measures of each of

the ____________ are the same.

A

BC

84

5541

5.8 cm

7.0 cm

4.6 cm

P

Q R

84

55 41

5.8 cm

7.0 cm

4.6 cm

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Similar TrianglesSimilar triangles have congruent ________________ but do not

necessarily have congruent _______________.

Properties of Similar Triangles

1. Corresponding angles of similar triangles are equal.

2. Corresponding sides of similar triangles are proportional.

ABC ADE

Corresponding angles:

Corresponding sides:A

B

C

D

E

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Example

A woman is standing next to a building on level ground. The woman, who is 5’6” tall, casts a shadow that is 3.0 feet long. At the same time, a building casts a shadow that is 92 feet long. How tall is the

building? Round ans to 2 sig digits.

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Section 2.3 Quadrilaterals

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A quadrilateral is a polygon with ______ sides and _____ interior angles.

The sum of the measures of the interior angles of a quadrilateral is ___________.

A ________________ of a polygon is a line segment joining any two non-adjacent vertices.

Special Types of Quadrilaterals

Parallelogram

Rhombus

Rectangle

Square

Trapezoid

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31

Determine if each statement is true sometimes, always, or never.

1. A rectangle is a parallelogram.

_______________

2. A rhombus is a square.

_______________

3. A square is a rhombus.

_______________

4. A trapezoid is a parallelogram.

_______________

5. A square is a rectangle.

_______________

6. A parallelogram is a rectangle.

_______________

7. A square is a trapezoid.

_______________

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Perimeter

The perimeter of a quadrilateral is the sum of the lengths of the four sides. (The distance around the figure.)

Some Special Formulas:

1. Perimeter of a Rectangle: P = ________________

2. Perimeter of a Square: P = ____________

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Area Formulas1. Area of a Parallelogram: A = ________________

2. Area of a Rectangle: A = ________________

3. Area of a Square: A = ________________

4. Area of a Trapezoid: A = ________________

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Examples

1) Text p. 62 # 30

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Examples

2) Text p. 62 # 32

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Examples

3) Text p. 63 # 38

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Section 2.4 Circles

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Some Circle Vocab

1. ________________: The set of all points equidistant from a fixed point, called the __________.

2. ________________: A line segment with endpoints on the circle that passes through its center.

3. ________________: A line segment from the center to a point on the circle.

4. ________________: A line segment with endpoints on the circle.

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More Circle Vocab

5. ________________: A line that intersects (touches) the circle at EXACTLY ONE POINT.

6. ________________: A line that passes through two points of the circle.

The radius is perpendicular to the tangent at the point of tangency.

o

40

An example

N

M

oP

Let O be the center of the circle.

MN is tangent to the circle.

If MNO = 14°, find MOP.

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Circumference (pi) is an irrational number that is approximately equal to ______

is the ratio of the circumference (perimeter) of a circle to its diameter.

C

d

Circumference of a Circle:

2C d or C r

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Area

Area of a Circle:

2A r

Example

Find the area of a circle with diameter 18.5 cm. Give answer to three significant digits.

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A circular walkway, with a uniform width of 3.0 ft, is to be installed around a traffic circle that has a diameter of 18 ft. Find the area of the walkway to two significant digits.

An example

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What is the area of the largest circle that can be cut from a rectangular plate 21.2 cm by 15.8 cm? How much waste is there? Give answers to 3 sig digits.

Another example

21.2 cm

15.8 cm

45

20 m

14 m

Neglecting waste, how much would it cost to lay down a hardwood floor on the indoor rink if flooring costs $22.50 per square meter? Round to nearest dollar.

One More Example

Note: Ends are semicircular.

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Arcs & Angles

1. A _________ __________ is an angle formed by two radii (its vertex is the center of the circle.)

2. An _________ is a part of the outside of the circle (a curved segment)

The measure of an arc is the same as the measure of the central angle that forms it. (Measured in degrees.)

AB

o

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Regions of a Circle

1. A ___________________ is a region bounded by two radii and the arc they intercept.

2. A ____________ is a region bounded by a chord and its arc.

AB

o

A

B

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More Angles

An ____________ ___________ has its vertex on the circle.

A

C

B

Inscribed angle

Intercepted arc

The measure of an inscribed angle is half of its intercepted arc.

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An ExampleIf the measure of an inscribed angle is 45°, find the length

(to 2 sig dig) of its intercepted arc, given that the radius of the circle is 8.0 cm.

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Section 2.6 Solid Geometric Figures

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Volume & Surface Area

Volume is the measure of space occupied by a solid

geometric figure.

Units used for volume:

Surface Area is the total area of all of the faces of a solid

geometric figure. Units used for surface area:

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Formulas for Volume & Surface Area

V = Volume

A = Total Surface AreaS = Lateral Surface Area (does not include area of the bases)

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Solid Geometric Figures

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Formulas for Volume & Surface AreaSolid Geometric Figure Volume Formula Surface Area Formula

Rectangular Solid

Cube

Right Circular Cylinder

Right Circular Cone

Regular Pyramid

Sphere

V lwh3V e

2V r h

21

3V r h

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3V r

1

3V Bh

2 2 2A lw lh wh

26A e

22 2A r rh

2A r rs

1

2A ps

24A re is length of edge of cube s is the lateral height

B is area of the base p is the perimeter of the base

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Example 1P 74 # 26

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Example 2P 74 # 28

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Example 3P 74 # 34