Chapter 3

Geometry and Measurement

What You Will Learn:

To identify, describe, and draw: Parallel line segments Perpendicular line segments

To draw: Perpendicular bisectors Angle bisectors

Generalize rules for finding the area of: Parallelograms Triangles

Explain how the area of a rectangle can be used to find the area of: Parallelograms Triangles

3.1 – Parallel and Perpendicular Line Segments

What you will learn:To identify, describe, and draw:

Parallel line segmentsPerpendicular line segments

Parallel

Describes lines in the same plane that never cross, or intersect

The perpendicular distance btw parallel line segments must be the same at each end of the line segments.

They are always marked using “arrows”

http://www.mathopenref.com/parallel.html

Some ways to create parallel line segments:

Using paper foldingUsing a ruler and a right triangle

Example:Draw a line segment, AB. Draw another line

segment, CD, parallel to AB.

Example:Draw a line segment, AB. Draw another line segment,

CD, parallel to AB.B

AC

D

B

AC

D

Label the endpoints (A, B, C, D).Mark the lines with arrows to show the lines are parallel.

B

A

Use a ruler todraw a linesegment.

Slide the triangle, draw a parallel line.

Perpendicular

Describes lines that intersect at right angles (90°)

They are marked using a small square

http://www.mathopenref.com/perpendicular.html

right angle

Using paper folding (p. 85)Using a ruler and protractor (p. 85)http://www.mathopenref.com/

constperplinepoint.html

Some ways to create perpendicular line segments:

Assignment

P. 86#1, 3-5, 7, 9, 11, Math LinkStill Good? #2, 8, 10, 12, 13ProStar? #14-16

right angle

3.2 – Draw Perpendicular Bisectors

Bisect:Bi means “two.” Sect means “cut.” So, Bisect

means to cut in two.

Perpendicular bisectorA line that divides a line segment in half and is

at right angles (90°) to the line segment.Equal line segments are marked with “hash”

marks

Some ways to create a perpendicular bisector:

Using a compass (p. 90)http://www.mathopenref.com/constbisectline.html

Using a ruler and a right triangle (p. 91)Using paper folding (p. 91)

Assignment

P. 92, # 1-5, 8Still Good? # 6, 7, 9, MathLinkProStar? #10

3.3 – Draw Angle Bisectors

Terms:Acute angle

An angle that is less than 90°Obtuse angle

An angle that is more than 90°Angle Bisector

A line that divides an angle into two equal partsEqual angles are marked with the same symbol

Less than 90°

Greater than 90°

Some ways to create an angle bisector include:

Using a ruler and compass (p. 95)http://www.mathopenref.com/constbisectangle.html

Using a ruler and protractor (p. 95)Using paper folding (p.95)

Assignment

P. 97, # 1 & 2, 5, 6, 8Still Good? # 3 & 4, 9, 11, 13, MathLinkProStar? #12, 14, 15

Greater than 90°: obtuse

Less than 90°: acute

Angle Bisector

3.4 – Area of a Parallelogram

Area of a rectangle: Area = length x width

ParallelogramA four-sided figure with opposite sides parallel

and equal in length

http://www.mathopenref.com/parallelogramarea.html

w

l

6 cm

4 cmA = l x w

A = 6 cm x 4 cm

A = 24 cm2

Making a Parallelogram from a Rectangle

cut

paste

BaseA side of a two-dimensional closed figureCommon symbol is b

HeightThe perpendicular distance from the base to the

opposite sideCommon symbol is h

Suggest a formula for calculating the area of a parallelogram.

b

h

Area of a Rectangle vs. Area of a Parallelogram

Area = length x width = 12 cm x 8 cm = 96 cm

Area = base x height = 12 cm x 8 cm = 96 cm

12 cm

8 cm

12 cm

8 cm

Are they the same? Try it!

2 2

b

h

Sometimes it is necessary to extend the line of the base to measure the height

Key Ideas

The formula for the area of a rectangle can be used to determine the formula for the area of a parallelogram.

The formula for the area of a parallelogram is A = b x h, where b is the base and h is the height.

The height of a parallelogram is ALWAYS perpendicular to its base.

h

b

Assignment

P. 104, # 1-3, 5, 7, 9, 11Still Good? # 13-18, MathLinkProStar? # 19, 20

b

h A = b x h

3.5 – Area of a Triangle

What you will learn:Develop the formula for the area of a triangleCalculate the area of a triangle

What we know:The area of a rectangle

A = l x w

The area of a parallelogramA = b x h

Key Ideas

The formula for the area of a rectangle or parallelogram can be used to determine the formula for the area of a triangle

The formula for the area of a triangle is A = b x h 2, or A = b x h, 2

where b is the base of the triangle and h is the height of the triangle. The height of the triangle is always measured perpendicular to its

base. http://www.mathopenref.com/trianglearea.html

h

b

A = b x h

h

b

Cut the rectangle in half

A = b x h 2

Cut the area in half

Your Assignment

P. 113, #1-3 as a class.Area of a Triangle, NotebookArea of a Triangle Questions, NotebookP. 113, #4a), 5b)No problem? #8, 10, 11Still good? #13-15Pro Star? #16-19