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Parallel and Perpendicular

- To discover how slopes of parallel and

perpendicular lines are related - To learn the definitions of various

quadrilaterals defined by their parallel and

perpendicular sides - To learn the definitions of inductive and

deductive reasoning

Parallel lines are lines in the same plane

that never intersect. They are always the same

distance apart.

Perpendicular lines are lines that meet at a

right angle, that is, at an angle that measures

90.

Slopes

- A rectangle has two pairs of parallel line

segments and four right angles. When you draw a

rectangle on the coordinate plane and notice the

slopes of its sides, you will discover how the

slopes of parallel and perpendicular lines are

related. - Draw coordinate axes centered on graph paper.

Each member of your group should choose one of

the following sets of points. Plot the points and

connect them, in order, to form a closed polygon.

You should have formed a rectangle. - a. A(6, 20), B(13, 11), C(-5, -3), D(-12, 6)
- b. A(3, 1), B(3, 7), C(9, 16), D(15, 8)
- c. A(11, 21), B(17, 11), C(12, 3), D(16, 7)
- d. A(3, 10), B(5, 22), C(7, 25), D(15, 7)

- The slope of a line segment is the same as the

slope of the line containing the segment.

- What conjecture can you make about the slopes of

parallel lines based on your answers to the

previous two steps?

- To find the reciprocal of a number, you write the

number as a fraction and then invert it. - For example, the reciprocal of 2/3 is 3/2.
- The product of reciprocals is 1.

- Express the slope values of AB and BC as reduced

fractions. - Express the slope values of AD and DC as reduced

fractions. - What conjecture can you make about the slopes of

perpendicular lines? What is their product? Check

your conjecture by finding the slopes of any

other pair of perpendicular sides in your

rectangle. - On the coordinate plane, draw two new pairs of

parallel lines that have the slope relationship

you discovered in in this investigation. What

figure is formed where the two pairs of lines

intersect?

In the investigation, you made conjectures based

on studying examples. When you do this, you are

using inductive reasoning.

Example A

- Triangle ABC (written as ?ABC) is formed by

connecting the points (1, 3), (9, 5), and (10,

1). Is it a right triangle?

In Example A you used the fact that perpendicular

lines have opposite reciprocal slopes to

determine that ABC is a right triangle. The

process of showing that certain statements or

conclusions follow logically from an initial

assumption or fact is called deductive reasoning.

Example B

- Classify as specifically as possible the polygon

formed by the pointsA(-4, 1), B(-2, 4), C(4,

0), and D(-1, -1).

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