• 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 thatnever intersect. They are always the same distanceapart.

Perpendicular lines are lines that meet at a right angle, that is, at an angle that measures 90°.

SlopesA 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?

AD BC•Find the slopes of and AB CD• Find the slopes of and

AD

You can write the segment between A and D as .

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.

3 21

2 3

• 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).