3.1: Conditional Statements
This activity is intended to supplement Geometry, Chapter 2, Lesson 2.
ID: 8746
Time required: 40 minutes
Activity Overview
In this activity, students will write logical statements related to the given conditional statement. They will explore whether the statements are true or false and find counterexamples for false statements. These explorations will involve the slopes of parallel and perpendicular lines and lengths of collinear and noncollinear segments.
Topic: Inductive & Deductive Reasoning
 Write the inverse, converse, and contrapositive statements corresponding to a given conditional statement.
 Use a counterexample to prove that a statement is false.
Teacher Preparation and Notes
 This activity is designed to be used in a high school or middle school geometry classroom.
 Before beginning this activity, students should be familiar with the terms inverse, converse, and contrapositive.

Students will discover the following concepts:
 Parallel lines have slopes that are equal; perpendicular lines have slopes that are opposite reciprocals (the product of the slopes is 1).

The Segment Addition Postulate states that
AB+BC=AC ifB is betweenA andC and the points are collinear. If the points are not collinear, thenAB+BC>AC .
 If desired, teachers can explore which of the statements in the activity are also biconditional statements (definitions that are always true).
 This activity is designed to be studentcentered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.
 To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
 To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=8746 and select COLSEG.8xv and NOCOLSEG.8xv.
Associated Materials
 Student Worksheet: Conditional Statements http://www.ck12.org/flexr/chapter/9687
 Cabri Jr. Application
 COLSEG.8xv and NOCOLSEG.8xv
Problem 1 – Slopes of lines
To begin, students should open a new Cabri Jr. file.
Step 1: Students will need to construct two parallel lines. First, a line needs to be constructed using the Line tool and a point not on the line using the Point tool.
Step 2: Using the Parallel tool, a line parallel to the existing line should be constructed through the point not on the original line.
Step 3: Students will find the slope of both lines by using the Slope tool (Measure > Slope). Students can now use the Hand tool to drag the original line or the point and observe the results.
What is true of the slopes of parallel lines?
Students should record their observations on the worksheet part
Students will next construct perpendicular lines in a new Cabri Jr. file.
Step 4: Again construct a line and a point not on the line. Have students select the Perp. tool to construct a line perpendicular to the existing line through the point not on the original line.
Step 5: Students will find the slope of both lines. They should drag the original line or the point and observe the results.
Step 6: Have students select Calculate to find the product of the slopes. They should again drag the line to observe the results.
What is true of the slopes of perpendicular lines?
Students should record their observations, write conditional statements, and determine the truth of the statements on the worksheet in part
On a new Cabri Jr. file, students will construct two lines that have the same
Step 7: Students should start by placing a point on the
Step 8: Make sure that students construct two different lines with the same
Note: If desired, students can display the equations of both lines using the Coord. & Eq. tool instead of the slopes.
Step 9: Direct students to drag the lines and the
Students will write conditional statements on the worksheet part
The screenshot at right shows a counterexample for the converse statement “If two lines have different slopes, then they have the same
Problem 2 – Collinear and noncollinear segments
Step 1: Distribute the file COLSEG to students. Points
Step 2: Students will drag the points with the Hand tool and observe the changes in the lengths.
Be sure that students drag all three points, and change the order of the points (they should investigate situations where
Students should record their observations on the worksheet and write a conditional statement.
Step 3: Distribute the file NOCOLSEG to students. Points
Step 4: Students should now drag the points and observe the changes in the distances. If desired, students can use the Calculate tool to find the sum of
Students should record observations on the worksheet and write a conditional statement.
Solutions
Problem 1
A. If two lines are parallel, then the slopes of the lines are equal.
Converse: If the slopes of the lines are equal, then the two lines are parallel.
Inverse: If the two lines are not parallel, then the slopes of the lines are not equal.
Contrapositive: If the slopes of the lines are not equal, then the two lines are not parallel.
Each conditional statement is true.
B. If two lines are perpendicular, then the slopes of the lines are equal to 1.
Converse: If the slopes of two lines are equal to 1, then the lines are perpendicular.
Inverse: If two lines are not perpendicular, then the slopes of the lines are not equal to 1.
Contrapositive: If the slopes of two lines are not equal to 1, then the lines are not perpendicular.
Each conditional statement is true.
C. Converse: If two lines have different slopes, then the lines have the same
Inverse: If two different lines do not have the same
Contrapositive: If two lines do not have different slopes, then the different lines do not have the same
Each conditional statement is false.
Problem 2
A. Answers will vary for distances of
The lengths
If
B. Answers will vary for distances of
\begin{align*}AB + BC > AC\end{align*}
If \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} are not collinear, then the sum of the lengths of \begin{align*}AB\end{align*} and \begin{align*}BC\end{align*} is greater than \begin{align*}AC\end{align*}.
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