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Connecting Geometry to Advanced Placement* Mathematics
A Resource and Strategy Guide
Logical Reasoning (“if-then” Statements)
Objective:
Students will write conditional statements, including converse, inverse, and contrapositive in
geometry.
Connections to Previous Learning:
Classification of quadrilaterals, points and angles.
Connections to AP*:
This lesson develops the logical reasoning capabilities students need to be successful in AP classes,
math in particular.
Materials:
Student Activity pages
Teacher Notes:
In AP calculus students have to deal with statements like “if a function is differentiable, then it is
continuous”. In Pre-AP geometry, students should have the opportunity to write conditional
statements and determine the truth of those statements, providing counterexamples when the
statement is false.
A counterexample is a situation when the hypothesis (if part) is true and the conclusion (then part) is
false. Students often have difficulty writing counterexamples.
Given the conditional “if p then q”
The converse of the conditional is “if q then p”.
The inverse of the conditional is “if not p then not q”.
The contrapositive of the conditional is “if not q then not p”.
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Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
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Student Activity
Logical Reasoning (“if-then” Statements)
I.
Determine if the following conditional is true or false, provide a counterexample if the
statement is false. All points are in the Euclidean Plane.
1.
If AM = MB, then M is the midpoint of AB .
2.
If AB = 2PB, then P is the midpoint of AB .
3.
If two angles are supplementary then they are equal.
4.
If a figure is a square, then it is a quadrilateral.
5.
If a quadrilateral has four right angles then it is a square.
6.
If a quadrilateral is a parallelogram then it is a rectangle.
7.
In a plane, if two lines are perpendicular to the same line, then they are parallel.
8.
If a quadrilateral is a square, then it is a rectangle.
II.
Write the converse, inverse and contra-positive conditionals for the true problems above. State
whether each conditional is true or false and provide a counterexample when the statement is
false.
III.
Notice in the conditionals in part 2 the contrapositive is true. In general a contrapositive is
true if the original if-then statement is true and is false if the original if-then statement is false.
Choose one of the false if-then statements above and write the contrapositive. Give a
counterexample to confirm the contrapositive is false.
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Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
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Connecting Geometry to Advanced Placement* Mathematics
A Resource and Strategy Guide
Logical Reasoning (“if-then” Statements)
Answers:
I.
1.
False
2.
False
3.
False
4.
True
5.
False – Rectangle
6. False - Rhombus
7.
True
8. True
4.
Converse - If a figure is a quadrilateral, then it is a square. False, one counterexample is
a trapezoid.
Inverse - If a figure is not a square, then it is not a quadrilateral. False, one
counterexample is a trapezoid
Contrapositive - If a figure is not a quadrilateral then it is not a square. True
Converse - In a plane, if two lines are parallel and intersect another line, then they are
perpendicular to the intersected line. False
Inverse - In a plane, if two lines are not perpendicular to a line, then they are not parallel.
False
Contrapositive - In a plane, if two lines are not parallel and intersect another line, then
they are not perpendicular to the intersected line. True
Converse - If a quadrilateral is a rectangle then it is a square. False
Inverse - If a quadrilateral is not a square then it is not a rectangle. False
Contrapositive - If a quadrilateral is not a rectangle, then it is not a square. True
II.
7.
8.
III.
Answers will vary depending upon the student’s choice of statements. Using f, the
contrapositive is “if a quadrilateral is not a rectangle then it is not a parallelogram”. The
counterexample is a parallelogram because it is not a rectangle but it is a parallelogram.
®
Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
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