GEOMETRY
Conditionals,
Biconditionals,
Counterexamples & the
Law of Detachment
Matching Activity
Print Resource
Summary
Goals
• Students will write conditional, converse, and biconditional statements.
• Students will determine whether a conditional statement is true, and
provide a counterexample if it is false.
• Students will use the Law of Detachment to draw conclusions.
Teacher Notes
• This resource contains 3 activities:
• Conditionals & Biconditionals (p. 3-5; KEY p. 8)
• Law of Detachment (p. 6; KEY p. 9)
• Converses & Counterexamples (p. 7; KEY p. 10)
• The Conditionals & Biconditionals activity is recommended for use
as partner or small group work.
• Review the vocabulary prior to assigning this activity.
Vocabulary
acute angle, biconditional, conclusion, conditional, converse, hypothesis,
linear pair, parallel, rectangle, square
Materials
• Scissors and glue (for Conditionals & Biconditionals matching activity
only)
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© 2021 Carrie King (Sum-It Learning)
Conditionals & Biconditionals
Directions
1.
Cut out each conclusion. Cut out the “if and only if”, “If”, and “then”
pieces.
2. Match the conclusion pieces with the corresponding hypothesis.
Lay them next to each other, but don’t glue yet!
For each statement, follow steps #3-5:
3. Determine whether you could combine the statements to write a
biconditional. Use the “If” and “then” pieces as you rearrange the
hypothesis and conclusion pieces to see if both the statement
and its converse are true.
4. Once you have decided whether the statement can be written as
a biconditional, glue the conclusion next to the hypothesis. Do
NOT glue the “If” and “then” pieces!
5. Write the words “if and only if” if it is a biconditional. Use the
words “if” and “then” if it is a conditional. See the example below.
Example:
Hypothesis- You play the flute
Conclusion- you are a musician
Conditional: If you play the flute, then you are a musician (TRUE)
Converse: If you are a musician, then you play the flue (FALSE)
Since the converse is false, this cannot be written as a biconditional.
Final answer: If you play the flute, then you are a musician.
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© 2021 Carrie King (Sum-It Learning)
Conditionals & Biconditionals
1) ______ Two segments are congruent
2) ______ An integer is a multiple of 10
3) ______ A figure is a square
4) ______ Two lines are equidistant
5) ______ A vegetable is a carrot
6) ______ A vehicle is a car
7) ______ A figure has six sides
8) ______ Two angles form a linear pair
9) ______ An angle has a measure less than 90°
10) ______ A figure is a rectangle
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© 2021 Carrie King (Sum-It Learning)
Conclusions
___________ the last digit is 0.
If
___________ it is an acute angle.
___________ they have the same measure.
___________ it is a hexagon.
___________ it is orange.
then
if and only if
___________ it has two pairs of parallel sides.
___________ it is a vehicle with four wheels.
___________ it has four congruent sides.
___________ the angles sum to 180°.
___________ the two lines are parallel.
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© 2021 Carrie King (Sum-It Learning)
Law of Detachment
Use the given conditional, hypothesis, and the Law of Detachment to draw a
conclusion.
1.
Conditional: If a figure is a square, then it has four congruent sides.
Hypothesis: Figure MATH is a square.
Conclusion:
Write the conclusion symbolically:
2.
Conditional: If an angle has a measure less than 90°, then the angle is
acute.
Hypothesis: Angle C has a measure of 88°.
Conclusion:
3.
Conditional: If two lines are equidistant, then the lines are parallel.
Hypothesis: Lines JA and CR are equidistant.
Conclusion:
Write the conclusion symbolically:
4.
Conditional: If a figure is a rectangle, then it has two pairs of parallel
sides.
Hypothesis: Figure STAR is a rectangle.
Conclusion:
Write the conclusion symbolically:
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© 2021 Carrie King (Sum-It Learning)
Converses & Counterexamples
Write the converse of each conditional statement from the Conditionals &
Biconditionals Activity.. Indicate whether the statement is true or false. If the
statement is false, provide a counterexample.
1)
Converse: __________________________________________________________________
True/False
Counterexample:
2) Converse: __________________________________________________________________
True/False
Counterexample:
3) Converse: __________________________________________________________________
True/False
Counterexample:
4) Converse: __________________________________________________________________
True/False
Counterexample:
5) Converse: __________________________________________________________________
True/False
Counterexample:
6) Converse: __________________________________________________________________
True/False
Counterexample:
7) Converse: __________________________________________________________________
True/False
Counterexample:
8) Converse: __________________________________________________________________
True/False
Counterexample:
9) Converse: __________________________________________________________________
True/False
Counterexample:
10) Converse: __________________________________________________________________
True/False
Counterexample:
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© 2021 Carrie King (Sum-It Learning)
Conditionals & Biconditionals
KEY
1) ______ Two segments are congruent if and only if they have the same measure.
2) ______ An integer is a multiple of 10
if and only if the last digit is 0.
3) If A figure is a square then it has four congruent sides.
4) ______ Two lines are equidistant if and only if the two lines are parallel.
5) If A vegetable is a carrot then it is orange.
6) If A vehicle is a car then it is a vehicle with four wheels.
7) ______ A figure has six sides if and only if it is a hexagon.
8) If Two angles form a linear pair then the angles sum to 180°.
9) ______ An angle has a measure less than 90° if and only if it is an acute angle.
10) If A figure is a rectangle then it has two pairs of parallel sides.
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© 2021 Carrie King (Sum-It Learning)
Law of Detachment
KEY
Use the given conditional, hypothesis, and the Law of Detachment to draw a
conclusion.
1.
Conditional: If a figure is a square, then it has four congruent sides.
Hypothesis: Figure MATH is a square.
Conclusion: Figure MATH has four congruent sides.
Write the conclusion symbolically: 𝑀𝐴 ≅ 𝐴𝑇 ≅ 𝑇𝐻 ≅ 𝐻𝐴
2.
Conditional: If an angle has a measure less than 90°, then the angle is
acute.
Hypothesis: Angle C has a measure of 88°.
Conclusion: Angle C is an acute angle.
3.
Conditional: If two lines are equidistant, then the lines are parallel.
Hypothesis: Lines JA and CR are equidistant.
Conclusion: Line JA and line CR are parallel.
Write the conclusion symbolically: 𝐽𝐴 ฮ 𝐶𝑅
4.
Conditional: If a figure is a rectangle, then it has two pairs of parallel
sides.
Hypothesis: Figure STAR is a rectangle.
Conclusion: Figure STAR has two pairs of parallel sides.
Write the conclusion symbolically: 𝑆𝑇 ฮ 𝐴𝑅 and 𝑇𝐴 ฮ 𝑅𝑆
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© 2021 Carrie King (Sum-It Learning)
Converses & Counterexamples
KEY
Write the converse of each conditional statement from the Conditionals &
Biconditionals Activity. Indicate whether the statement is true or false. If the
statement is false, provide a counterexample. Note: Counterexamples will vary.
1)
Converse: If two segments have the same measure, then they are congruent.
True/False
Counterexample:
2) Converse: If an integer has a last digit of 0, then it is a multiple of 10.
True/False
Counterexample:
3) Converse: If a figure has four congruent sides, then it is a square
True/False
Counterexample: A rhombus also has 4 congruent sides,
4) Converse: If two lines are parallel, then they are equidistant.
True/False
Counterexample:
5) Converse: If a vegetable is orange, then it is a carrot.
True/False
Counterexample: Other orange vegetables include sweet potato and pumpkin.
6) Converse: If a vehicle has four wheels, the vehicle is a car.
True/False
Counterexample: Other vehicles with 4 wheels include: go-cart, golf cart, truck.
7) Converse: If two lines are parallel, then they are equidistant.
True/False
Counterexample:
8) Converse: If two angles sum to 180°, then the angles form a linear pair,
True/False
Counterexample: The two angles don’t have to be adjacent. See diagram below:
9) Converse: If an angle is acute, then it has a measure less than 90°.
True/False
Counterexample:
10) Converse: If a figure has two pairs of parallel lines, then it is a rectangle.
True/False
Counterexample: A parallelogram also has two pairs of parallel lines.
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© 2021 Carrie King (Sum-It Learning)
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fun! If you enjoyed this resource, please leave feedback in my TpT store. If you
have questions or comments, please email me at carrie.sum.it@gmail.com.
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© 2021 Carrie King (Sum-It Learning)