Download Deductive Reasoning and Proof Worksheets

Deductive Reasoning
Using symbolic notation to write conditional statements.
p – represents the hypothesis
q – represents the conclusion
conditional statement: “If p, then q” can be written as “ p → q. ”
converse: “If q, then p” can be written as “ q → p. ”
biconditional statement: “p if and only if q” can be written as “ p ↔ q. ”
negation: “Not p” can be written as “ ~ p. ”
inverse: “If not p, then not q” can be written as “ ~ p→ ~ q. ”
contrapositive: “If not q, then not p” can be written as “ ~ q→ ~ p. ”
You try it! Write the converse, q → p, the inverse, ~ p→ ~ q, and the
contrapositive, ~ q→ ~ p. Is the biconditional statement, p ↔ q, true?
Let p be “the value of x is –4” and q be “the square of x is 16.”
converse – If the square of x is 16, then the value of x is –4.
inverse – If the value of x is not –4, then the square of x is not 16.
contrapositive – If the square of x is not 16, then the value of x is not –4.
The biconditional statement is not true since the converse is not true.
A counterexample is the case where x = 4.
deductive reasoning – uses facts, definitions, and accepted properties in order to write a
logical argument.
Example:
Josh knows that Dell Computers cost less than Sony Vaios. He also
knows that Vaio computers cost less than Macs. Josh reasons that
Dells costs less than Macs.
inductive reasoning – uses previous examples and patterns to form a conjecture.
Example:
Josh knows that Dell computers cost less than Vaios. All other brands
of computers that Josh knows of cost less than Dells. Josh reasons that
Vaios costs more than all other brands.
Law of Detachment: If p→ q is a true conditional statement and p is true, then q is true.
Example:
If two angles form a linear pair, then they are supplementary;
∠A and ∠B are a linear pair. So, ∠A and ∠B are supplementary.
Law of Syllogism: If p→ q and q→ r are true conditional statements, then p→ r is true.
Example:
Use the following true statements to write a conditional statement using
the Law of Syllogism.
If a fish swims at 68 mi/h, then it swims at 110 km/h.
If a fish can swim at 110 km/h, then it is a sailfish.
If a fish is the largest species of fish, then it is a great white shark.
If a fish weighs over 2000 pounds, then it is the largest species of fish.
If a fish is the fastest species of fish, then it can reach speeds of 68 mi/h.
CONNECTION: Deductive reasoning uses the Law of Syllogism to form logical
arguments.
Name:__________________________
Date:___________________________
Period:_________________________
Deductive Reasoning Worksheet
Single underline the hypothesis and double underline the conclusion of each conditional.
1. If
, then
.
2. I can’t sleep if I’m not tired.
3. I’ll try if you will.
4. If
then
5.
implies
6. If
, then
is a right angle.
Rewrite the pair of conditionals as a biconditional.
7. If B is between A and C, then AB + BC =AC.
If AB + BC =AC, then B is between A and C.
___________________________________________________________________________________
8. If
If
, then
is a straight angle.
is a straight angle, then
.
___________________________________________________________________________________
Write each biconditional as two conditionals that are converses of eachother.
9. Points are collinear if and only if they all lie in one line.
______________________________________________________________________________________
______________________________________________________________________________________
10. Points lie in one plane if and only if they are coplanar.
______________________________________________________________________________________
______________________________________________________________________________________
Provide a counter example to show that that each statement is false. You may use words or a diagram.
11. If
12. If
, then
, then
13. If point G is on
14. If
, then
.
.
, then G is on
.
.
15. If a four sided figure has four right angles, then it has four congruent sides.
16. If a four sided figure has four congruent sides, then it has four right angles.
Tell whether each statement is true or false. Then write the converse and tell whether its true or false.
17. If
, then
.
18. If
, then
.
19. If
20. If
, then
.
, then
is not obtuse.
21. If Pam lives in Chicago, then she lives in Illinois.
22. If
, then
.
23.
if
.
24. If
, then
25. If
, then
26.
implies that
.
.
or
.
27. If points D, E, and F and collinear, then DE + EF = DF.
28. P is the midpoint of
implies that GH = 2PG.
Write a definition of congruent angles as a biconditional.
______________________________________________________________________________________
Write a definition of a right angle as a biconditional.
______________________________________________________________________________________
What can you conclude if the following sentences are all true.
(i) If p, then q.
(ii) p
(iii) If q, then not r.
(iv) s or r.
Reasoning with Properties from Algebra
Review of Algebraic Properties of Equality
Name
Property
Addition Property
If a = b, then a + c = b + c.
Subtraction Property
If a = b, then a – c = b – c.
Multiplication Property If a = b, then a • c = b • c.
Division Property
If a = b and c ≠ 0, then a ÷ c = b ÷ c.
Reflexive Property
For any real number a, a = a.
Substitution Property
If a = b, then a can be substituted for b in
any equation or expression.
If a = b and b = c, then a = c.
Applying Algebraic Properties to Geometry
Property
Segment Length
Angle Measure
Reflexive
For any segment AB, AB=AB.
For any ∠A, m∠A=m∠A.
Symmetric
If AB=CD, then CD=AB.
If m∠A=m∠B, then m∠B=m∠A.
Substitution
If AB=CD and CD=EF,
then AB=EF.
If m∠A=m∠B and m∠B=m∠C,
then m∠A=m∠C.
Example:
The formula to convert Fahrenheit to Celsius is
. Solve the
equation for F and write the reason for each step. Then use the result to find
the Fahrenheit temperature at 24° C.
Solve for F:
Find °F at 24°C.
Reasons: