REASONING AND PROOF WORKSHEETS

About "Reasoning and proof worksheets"

Reasoning and Proof Worksheets :

Worksheet given in this section is much useful to the students who would like to practice problems on "Reasoning and Proof"

Reasoning and proof worksheets - Problems

Problem 1 :

Sketch the next figure in the pattern.

Problem 2 :

Write (a) inverse, (b) converse, (c) contrapositive of the following statement. 

Statement :

"If there is snow on the ground, the flowers are not in bloom"   

Problem 3 :

Each of the following statements is true. Write the converse of each statementand decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample.

(i) If two points lie in a plane, then the line containing them lies in the plane.

(ii) If a number ends in 0, then the number is divisible by 5.

Problem 4 :

Let p be "the value of x is -5" and let q be "the absolute value of x is 5". 

(i) Write p -> q in words. 

(ii) Write q -> p in words. 

(iii) Decide whether the biconditional statement p <-> q is true. 

Problem 5 :

Solve 52y - 3(12 + 9y)  =  64 and write a reason for each step. 

Problem 6 : 

In the diagram given below, Q is the midpoint of PR. 

Show that PQ and QR are each equal to 1/2 ⋅ PR. 

Problem 7 :

In the diagram shown below, 

∠1 and ∠2 are supplements,

∠3 and ∠4 are supplements,

∠1 ≅ ∠4

Prove ∠2 ≅ ∠3

Reasoning and proof worksheets - Solution

Problem 1 :

Sketch the next figure in the pattern.

Solution : 

Each figure in the pattern looks like the previous figure with another row of squares added to the bottom. Each figure looks like a stairway. 

So, the sixth figure in the pattern must have six squares in the bottom row.  

Problem 2 :

Write (a) inverse, (b) converse, (c) contrapositive of the following statement. 

Statement :

"If there is snow on the ground, the flowers are not in bloom"   

Solution : 

(a) Inverse : 

"If there is no snow on the ground, the flowers are in bloom" 

(b) Converse : 

"If flowers are not in bloom, then there is snow on the ground"

(b) Contrapositive : 

"If flowers are in bloom, then there is no snow on the ground"

Problem 3 :

Each of the following statements is true. Write the converse of each statementand decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample.

(i) If two points lie in a plane, then the line containing them lies in the plane.

(ii) If a number ends in 0, then the number is divisible by 5.

Solution :

Solution (i) :

Converse :

(i) If a line containing two points lies in a plane, then the points lie in the plane.

The converse is true, as shown in the diagram. So, it can be combined with the original statement to form the true biconditional statement written below.

Biconditional statement :

Two points lie in a plane, if and only if the line containing them lies in the plane.

Solution (ii) :

Converse :

If a number is divisible by 5, then the number ends in 0. The  converse is false. As a counterexample, consider the number 15. It is divisible by 5, but it does not end in 0, as shown below.

20 ÷ 5  =  4

25 ÷ 5  =  5

30 ÷ 5  =  6

Problem 4 :

Let p be "the value of x is -5" and let q be "the absolute value of x is 5". 

(i) Write p -> q in words. 

(ii) Write q -> p in words. 

(iii) Decide whether the biconditional statement p <-> q is true. 

Solution : 

(i) If the value of x is -5, then the absolute value of x is 5.

(ii) If the absolute value of x is 5, then the value of x is -5.

(iii) The conditional statement in part (a) is true, but its converse in part (b) is false. So, the biconditional statement p <-> q is false. 

Problem 5 :

Solve 52y - 3(12 + 9y)  =  64 and write a reason for each step. 

Solution : 

Given :

52y - 3(12 + 9y)  =  64

Distributive Property :

Distribute 3 to 12 and 9y. 

52y - 36 - 27y  =  64

Simplify :

25y - 36  =  64

Addition property of equality : 

Add 36 to each side. 

25y  =  100

Division property of equality : 

Divide both sides by 25. 

y  =  4

Problem 6 : 

In the diagram given below, Q is the midpoint of PR. 

Show that PQ and QR are each equal to 1/2 ⋅ PR. 

Solution : 

Given : Q is the midpoint of PR

Prove : PQ  =  1/2 ⋅ PR and QR  =  1/2 ⋅ PR

Statements

aaaa Q is the aaaa a midpoint of PR

PQ = QR

PQ + QR = PR

PQ + PQ = PR

⋅ PQ = PR

PQ = 1/2 ⋅ PR

QR = 1/2 ⋅ PR

Reasons

Given aaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaa

Definition of midpoint

Segment Addition Postulate

Substitution property of equality

Distributive property

Division property of equality

Substitution property of equality

Problem 7 :

In the diagram shown below, 

∠1 and ∠2 are supplements,

∠3 and ∠4 are supplements,

∠1 ≅ ∠4

Prove ∠2 ≅ ∠3

Solution :

Statements

∠1 and ∠2 are supplements

∠3 and ∠4 are supplements

∠2 ≅ ∠3

 m∠1 + m∠2 = 180°   m∠3 + m∠4 = 180° 

m∠1  =  m∠4

a ∠1 + ∠2 = ∠3 + ∠1 aaaaaa 

m∠2 = m∠3

∠2 ≅ ∠3

Reasons

aaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa

Given aaaaaaaaaaaaaaaaaaaaaa aaaaaa


Definition of Supplementary angles aaaaaaaaaaaaaaaaaaaa 

Definition of congruent angles

Substitution property of equality aaaaaaaaaaaaaaaaaa

Subtraction property of equality

Definition of congruent angles

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ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

Solving quadratic equations by factoring

Solving quadratic equations by quadratic formula

Solving quadratic equations by completing square

Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equations 

Algebraic identities

Solving absolute value equations 

Solving Absolute value inequalities

Graphing absolute value equations  

Combining like terms

Square root of polynomials 

HCF and LCM 

Remainder theorem

Synthetic division

Logarithmic problems

Simplifying radical expression

Comparing surds

Simplifying logarithmic expressions

Negative exponents rules

Scientific notations

Exponents and power

COMPETITIVE EXAMS

Quantitative aptitude

Multiplication tricks

APTITUDE TESTS ONLINE

Aptitude test online

ACT MATH ONLINE TEST

Test - I

Test - II

TRANSFORMATIONS OF FUNCTIONS

Horizontal translation

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Reflection through x -axis

Reflection through y -axis

Horizontal expansion and compression

Vertical  expansion and compression

Rotation transformation

Geometry transformation

Translation transformation

Dilation transformation matrix

Transformations using matrices

ORDER OF OPERATIONS

BODMAS Rule

PEMDAS Rule

WORKSHEETS

Converting customary units worksheet

Converting metric units worksheet

Decimal representation worksheets

Double facts worksheets

Missing addend worksheets

Mensuration worksheets

Geometry worksheets

Comparing  rates worksheet

Customary units worksheet

Metric units worksheet

Complementary and supplementary worksheet

Complementary and supplementary word problems worksheet

Area and perimeter worksheets

Sum of the angles in a triangle is 180 degree worksheet

Types of angles worksheet

Properties of parallelogram worksheet

Proving triangle congruence worksheet

Special line segments in triangles worksheet

Proving trigonometric identities worksheet

Properties of triangle worksheet

Estimating percent worksheets

Quadratic equations word problems worksheet

Integers and absolute value worksheets

Decimal place value worksheets

Distributive property of multiplication worksheet - I

Distributive property of multiplication worksheet - II

Writing and evaluating expressions worksheet

Nature of the roots of a quadratic equation worksheets

Determine if the relationship is proportional worksheet

TRIGONOMETRY

SOHCAHTOA

Trigonometric ratio table

Problems on trigonometric ratios

Trigonometric ratios of some specific angles

ASTC formula

All silver tea cups

All students take calculus 

All sin tan cos rule

Trigonometric ratios of some negative angles

Trigonometric ratios of 90 degree minus theta

Trigonometric ratios of 90 degree plus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 180 degree minus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 270 degree minus theta

Trigonometric ratios of 270 degree plus theta

Trigonometric ratios of angles greater than or equal to 360 degree

Trigonometric ratios of complementary angles

Trigonometric ratios of supplementary angles 

Trigonometric identities 

Problems on trigonometric identities 

Trigonometry heights and distances

Domain and range of trigonometric functions 

Domain and range of inverse  trigonometric functions

Solving word problems in trigonometry

Pythagorean theorem

MENSURATION

Mensuration formulas

Area and perimeter

Volume

GEOMETRY

Types of angles 

Types of triangles

Properties of triangle

Sum of the angle in a triangle is 180 degree

Properties of parallelogram

Construction of triangles - I 

Construction of triangles - II

Construction of triangles - III

Construction of angles - I 

Construction of angles - II

Construction angle bisector

Construction of perpendicular

Construction of perpendicular bisector

Geometry dictionary

Geometry questions 

Angle bisector theorem

Basic proportionality theorem

ANALYTICAL GEOMETRY

Analytical geometry formulas

Distance between two points

Different forms equations of straight lines

Point of intersection

Slope of the line 

Perpendicular distance

Midpoint

Area of triangle

Area of quadrilateral

Parabola

CALCULATORS

Matrix Calculators

Analytical geometry calculators

Statistics calculators

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Chemistry periodic calculator

MATH FOR KIDS

Missing addend 

Double facts 

Doubles word problems

LIFE MATHEMATICS

Direct proportion and inverse proportion

Constant of proportionality 

Unitary method direct variation

Unitary method inverse variation

Unitary method time and work

SYMMETRY

Order of rotational symmetry

Order of rotational symmetry of a circle

Order of rotational symmetry of a square

Lines of symmetry

CONVERSIONS

Converting metric units

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WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

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Algebra word problems

Word problems on trains

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Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

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Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6