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Conditional statements and converse - class-VIII

Description: conditional statements and converse
Number of Questions: 57
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Tags: proof of mathematical statements business maths mathematical logic introduction to euclid's geometry maths basic concepts in geometry the elements of geometry concept of vertically opposite angles axioms, postulates and theorems english discrete mathematics vocabulary similar triangles
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$\angle A=\angle B$ and $\angle B=\angle C$, According to which axiom of Euclid the relation between $\angle A$ and $\angle C$ is established?

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. I

  2. II

  3. III

  4. IV

Show answer
Correct Option: A
Explanation:

Given that $\quad \angle A=\angle B\quad & \quad \angle B=\angle C.\quad $

Then, according to Euclid's first axiom, which states that 
"things which are   equal to the same thing are also equal to each other",
 $\quad \angle A=\angle C\quad $
Ans- Option A.

Euclid's fourth axiom says that everything equals itself.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: A
Explanation:

Euclid's fourth axiom states that "things which coincide with one another are equal to one another."

So the given statement is true by the axiom IV.
Ans- Option A.

The boundaries of the solids are called curves.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Ambiguous

  4. Data Insufficient

Show answer
Correct Option: B
Explanation:

The boundaries of the solids are called surfaces.

While the boundaries of the surfaces are called curves.

The Euclidean geometry is valid only for figures in the plane.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Ambiguous

  4. Data Insufficient

Show answer
Correct Option: A
Explanation:

The given statement is true. Because,  by Einstein's  theory of  general  relativity, physical space itself is not Euclidean. Euclidean space is a good approximation  for it where the gravitational field is weak.  

So, in space or in multidimensional space the Euclidean axioms are not applicable.
Ans- Option A.

The total number of propositions in the Elements are

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $465$

  2. $460$

  3. $13$

  4. $55$

Show answer
Correct Option: A
Explanation:
Elements is the book written by a mathematician Euclid's and it has $465$ propositions , $131$ definitions $5$ postulates of Euclid's and $5$ common notions are written 
Therefore, there are $465$ propositions in the elements.

Euclid belongs to the country

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Babylonia

  2. Egypt

  3. Greece

  4. India

Show answer
Correct Option: C
Explanation:

Euclid belongs to Greece.

Ans- Option C.

The edges of a surface are called curves.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Ambiguous

  4. Data Insufficient

Show answer
Correct Option: A
Explanation:

The edges of a surface are called curves.

For example, circle is a plane surface and its edge is the circumference, which is a curve

Two salesmen make equal during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Equal sales in September

  2. Unequal sales in September

  3. Ambiguous

  4. None of the above

Show answer
Correct Option: A
Explanation:

Let each salesman make Rs. $x$ in August.
In September, the sale of each salesman is Rs. $2x$.
According to Euclids sixth axiom, things which are double of the same thing are equal to one another, the wholes are equal.
So, the sales of each salesman are equal.

The things which are double of the same thing are equal to one another.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Ambiguous

  4. Data Insufficient

Show answer
Correct Option: A
Explanation:

Let a=2x   and  b=2x  when a, b & x are arbritary numbers or things. 

The first axiom of Euclid states that
 "things which are equal to the same thing are also equal to each other.
So a=b.
i.e the given statement is true.
Ans- Option A.

If equals are added to equals, then the wholes are .......

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. unequal

  2. equal

  3. sometimes equal sometimes unequal

  4. nearest to each other

Show answer
Correct Option: B
Explanation:

According to the first axiom of Euclid " if equals are added to equals the wholes are equal."

Ans- Option B.

State whether the following statements are true or false
A finite line can be extended on its both sides endlessly to get a straight line

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

True,

As per Euclid conceived idea in second axiom a finite line can be extended on its both sides endlessly to get a straight line

State whether the following statements are true or false:
Only one line can pass through a given point.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Infinite lines can pass through a given point.

So, the statement is false.

Two distinct points in a plane determine ______ lines.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Unique

  2. Two

  3. Three

  4. None of these

Show answer
Correct Option: A
Explanation:

According to Euclid's Axioms, For every two points, $A,\,B$ there exists no more than one line that contains each of the points $A,B$.

Therefore, a unique line can be made from two distinct points.

Things which are equal to the same thing are _____ to one another.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Perpendicular

  2. Not equal

  3. Equal

  4. Parallel

Show answer
Correct Option: C
Explanation:

Let $A$ and $B$ both be equal to $C$

$\Rightarrow A=C ; B=C$

From this we can clearly say that, $A=B=C$
Hence, things which are equal to the same thing must be equal to one another.

By applying Euclid's division lemma $72$ and $28$ can be expressed as

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $28 = (72 - 16) \times 2$

  2. $72 = (28 \times 2) + 16$

  3. $72 = (28 \times 2) - 16$

  4. $16 = 72 - (28 + 2)$

Show answer
Correct Option: B
Explanation:

Solution:

According to Euclid's division lemma if $a$ and $b$ are two numbers then they can be expressed as $b=ap+r.$
Therfore,
$72$ and $28$ can be expressed as
$72=(28\times2)+16$
So, $B$ is the correct option. 

Euclidean geometry is valid only for curved surfaces.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Sometimes True

  4. Data Insufficient

Show answer
Correct Option: B
Explanation:

Euclid's postulates:

$\rightarrow$A straight line can be drawn joining any two points.
$\rightarrow$A straight line segment can be extended indefinitely in a straight line.
$\rightarrow$A circle can be drawn having segment as radius and one endpoint as center. 
$\rightarrow$All right angles are congruent and equal.
$\rightarrow$Parallel postulate.

According to Euclid's axioms, the _____ is greater than the part.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Half

  2. Large

  3. Whole

  4. None of these

Show answer
Correct Option: C
Explanation:

According to axiom $5$ of Euclid, whole is greater than the part and it is a universal truth.


Proof:
Let's take whole $3$ and part $\dfrac{1}{3}$
Subtracting, we get
$3-\dfrac{1}{3} = \dfrac{8}{3} > 0$

Hence, Proved that whole is greater than part.

Using Euclid's Division Lemma, for any positive integer $n, n^3-n$ is always divisible by 

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $6$

  2. $4$

  3. $3$

  4. $8$

Show answer
Correct Option: A
Explanation:

$n^{3}-n=n(n^{2}-1)=n(n-1)(n+1)$ is divisible by $3$ then  possible remainder is $0, 1$ and $2$


[$\because$ if $P=ab+r$, Then $0\le r < a$ by Euclid lemma]


$\therefore$ Let $n=3r, 3r+1,3r+2$ where $r$ is an integer

Case $1$: When $n=3r$

Then, $n^{3}-n$ is divisible by $3$  [$\because n^{3}-n=n(n-1)(n+1)=3r(3r-1)(3r+1)$, early shown it is divisible by $3$]

Case $2$: When $n=3r+1$

$n-1=3r+1-1=3r$

Then, $n^{3}-n=(3r+1)(3r)(3r+2)$ it is divisible by $3$

Case: when $n=3r-1$

$m+1=3r-1+1=3r$

Then, $n^{3}-n=(3r-1)(3r-2)(3r)$ it is divisible by $3$

Now out of three $(n-1)^{n}$ and $(n+1)$ are must be even so it is divisible by $2$

$n^{3}-n$ is divisible by $2\times 3=6$

Euclid stated that if equals are subtracted from equals, the remainders are equals in the form of :

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. an axiom

  2. a postulate

  3. a definition

  4. a proof

Show answer
Correct Option: A
Explanation:

The above statement is Euclid's third axiom. So, $A$ is correct.

The things which coincide with one another are:

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. equal to another

  2. unequal

  3. double of same thing

  4. Triple of same things

Show answer
Correct Option: A
Explanation:

According to Euclid's postulates, $equal$ things coincide with each other.

Hence, $A$ is correct.

Euclid's stated that all right angles are equal to each other in the form of :

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. an axiom

  2. a definition

  3. a postulate

  4. a proof

Show answer
Correct Option: C
Explanation:

One of Euclid's five postulates is:

$All$ $right$ $angles$ $are$ $CONGRUENT$.
So, the correct option is  $C$.

Which of the following is Euler's formula?

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $F+V=E+2$

  2. $F+E=V+2$

  3. $F+E-V=2$

  4. $F+2=E+V$

Show answer
Correct Option: A
Explanation:

Euler's formula $F + V = E + 2.$


Option $A$ is the correct answer.

Euclid stated that all right angles are equal to one another in the form of a/an ..........

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Axiom

  2. Defination

  3. Postulate

  4. Proof

Show answer
Correct Option: C
Explanation:

Postulates


1. A straight line may be drawn from any point to any other point.


2. A terminated line (line segment) can be produced indefinitely.
 3. A circle may be described with any centre and any radius.


4. All right angles are equal to one another.


5. If a straight line falling on two straight lines makes the interior angles on the same

side of it, taken together less than two right angles, then the the two straight lines if

produced indefinitely, meet on that side on which the sum of angles is taken together

less than two right angles.

Euclid used the term postulate for the assumptions that were specific to geometry

and otherwise called axioms. A theorem is a mathematical statement whose truth

has been logically established.
Answer (C) Postulate

Select the correct match.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Postulate III $\quad$ A terminated line can be produced indefinitely

  2. Postulate II $\quad$ All right angles are equal to one another

  3. Postulate IV $\quad$ A circle can be drawn with any centre and any radius

  4. Postulate I $\quad$ A straight line may be drawn from any one point to any other point.

Show answer
Correct Option: D
Explanation:

Postulates 
1. A straight line may be drawn from any point to any other point. 
2. A terminated line (line segment) can be produced indefinitely.
 3. A circle may be described with any centre and any radius. 
4. All right angles are equal to one another. 
5. If a straight line falling on two straight lines makes the interior angles on the same side of it, taken together less than two right angles
So, (D) option is correct.
Answer (D) Postulate I  A straight line may be drawn from any one point to any other point.

Which of the following is NOT a Euclid's postulate?

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. We can describe a circle with any center and radius

  2. All right angles are equal to one another

  3. There is a unique line that passes through two given points

  4. Through a point not on a given line, exactly one parallel line may be drawn to the given line

Show answer
Correct Option: D
Explanation:

Following statement is not a Euclid's postulate:-

Through a point not on a given line, exactly one parallel line may be drawn to the given line.

Two lines can intersect in _____ points.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $0$

  2. $1$`

  3. $2$

  4. infinite

Show answer
Correct Option: A,B,D
Explanation:

Two lines can intersect at one point

EX:
$3x+4y=2$

$3x+9y=8$
Two lines can intersect at no point (parallel lines)
$3x+4y=2$
$3x+4y=5$
Two lines can intersect at infintie points 
Both lines lie on each other (same lines)
$3x+4y=2$
$6x+8y=4$

Complete the sentence with the most suitable option:
She wishes she _________ younger than she is.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Were

  2. Would be

  3. Had been

  4. Is

Show answer
Correct Option: A

Complete the sentence with the most suitable option:
I wish we _____________ late for this film. I can't follow the story.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Didn't arrive

  2. Hadn't arrived

  3. Wouldn't arrive

  4. Wouldn't have arrived

Show answer
Correct Option: B

Complete the sentence with the most suitable option:
I wish our school __________ a bigger library than it has.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Had had

  2. Would have

  3. Had

  4. Will have

Show answer
Correct Option: C

Complete the sentence with the most suitable option:
If only you _________ less noise.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Had made

  2. Made

  3. Would make

  4. Might make

Show answer
Correct Option: B

Complete the sentence with the most suitable option:
He wished his father ______________ him some shoes before the new term starts.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Was buying

  2. Was going to buy

  3. Would buy

  4. Had bought

Show answer
Correct Option: C

Complete the sentence with the most suitable option:
A: Janet couldn't come to the concert last night.
B: I wish she _______________.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Is able to come

  2. Was able to come

  3. Could have come

  4. Had come

Show answer
Correct Option: C

Complete the sentence with the most suitable option:
The party was so bad that they left early. They wish __________________.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. They didn't go

  2. They wouldn't have gone

  3. They didn't have to go

  4. They hadn't gone

Show answer
Correct Option: D

Complete the sentence with the most suitable option:
It is a pity that we rejected their proposal.
If only we ____________ their proposal.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Won't reject

  2. Hadn't rejected

  3. Wouldn't reject

  4. Didn't reject

Show answer
Correct Option: B

Complete the sentence with the most suitable option:
My father wasn't able to pay the money back.
I wish my father _____________ the money back.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Could have paid

  2. Were able to pay

  3. Could pay

  4. Would be able to pay

Show answer
Correct Option: A

Complete the sentence with the most suitable option:
I couldn't get through the traffic in time.
I wish I ______________ through the traffic in time.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Am able to get

  2. Had been able to get

  3. Was able to get

  4. Would have got

Show answer
Correct Option: B

Complete the sentence with the most suitable option:
A: Don't you have enough time to learn English?
B: No, I don't. I wish I __________ enough time to learn English.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Did have

  2. Had had

  3. Do have

  4. Will have

Show answer
Correct Option: A

Complete the sentence with the most suitable option:
A: Sandy had an accident because she wasn't careful.
B: If only she ________________.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Had been careful

  2. Were careful

  3. Would be careful

  4. Would have been careful

Show answer
Correct Option: A

Complete the sentence with the most suitable option:
A: I must go home now. 
B: I wish you ______________ home now.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Don't have to go

  2. Mustn't have gone

  3. Didn't have to go

  4. Needn't have gone

Show answer
Correct Option: C

Complete the sentence with the most suitable option:
It is a pity you drink too much.
If only you __________ so much.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Don't drink

  2. Won't drink

  3. Didn't drink

  4. Hadn't drunk

Show answer
Correct Option: C

Complete the sentence with the most suitable option:
The bus always stops at every bus-stop. I wish it _____________ at every bus-stop.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Didn't stop

  2. Would stop

  3. Hadn't stopped

  4. Won't stop

Show answer
Correct Option: A

Complete the sentence with the most suitable option:
A: We have to cancel the match because it is raining.
B: If only it _______________.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Won't be raining

  2. Isn't raining

  3. Didn't rain

  4. Weren't raining

Show answer
Correct Option: D

Complete the sentence with the most suitable option:
A: Robert is unhappy because he can't find his passport.
B: If only he __________________.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Could have found it

  2. Had been able to find

  3. Could find it

  4. Had found it

Show answer
Correct Option: C

Complete the sentence with the most suitable option:
A: Mary is sorry that she bought such an expensive dress.
B: If only she ________________ such an expensive dress.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Didn't but

  2. Wouldn't have bought

  3. Hadn't bought

  4. Could have bought

Show answer
Correct Option: C

Identify the conditional phrase in the following sentence from the options given below:
If the police had arrived a bit early, the thief would have easily been caught.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Future Result

  2. Imaginative Counterfactual

  3. Factual Conditions

  4. Imaginative Hypothetical

Show answer
Correct Option: B
Explanation:

Since the situation posited is imaginative and expresses a speculative condition, Option B is correct. The rest of the options do not present an unlikely condition, hence incorrect. 

Identify the conditional phrase in the following sentence from the options given below:
If you give the details by today evening, the payment would be credited tomorrow.

english basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. Imaginative Counterfactual

  2. Factual Timeless

  3. Future Result

  4. None of these

Show answer
Correct Option: C
Explanation:

Since the situation posited and its consequences are to happen in future, Option C is correct. The rest of the options do not present a futurist condition, hence incorrect. 

What is true about the statement "If two angles are right angles the angles have equal measure" and its converse "If two angles have equal measure then the two angles are right angles"?

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. The statement is true but its converse is false

  2. The statement is false but its converse is true

  3. Both the statement and its converse are false

  4. Both the statement and its converse are true

Show answer
Correct Option: A
Explanation:

Two right angles are always equal, each measuring 90 degrees.

However,  two equal angles can be anything not necessarily equal to 90 degrees always.
Hence $A$ is correct.

The converse of "if $x\in A\cap B$ then $x\in A$ and $x\in B$", is

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If $x\in A$ and $x\in B$, then $x\in A\cap B$.

  2. If $x\not\in A\cap B$, then $x\not\in A$ or $x\not\in B$.

  3. If $x\not\in A$ or $x\not\in B$, then $x\not\in A\cap B$.

  4. If $x\not\in A$ or $x\not\in B$, then $x\in A\cap B$.

Show answer
Correct Option: A
Explanation:

The converse of "If P then Q" is "If Q then P"
Hence, Option A

The converse of "If in a triangle $ABC, AB=AC$, then $\angle B=\angle C$", is

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. lf in a triangle $ABC, \angle B=\angle C$, then $AB=AC$.

  2. lf in a triangle$ABC, AB\neq AC$, then $\angle B\neq\angle C$.

  3. lf in a triangle $ABC, \angle B\neq\angle C$, then $AB\neq AC$.

  4. lf in a triangle $ABC, \angle B\neq\angle C$, then $AB=AC$ .

Show answer
Correct Option: A
Explanation:

Take $p:AB=AC$

and $q:\angle B=\angle C$
So the given statement is symbolically represented as $p\rightarrow q$
Now by definition, Converse of a conditional statement $p\rightarrow q$ is $q\rightarrow p$
So $q\rightarrow p$ is given by 
"If in a triangle $ABC, \angle B=\angle C,$ then $AB=AC.$"

Which of the following is the converse of the statement: "If x>4 then x+2>5"?

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If x+2<5 then x<4

  2. If x is not greater than 4 then x+2 is not greater than 5

  3. If x+2>5 then x>4

  4. If x+2 is not greater than 5 then x is not greater than 4

Show answer
Correct Option: C
Explanation:
Converse of  "If  $A$  then  $B$"   is   "If  $B$  then  $A$".    Hence,
Converse of  "If  $x>4$  then  $x+2>5$"  will be  "If  $x+2>5$   then  $x>4$"
So, $C$ is correct.

The converse of "If $x$ has courage, then $x$ will win", is

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If $x$ wins, then $x$ has courage.

  2. If $x$ has no courage, then $x$ will not win.

  3. If $x$ will not win, then $x$ has no courage.

  4. If $x$ will not win, then $x$ has courage.

Show answer
Correct Option: A
Explanation:

Take $p:x$ has courage

and $q:x$ will win
So the given conjugation is $p\Rightarrow q$

Now we need to find converse of this.
Be definition, Converse will be $q\Rightarrow p$
This is symbolic for "If $x$ wins then $x$ has courage

The converse of "if in a triangle $ABC, AB>AC$, then $\angle C=\angle B$", is

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. lf in a triangle $ABC, \angle C=\angle B$, then $AB>AC$.

  2. lf in a triangle$ABC, AB\not\simeq AC$, then $\angle C\not\simeq \angle B$.

  3. lf in a triangle $ABC, \angle C\not\simeq \angle B$, then $ AB\not\simeq AC$.

  4. lf in a triangle $ABC, \angle C\not\simeq \angle B$, then $AB>AC$.

Show answer
Correct Option: A
Explanation:

Take $p:AB>AC$

and $q: \angle C=\angle B$
So the given statement is symbolically represented as $p\rightarrow q$
Now by definition, Converse of a conditional statement $p\rightarrow q$ is $q\rightarrow p$
Thus $q\rightarrow p$ is given by
"If in a $\triangle ABC, \angle C=\angle B$ then $AB>AC$."

Which of the following statements is the converse of "If the moon is full, then the vampires are prowling."?

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If the vampires are prowling, then the moon is full.

  2. If the moon is not full, then the vampires are prowling

  3. If the vampires are not prowling, then the moon is not full.

  4. None of these

Show answer
Correct Option: A
Explanation:

Converse of  "If $P$, then $Q$"  is  "If $Q$, then $P$"

Similarly, option "$A$" is converse of the given statement  "If the moon is full, then the vampires are prowling."

Which of the following statements is the converse of "You cannot skateboard if you do not have a sense of balance."?

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If you cannot skateboard, then you do not have a sense of balance.

  2. If you do not have a sense of balance, then you cannot skateboard.

  3. If you skateboard, then you have a sense of balance.

  4. None of these

Show answer
Correct Option: B
Explanation:

Converse of "If $P$, then $Q$"  is  "If $Q$, then $P$". 

Now, the given statement  "You cannot skateboard if you do not have a sense of balance." can be re-written as "If you do not have a sense of balance, then you cannot skateboard."
The converse of this statement is option $B$.

Which of the following statements is the contrapositive of the statement, You win the game if you know the rules but are not overconfident.

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If you lose the game then you dont know the rules or you are overconfident.

  2. A sufficient condition that you win the game is that you know the rules or you are not overconfident.

  3. If you dont know the rules or are overconfident you lose the game

  4. If you know the rules and are overconfident then you win the game.

Show answer
Correct Option: A
Explanation:

Contrapositive is the inverse of the converse of the statement.

It is obtained by first interchanging the hypothesis and conclusion and then adding "not" to both
In this case, converse is "If you win the game, then you know the rules but are not overconfident."
Inverse of this statement gives answer as A.

The converse of $p \rightarrow (q \rightarrow r)$ is

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. $(q \wedge \sim r) \vee p$

  2. $(\sim q \vee r) \vee p$

  3. $(q \wedge \sim r) \wedge \sim p$

  4. $(q \wedge \sim r) \wedge p$

Show answer
Correct Option: A
Explanation:

The converse of $p \rightarrow (q \rightarrow r)$ is,

$\equiv (q \to r) \to p \equiv (\sim q \vee r) \to p \equiv  \sim (\sim q \vee r) \vee p\equiv (q \wedge \sim r) \vee p $

Which of the following statements is the contrapositive of "If a polygon has four sides, then it is called a quadrilateral."?

business maths basic concepts in geometry conditional statements and converse euclid's postulates introduction to euclid's geometry

  1. If a polygon is called a quadrilateral, then it has four sides.

  2. If a polygon is not called a quadrilateral, then it does not have four sides

  3. If a polygon does not have four sides, then it is not called a quadrilateral.

  4. None of these

Show answer
Correct Option: B
Explanation:

Contrapositive will switch if and then and also add not to both parts.
Option B is correct.

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