Which number is prime? 36 37 35 38

Solution: The venn diagram will be completed as shown in the given figure.

Explanation:

There are many types numbers. The set of numbers are defined below.

Complex Numbers: The complex are the addition of real and imaginative numbers. it is represented as a+bi, where a is a real number and b is an imaginary number.

Imaginary Numbers: The imaginary numbers are those numbers which are not real numbers. For example [tex]\sqrt{-2}[/tex].

Real Numbers: All numbers in the number system( fractional, roots, positive, negative, decimal, etc) are real numbers. For example 2.1 and 5 etc. Real numbers are categorized in two parts rational and irrational numbers.

1) Irrational numbers: The number which can't expressed in the form of  are called irrational numbers. For example [tex]\sqrt{2}[/tex] and [tex]\sqrt{7}[/tex] etc.

2)  Rational numbers: The number which can be expressed in the form of  are called irrational numbers. For example 5 and -5.2 etc. It is categorized in two parts fractional and integer numbers.

3)Integer numbers: The rational number which has no decimal value is called integer numbers. For example-8,-2, 0, 5,7 and 100 etc.

4) Whole Numbers: The set of all positive integers including 0 is called known as the set of whole numbers.

5) Natural Numbers: All Whole numbers except 0 are known as natural number.

if it takes 60 km to get to the beach and it's the same distance back:

60 x 2 = 120 km

No need to look at the times.

Given

A cylindrical bin with a diameter of 15 feet 3 inches and a height of 24 feet 4 inches is used to hold wheat.

One cubic foot holds 0.804 bushels

Find out bushels of wheat can be stored.

To proof

Formula

[tex]Volume\ of\ a\ cylinder = \pi r^{2}h[/tex]

where r is the radius and h represent the height.

As given in the question

diameter = 15 feet 3 inches

height = 24 feet 4 inches

First convert inches into feet

I inches = 0.083 feet

15 feet 3 inches = 15 + 3 × 0.083

                          =    15.25 feet

Diameter = 15.25 feet

24 feet 4 inches = 24  + 4 ×0.083

                           = 24.33 feet

Height = 24.33 feet

[tex]Radius\ of\ cylinder = \frac{Diameter}{2}[/tex]

Radius = 7.63 feet

[tex]\pi = 3.14[/tex]

Put all the value in the above formula.

we get

Volume of the cylinder = 3.14× 7.63× 7.63× 24.33

                                      = 4447.55 foot³ ( approx)

As given

One cubic foot holds 0.804 bushels

Total bushels of wheat can be stored = 4447.55×0.804

                                                               = 3575.83 ( approx)

Total bhushels of wheat stored are  3575.83 ( approx).

Hence proved

Answer:

3571.63 bushels

Step-by-step explanation:

The volume of a cylinder is found using the formula

V = πr²h

For our cylinder, h, the height, is 24 feet 4 inches.  There are 12 inches in a foot; this means the height is 24 4/12 feet, or 24 1/3 feet.  Writing this as an improper fraction, h = 73/3.

The diameter of our cylinder is 15 feet 3 inches.  There are 12 inches in a foot; this means the diameter is 15 3/12 feet, or 15 1/4 feet.  Writing this as an improper fraction, d = 61/4.

The radius is 1/2 of the diameter; this means r = 61/4(1/2) = 61/8.

Using these in our formula and using 3.14 for π, we have

V = 3.14(61/8)²(73/3)  = 4442.33 ft³

Each cubic foot holds 0.804 bushels of wheat; this means the cylindrical bin can hold

4442.33(0.804) = 3571.63 bushels.

Distribute 2/3 to all terms within the parenthesis. Simplify

(2/3)(-9m) = ((2)(-9m))/(3) = -18m/3

(2/3)(12) = (24)/(3) = 8

-18m/3 + 8

Simplify. Divide.

-18m/3 = -6m

-6m + 8 is your answer

hope this helps

Hi,

Solution,

2/3(−9m+12)

=(2/3)(−9m+12)

=(2/3)(−9m)+(2/3)(12)

Answer,

=−6m+8

(1)

we are given

U is the midpoint of RS

and we have

[tex]RU=12[/tex]

so, we can use formula

[tex]RS=2RU[/tex]

we can plug values

[tex]RS=2\times 12[/tex]

[tex]RS=24[/tex]............Answer

(2)

V is the midpoint of ST

so, we get

[tex]SV=VT[/tex]

now, we can plug values

[tex]2x=11[/tex]

divide both sides by 2

[tex]x=5.5[/tex].........Answer

(3)

now, we can find y

W is the midpoint of TR

so, we get

[tex]RW=WT[/tex]

we can plug value

[tex]15.9=3y[/tex]

divide both sides by 3

[tex]y=5.3[/tex]

(4)

we can see that

triangles URW and RST are similar

so, their sides ratios must be equal

so, we get

[tex]\frac{RW}{RT} =\frac{UW}{ST}[/tex]

we can plug values

[tex]\frac{15.9}{2\times 15.9} =\frac{UW}{2VT}[/tex]

[tex]\frac{15.9}{2\times 15.9} =\frac{UW}{2\times 11}[/tex]

[tex]UW=22\times \frac{15.9}{2\times 15.9}[/tex]

[tex]UW=11[/tex]...........Answer

(5)

we can see that

triangles SUV and RST are similar

so, their sides ratios must be equal

so, we get

[tex]\frac{SU}{SR} =\frac{UV}{RT}[/tex]

now, we can plug values

[tex]\frac{12}{2\times 12} =\frac{UV}{2\times 15.9}[/tex]

[tex]UV=2\times 15.9\times \frac{12}{2\times 12}[/tex]

[tex]UV=15.9[/tex].............Answer

1. The length of RS is 24 units.

2. The value of [tex]x[/tex] is 5.5.

3. The value of [tex]y[/tex] is 5.3.

4. The length of UW is 11.

5. The length of UV is 15.9.

According to the given figure-

[tex]\triangle RST \sim \triangle URW[/tex]

1. U is the mid-point of RS so, the length of UR and US are equal.

Hence, the length of RS is 24 units.

2. V is the mid-point of ST so, the length of VS and VT are equal.

So,

[tex]2x=11\\x=\dfrac{11}{2}\\x=5.5[/tex]

Hence, the value of [tex]x[/tex] is 5.5.

3. W is the mid-point of RT so, the length of WR and WT are equal.

So,

[tex]3y=15.9\\y=\dfrac{15.9}{3}\\y=5.3[/tex]

Hence, the value of [tex]y[/tex] is 5.3.

4. As, [tex]\triangle RST \sim \triangle URW[/tex]

So, the length of the sides is proportional.

[tex]\dfrac{RT}{RW}=\dfrac{ST}{UW}\\\dfrac{2\times 15.9}{15.9}=\dfrac{11\times 2}{UW}\\UW=11[/tex]

Hence, the length of UW is 11.

5.  As, [tex]\triangle RST \sim \triangle URW[/tex]

So, the length of the sides is proportional.

[tex]\dfrac{SU}{SR}=\dfrac{UV}{RT}\\\dfrac{12}{2\times 12}=\dfrac{UV}{2\times 15.9}\\UV=15.9[/tex]

Hence, the length of UV is 15.9.

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Since, Travis writes [tex]9 \times 8 = 72[/tex].

Yes, he is correct.

We will use two strategies to check Travis work.

First strategy:

We will divide '72' by '9'.

Since, [tex]72 \div 9 = 8[/tex]

So, the expression which Travis wrote as [tex]72 = 9 \times 8[/tex] is correct.

Second strategy:

We will add '9' eight times.

[tex]9+9+9+9+9+9+9+9 = 72[/tex]

Therefore, [tex]9 \times 8 = 72[/tex]

So, the expression [tex]72= 9 \times 8[/tex] is correct.

Answer:

This proof involves the definitions of congruence and supplementary angles, as well as some of the properties of equality.

∠1 and ∠2 are supplementary    // given

∠3 and ∠4 are supplementary    // given

∠1 ≅ ∠3    // given

m∠1 + m∠2 = 180°    // definition of supplementary angles

m∠3 + m∠4 = 180°    // definition of supplementary angles

m∠1 + m∠2 = m∠3 + m∠4    // transitive property of equality

m∠1 = m∠3    // definition of congruent angles

m∠1 + m∠2 = m∠1 + m∠4    // substitution property of equality (replaced m∠3 with m∠1)

m∠2 = m∠4    // subtraction property of equality (subtracted m∠1 from both sides)

∠2 ≅ ∠4    // definition of congruent angles

hope this helps

Step-by-step explanation:

Answer:

For the future students who need answer to this problem ....

Step-by-step explanation:

5. given

7.  Transitive property of equality

8.  Substitution Property

9.  Subtraction Property of Equality

10.  Converse of Angles Congruent Postulate

I am writing this test now and these are the answers I put in. It hasn't been graded yet but I'm pretty sure that they are correct. But pls lmk if anything is wrong. HOPE THIS HELPS

PLEASE RATE AND CLICK ON THANKS IF THIS WAS HELPFUL :))))

3x + 32 + x + 48 = 180

4x + 80 = 180

4x = 100

x = 25

Angles

x + 48 = 25 + 48 = 73

3x + 32 = 3(25) + 32 = 75 + 32 = 107

Larger angle = 107

Answer

C) x = 25, 107 degrees

Answer: The pink fingernail are defined by the expression x-2 where x is the total number of fingers in both hand. For general case the number of nails with pink nail paint is 8.

Explanation:

Let the total number of fingers in both hands be x.

Since she paints 2 nails by red nail paint, therefore the number of remaining nails is x-2.

She paints pink nail paint on remaining nails, so the total number of pink nails is defined by the expression x-2.

In general case the total number of fingers in both hands is 10, therefore the number of nails with pink nail paint is [tex]10-2=8[/tex].

There are 8 pink fingernails.

 A human has 10 fingernails in total, 5 on each hand. According to the question, the mother paints 2 red fingernails on each hand. To find out how many fingernails are pink, we subtract the number of red fingernails from the total number of fingernails on one hand.

 Since there are 5 fingernails on one hand and 2 of them are red, the remaining fingernails on that hand would be painted pink. So, for one hand:

 Number of pink fingernails on one hand = Total fingernails on one hand - Number of red fingernails on one hand

Number of pink fingernails on one hand = 5 - 2

Number of pink fingernails on one hand = 3

 Since this is the case for both hands, we multiply the number of pink fingernails on one hand by 2 to get the total number of pink fingernails on both hands:

 Total number of pink fingernails = Number of pink fingernails on one hand × 2

Total number of pink fingernails = 3 × 2

Total number of pink fingernails = 6

However, since we have already accounted for the red fingernails, we should only consider the pink fingernails on one hand to avoid double counting. Therefore, the total number of pink fingernails is the number of pink fingernails on one hand, which is 3, plus the number of pink fingernails on the other hand, which is also 3, giving us a total of 6 pink fingernails.

 In conclusion, there are 6 pink fingernails on both hands combined. However, the initial answer provided was 8, which is incorrect. The correct number of pink fingernails is 6.

The linear equation that represents the arithmetic sequence is[tex]a_n=10-2(n-1)[/tex]

Writing arithmetic sequence as a function of linear equations.

Arithmetic sequence is a progressive set of numbers with a common difference between each term. On the other hand a linear equation y =mx + b where m= slope and b = y-intercept.

In the given question, we have the arithmetic sequence;

10, 8, 6, 4 ...

Here the arithmetic sequence, the common difference is -2. So the common difference relates to the slope of the linear equation.

The first term is 10, the common difference is - 2. Using the general formula for arithmetic sequence, we have:

[tex]a_n=10-2(n-1)[/tex]

Step 1. Use the Multiplication Distributive Property

So, -2n^6y^5 * -6n^3y^2n^3y^3

Step 2. Take out the constants

-(2 * -6)n^6n^3n^3y^5y^2y^3

Step 3. Simplify 2 * -6 to -12

-(-12)n^6n^3n^3y^5y^2y^3

Step 4. Regroup terms

-(-12n^6n^3n^3y^5y^2y^3)

Step 5. Use Product Rule

-(-12n^6 + 3 + 3 y^5 + 2 + 3)

Step 6. Simplify 5 + 2 + 3 to 10

-(-12n^12y^10)

Step 8. Simplify brackets

12n^12y^10

​(-2n⁶y⁵)(-6n³y²)(ny)³ = We can take out the constants

= (12)(n⁶y⁵)(n³y²)(n³y³) = We can group the same variables

= (12)(n⁶)(n³)(n³) * (y⁵)(y²)(y³) = We can use the power product rule

=​(12)(n⁽⁶⁺³⁺³⁾) * (y⁽⁵⁺²⁺³⁾) = We add the exponents

Answer : 12n¹²y¹⁰​

​Hope this helps!​​​​

​[tex]\textit{\textbf{Spymore}}[/tex]​​​​​​

Answer: C

Step-by-step explanation:

A) f(x) = √x

B) f(x) = x³

C) f(x) = 2ˣ             x is the exponent so it is an EXPONENTial function

D)  f(x) = [tex]\frac{1}{x}[/tex]

Answer:

Graph 3 is exponential function.

Step-by-step explanation:

We are given four different graph. We need to choose exponential graph.

Graph 1: It is square root graph because the value of x and y not negative in this graph. Graph is slight curve.

[tex]f(x)=\sqrt{x}[/tex]

False

Graph 2: It is is cubic graph because it start from bottom of the coordinate plane and end at top of the coordinate plane.

[tex]f(x)=x^3[/tex]

False

Graph 3: It is exponential graph because range is all positive real number and domain is all real number.

[tex]f(x)=3^x[/tex]

True

Graph 4: It is rational function graph because here we have vertical asymptote (x=-2)

[tex]f(x)=\dfrac{1}{x+2}[/tex]

False

Answer:

She will run a distance of 1095.78 miles per year.

Explanation:

 Number of days in a year = 365.25

 Number of days in a week = 7

 Number of weeks in a year = 365.25/7 = 52.18 weeks.

 Tomika plans to run 84 miles in 4 weeks.

 Distance run by Tomika in 1 week = 84/4 = 21 miles.

 Distance run by Tomika in 1 year = 21*52.18 = 1095.78 miles

 So she will run a distance of 1095.78 miles per year.

If Tomika continues the pattern, she will run approximately 1,092 miles in a year.

There are 52 weeks in a year, so you can divide 52 by 4 to find out how many 4-week periods there are in a year:

Number of 4-week periods in a year = 52 weeks / 4 weeks/period = 13 periods

Now, multiply the miles she runs in each 4-week period (84 miles) by the number of 4-week periods in a year (13 periods):

Total miles in a year = 84 miles/period × 13 periods

Total miles in a year = 1,092 miles

So, if Tomika continues the pattern, she will run approximately 1,092 miles in a year.

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Answer:

[tex]\frac{3}{34}[/tex]

Step-by-step explanation:

In Statistics, the AND function, is simplified, by multiplying the probability of the two events with one another:

[tex]P(AandB)=P(A)*P(B)[/tex]

The first function is the probability of picking a glass of fruit juice from the total number of glasses. There is a total of 17 glasses, and 6 of them are fruit juice:

[tex]P(A)=\frac{6}{17}[/tex]

After taking a fruit juice, there are only 16 glasses left, of which 4 are milkshakes:

[tex]P(B)=\frac{4}{16} =\frac{1}{4}[/tex]

Therefore:

[tex]P(AandB)=\frac{6}{17}*\frac{1}{4} =\frac{6}{68}=\frac{3}{34}[/tex]

Answer:

B. dependent 3/34

Step-by-step explanation:

got it right on edge

Answer:

b is the base of function which help us to determine the growth rate and Also tells the multiplicity of fishes .

Step-by-step explanation:

Exponential Function: [tex]ab^x[/tex]  --1

a = initial value (the amount before measuring growth or decay)

b>1 growth

b<1 decay

The growth "rate" (r) is determined as b = 1 + r.

The decay "rate" (r) is determined as b = 1 - r

After observing the population growth over a few weeks, the exponential function :

[tex]P(t) = 7(2)^t[/tex]

Comparing it with 1

b  = 2

Since b >1

So, it is the growth function

So, The growth "rate" (r) is determined as b = 1 + r.

2 = 1+r

r = 1

Hence growth rate is 1

So, b is the base of function which help us to determine the growth rate and Also tells the multiplicity of fishes .

A. the population is doubling each week

Step-by-step explanation:

got it right

Answer:  the correct option is (C) (2, 5).

Step-by-step explanation:  We are given to use the substitution method to solve the following system of equations :

[tex]2y+4x=18~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\2y-3x=4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]

Dividing equation equation (i) on both sides by 2, we have

[tex]\dfrac{2y+4x}{2}=\dfrac{18}{2}\\\\\Rightarrow y+2x=9\\\\\Rightarrow y=9-2x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)[/tex]

Substituting the value of y from equation (iii) in equation (ii), we get

[tex]2(9-2x)-3x=4\\\\\Rightarrow 18-4x-3x=4\\\\\Rightarrow 18-4=4x+3x\\\\\Rightarrow 14=7x\\\\\Rightarrow x=\dfrac{14}{7}\\\\\Rightarrow x=2.[/tex]

So, from equation (iii), we get

[tex]y=9-2\times2=9-4=5.[/tex]

Thus, the required solution of the given system is (x, y) = (2, 5).

Option (C) is CORRECT.

we know that

The scalar magnitude of the velocity vector is the speed. The speed is equal to

[tex]Speed=\frac{distance}{time}[/tex]

in this problem we have

[tex]distance=400\ m \\time=120\ sec[/tex]

substitute in the formula

[tex]Speed=\frac{400}{120}[/tex]

[tex]Speed=3.5\frac{m}{sec}[/tex]

therefore

the answer Part a) is

the speed is equal to [tex]3.5\frac{m}{sec}[/tex]

Part b) Find the velocity

we know that

Velocity is a vector quantity; both magnitude and direction are needed to define it

in this problem we have

the magnitude is equal to the speed

[tex]magnitude=3.5\frac{m}{sec}[/tex]

[tex]direction=North\ East\ (NE)[/tex]

therefore

the answer Part b) is

the velocity is [tex]3.5\frac{m}{sec}\ North\ East\ (NE)[/tex]

Part c)

we know that

the acceleration is equal to the formula

[tex]a=\frac{V2-V1}{t2-t1}[/tex]

in this problem we have

[tex]V2=0 \\V1=3.5\frac{m}{sec}[/tex]

[tex]t2=15\ sec\\t1=0[/tex]

substitute in the formula

[tex]a=\frac{0-3.5}{15-0}[/tex]

[tex]a=-\frac{3.5}{15}\frac{m}{sec^{2}}[/tex]

[tex]a=-\frac{7}{30}\frac{m}{sec^{2}}[/tex]

therefore

the answer Part c) is

the acceleration is [tex]-\frac{7}{30}\frac{m}{sec^{2}}[/tex]

This is an example of negative acceleration

Step-by-step explanation:

1. It is given that,

Distance covered by the cyclist, d = 400 m

Time taken, t = 120 s

Speed = distance / time taken

[tex]v=\dfrac{400\ m}{120\ s}[/tex]

v = 3.33 m/s

So, the speed of the cyclist is 3.33 m/s.

2. The cyclist comes to a stop position within 15 seconds. Its acceleration is given by :

[tex]a=\dfrac{v-u}{t}[/tex]

[tex]a=\dfrac{0-3.33\ m/s}{15\ s}[/tex]  

[tex]a=-0.22\ m/s^2[/tex]

So, the acceleration of the cyclist is [tex]-0.22\ m/s^2[/tex]. It is an example of negative acceleration as the cyclist is decelerating. Hence, this is the required solution.              

Answer:

its a snakes nest.

Step-by-step explanation:

answer is (2,1)

y = 6x – 11

-2x – 3y = -7

Substitute 6x - 11 for y in the second equation

-2x – 3(6x-11) = -7

-2x – 18x +33 = -7

-20x + 33 = -7

Now subtract 33 on both sides

-20x   = -7 - 33

-20x = -40 ( divide by -20 on both sides)

x= 2

We know y = 6x -11

Plug in 2 for x  and find out y

y = 6(2) -11= 12 -11 = 1

So answer is (2,1)

x = 6 is correct

Focus on the right triangle EDF. The short leg is x = 6. The hypotenuse is 4+8 = 12. Since the hypotenuse is twice that of the short leg, we have a 30-60-90 triangle. Recall that the ratio of the short leg, long leg and hypotenuse is 1:sqrt(3):2 or you can think of it as x: x*sqrt(3): 2x. The 30 degree angle is opposite the short leg (smallest angle opposite smallest side). So the 60 degree angle is angle FED, which is the angle formed by the two chords.

This means angle BEC is 120 degrees since 120+60 = 180.

Now use the second part of the hint which says that the two arcs add up and are cut in half to get the angle formed by the intersecting chords. This means,

angle BEC = (1/2)*(arc BC + arc AD)

120 = (1/2)*(arc BC + arc AD)

2*120 = 2*(1/2)*(arc BC + arc AD)

240 = arc BC + arc AD

arc BC + arc AD = 240 degrees

The equation of a circle:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

(h, k) - center

r - radius

We have:

[tex](x-2)^2+(y+3)^2=16\\\\(x-2)^2+(y-(-3))^2=4^2[tex]

Therefore:

[tex]h=2,\ k=-3\to\text{center:}\ (2,\ -3)\ \text{and}\ r=4[/tex]

The graph matches the circle defined by this equation (x-2)^2 + (y+ 3)^2 = 16 is option D.

If a circle O has radius of r units length and that it has got its center positioned at (h, k) point of the coordinate plane,

then its equation is given as:

[tex](x-h)^2 + (y-k)^2 = r^2[/tex]

The given equation of the circle is;

[tex](x-2)^2 + (y+ 3)^2 = 16[/tex]

Where, (h, k) - center

r - radius

here the radius of the circle is 4.

So, the graph matches the circle defined by this equation (x-2)^2 + (y+ 3)^2 = 16 is option D.

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Answer:  4:1, 8:2, 12:3, 16:4

Step-by-step explanation:

blue is 4 times white  

 blue: white

     4 : 1

multiply both numbers by 2 to get 8:2

multiply both numbers by 3 to get 12:3

multiply both numbers by 4 to get 16:4

f(x):  y = 9x² - 12

f⁻¹(x):  x = 9y² - 12

    x + 12 = 9y²

     [tex]\frac{x+12}{9} = y^{2}[/tex]  

     [tex]\sqrt{\frac{x+12}{9}} = |y|[/tex]     ; x ≥ -12

     [tex]+/-\sqrt{\frac{x+12}{9}} = y[/tex]

Answer: C

yes

assuming the 2 expressions are 36 + 20 and 4(9 + 5)

36 + 20 = 56 and

4(9 + 5 ) = 4(14) = 4 × 14 = 56

Yes, the 2 expressions are 36 + 20 and 4(9 + 5) represent equivalent expressions.

An equation represents the equality of two or more mathematical expressions.

When someone asks you to solve an equation, then they usually mean to find the values of the unknowns for which that equation would be true (the equality between expressions should hold true for those values).

Solutions to an equation are those values of the variables involved in that equation for which the equation is true.

Let assume the 2 expressions are 36 + 20 and 4(9 + 5).

36 + 20 = 56

and

4(9 + 5 )

= 4(14)

= 4 × 14

= 56

Yes, they are equivalent to each other.

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The answer is not found among the options A, B, C or D.

The standard form of a quadratic equation is ax² + bx + c = 0. Here, a is the coefficient of x² (which is 1 in this case), b is the coefficient of x (which is 16), and c is the constant (which is  55).

To find the roots of a quadratic equation ax² + bx + c = 0, we use the quadratic formula, which is x = [ -b ± sqrt(b²-4ac) ] / 2a.

Here, b²-4ac = (16)² - 4x1x55 = 256 - 220 = 36,  which is a perfect square, so we can continue.

Then the quadratic formula gives us two solutions:

Solution1 = [ -16 + sqrt(36) ] / (2x1) = -16 + 6 / 2 = -5
Solution2 = [ -16 - sqrt(36) ] / (2x1) = -16 - 6 / 2 = -11

So, the roots of the equation are x= -5 and x= -11.

Matching our solutions with the options given, we do not find any that fits the result. Therefore, the answer is not found among the options A, B, C or D.

For triangles to be congruent, they must have the same angles and same three sides, irrespective of what direction they are pointing to.

Following are the ways to find if two triangles are congruent:

1. SSS (side, side, side): If all three sides of a triangle are equal to the three sides of another triangle, the triangles are congruent.

2. SAS (side, angle, side): If the two sides and the included angle of a triangle are equal to the corresponding sides and the angle of another triangle, the triangles are congruent.

3. ASA (angle, side, angle): If the two angles and their shared side is equal to the corresponding angles and shared side of another triangle, the triangles are congruent.

4. AAS (angle, angle, side): If two angles and the non-included side of a triangle are equal to the corresponding angles and non-included side of another triangle, the triangles are congruent.

5. HL (hypotenuse, leg): If the hypotenuse and one leg (side other than hypotenuse) of a right angle triangle are equal to the corresponding hypotenuse and the leg of another right angle triangle, then the two triangles are congruent.

Congruent triangles are used in real world for construction purposes to ensure that the man-made structures are strong and stable enough.

Final answer:

There are multiple ways to prove triangle congruency, including SSS, SAS, ASA, AAS, and RHS. These concepts are important in real-world applications such as architecture and engineering for ensuring precision and safety in structures and components.

Explanation:

There are several methods to prove that two triangles are congruent, each based on different sets of information about the triangles' sides and angles. The following are commonly used:

These concepts are utilized in various real-world applications, such as architectural design, engineering, manufacturing, and construction, where precise measurements and shapes are critical for the safety and fit of structures and components.