Solve The Equation For The Variable Given. Solve For X (2024)

The length of PQ is 3√5 and its slope is -2

The length of SR is 3√5 and its slope is -2

The length of SP is 5√2 and its slope is -7

The length of RQ is 5√2 and its slope is -1

So PQ ≅ SR and SP ≅ RQ. By the Perpendicular Bisector theorem, adjacent sides are perpendicular. By the selection of side, ∠PSR, ∠SRQ, ∠RQP and ∠QPS are right angles. So, the quadrilateral is a rectangle.

Understanding Quadrilateral

To find the lengths and slopes of the sides of the quadrilateral PQRS, we apply the distance formula:

D = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

and the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

1. Length PQ:

Using the distance formula, the length PQ can be calculated as follows:

PQ = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

= √((3 - 0)² + (-4 - 2)²)

= √(3² + (-6)²)

= √(9 + 36)

= √45

= 3√5

2. Length SR:

Using the distance formula, the length SR can be calculated as follows:

SR = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

= √((1 - (-2))² + (-5 - 1)²)

= √((1 + 2)² + (-6)²)

= √(3² + 36)

= √(9 + 36)

= √45

= 3√5

3. Length SP:

Using the distance formula, the length SP can be calculated as follows:

SP = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

= √((1 - 0)² + (-5 - 2)²)

= √(1² + (-7)²)

= √(1 + 49)

= √50

= 5√2

4. Length RQ:

Using the distance formula, the length RQ can be calculated as follows:

RQ = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

= √((-2 - 3)² + (1 - (-4))²)

= √((-2 - 3)² + (1 + 4)²)

= √((-5)² + 5²)

= √(25 + 25)

= √50

= 5√2

Now, let's calculate the slopes of the sides:

1. Slope PQ:

The slope of PQ can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

= (-4 - 2) / (3 - 0)

= -6 / 3

= -2

2. Slope SR:

The slope of SR can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

= (-5 - 1) / (1 - (-2))

= -6 / 3

= -2

3. Slope SP:

The slope of SP can be calculated using the slope formula:

m =[tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

= (-5 - 2) / (1 - 0)

= -7 / 1

= -7

4. Slope RQ:

The slope of RQ can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

= (1 - (-4)) / (-2 - 3)

= 5 / (-5)

= -1

Therefore, the lengths and slopes of the sides of the quadrilateral PQRS are:

Length PQ: 3√5

Length SR: 3√5

Length SP: 5√2

Length RQ: 5√2

Slope PQ: -2

Slope SR: -2

Slope SP: -7

Slope RQ: -1

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To express the value 6 × 10^(-10), you can use scientific notation or decimal notation. Here are some valid expressions that equal 6 × 10^(-10):

0.0000000006

6E-10

6 × 10^(-10)

These expressions represent the same value, which is 6 multiplied by 10 raised to the power of -10.

Decimal is a number system used in everyday arithmetic and mathematics. It is also known as the base-10 system because it uses ten digits (0-9) to represent numbers. Each digit's position in a decimal number represents a power of 10.

In a decimal number, the digits to the left of the decimal point represent whole numbers, while the digits to the right of the decimal point represent fractions or parts of a whole. For example, in the decimal number 123.45, the digit "1" represents 100, the digit "2" represents 10, the digit "3" represents 1, the digit "4" represents 0.1, and the digit "5" represents 0.01.

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The probability of selecting a girl and a boy for the class committee can be calculated by considering the total number of outcomes and the number of favorable outcomes.

Identify the number of girls and boys in the class. In this case, there are 5 girls and 7 boys.

Determine the total number of students in the class. That is 5 + 7 = 12.

Determine the number of ways to select two students from the class.

Here we can use the combination formula, which is written as C(n, r), where n is the total number of items and r is the number of items to be chosen.

In our case, n = 12 (total number of students) and r = 2 (number of students to be selected).

C(12, 2) = 12! / (2!(12-2)!) = 66.

Determine the number of favorable outcomes.

In this case, we want to select one girl and one boy. We multiply the number of girls by the number of boys: 5 x 7 = 35.

To find the probability, we divide the number of favorable outcomes (35) by the total number of outcomes (66):

Probability = Number of favorable outcomes / Total number of outcomes = 35 / 66 = 5/6.

So, the probability of selecting a girl and a boy for the class committee is 5/6.

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Answer:

x+4y=7

Step-by-step explanation:

John's z-score is 0.75.

John's score of 79 on the math test corresponds to a z-score of 0.75 and a percentile of 77.82%.

What is John's z-score and percentile?

To get John's z-score, we will use the formula: z = (x - μ) / σ

Data

x = John's score (79)

μ = mean score (73)

σ = standard deviation (8)

z = (79 - 73) / 8

z = 6 / 8

z = 0.75.

To find John's percentile, we can use a standard normal distribution table or calculator.

The percentile represents the percentage of scores that fall below a given value. Using standard normal distribution table, we find that a z-score of 0.75 equals to percentile of 77.82%.

Full question:

The scores on a math test have a mean of 73 points and a standard deviation of 8 points. John scores a 79 on the test. Convert this score to a z-score and a percentile..

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Answer:

increasing exponential function.

Step-by-step explanation:

We can see from the table that the function is not increasing by the same amount as it goes up by 1 in x, so it is not linear.

It must be exponential.

It is increasing at it increases in x, so it is an increasing exponential function.

a. It will take 7 months to pay off the credit card.

b. it will take 4 months to pay off the credit card.

Since, APR stands for Annual Percentage Rate. It is the interest rate charged on a loan or credit card, expressed as a yearly percentage rate. The APR takes into account not only the interest rate, but also any fees or charges associated with the loan or credit card.

a. If you put half of the available money each month toward the credit card, then you are paying $150.00 per month towards the credit card balance.

We can use the formula for the present value of an annuity to find how many months it will take to pay off the credit card:

PV = PMT × ((1 - (1 + r)⁻ⁿ) / r)

where:

PV is the present value of the debt

PMT is the payment amount per period

r is the monthly interest rate

n is the number of periods

Substituting the values, we get:

754.43 = 150 × ((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)

Simplifying and solving for n, we get:

n = log(1 + (PV ×r / PMT)) / log(1 + r)

n = log(1 + (754.43×0.011333 / 150)) / log(1 + 0.011333)

n = 6.18

Therefore, it will take 7 months to pay off the credit card if you put half of the available money each month toward the credit card.

b. If you put all of the available money each month toward the credit card, then you are paying $300.00 per month towards the credit card balance.

754.43 = 300 ×((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)

Simplifying and solving for n, we get:

n = log(1 + (PV × r / PMT)) / log(1 + r)

n = log(1 + (754.43× 0.011333 / 300)) / log(1 + 0.011333)

n = 3.43

Therefore, it will take 4 months to pay off the credit card if you put all of the available money each month toward the credit card.

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You should leave at 7:34 to catch the bus that comes at 8:05.

We have,

To catch the bus that arrives at 8:05, you should leave with enough time to reach the bus stop 31 minutes before the bus's arrival time.

To catch the bus that arrives at 8:05, you need to be at the bus stop before the bus arrives. Since it takes you 31 minutes to get to the bus stop, you should leave your starting point early enough to allow for that travel time.

By subtracting 31 minutes from the bus's arrival time of 8:05, you determine the time at which you should depart.

In this case, subtracting 31 minutes gives you 7:34, meaning you should leave at 7:34.

To calculate the departure time, subtract 31 minutes from the bus's arrival time:

= 8:05 - 31 minutes

= 7:34

Thus,

You should leave at 7:34 to catch the bus that comes at 8:05.

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The minimum value of [tex]f(x) = -2^3 \sqrt\(5-4x+3)[/tex] on the interval [1, 8] is -3 and the maximum value is 5. To find the minimum value, we can start by finding the critical points of the function.

The critical points are the points where the derivative of the function is equal to zero. In this case, the derivative of the function is

[tex]f'(x) = -2^3 \times (5-4x+3) ^(-3/2) \times (-4)[/tex]

The critical points of the function are x = 1 and x = 5.

We can now evaluate the function at each critical point and at the endpoints of the interval to find the minimum and maximum values. The values of the function at the critical points and at the endpoints are

x | f(x)

-- | --

1 | -3

5 | 5

8 | 9

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Answer:

Set your calculator to degree mode.

x^2 = 4^2 + 10^2 - 2(4)(10)(cos 29°)

x^2 = 46.0304

x = 6.78

The number that belongs in the green box is 6.78.

Fixed costs are expenses that do not change with the level of output or production. Examples of fixed costs in Manuel's case might include the permit cost, rent for the hot dog stand, or insurance premiums.

In order to answer your question accurately, I would need the specific values for Manuel's profit and cost functions. The information you provided is incomplete, as you mentioned Manuel's initial profit hill is plotted in green on a graph, but the graph itself is not available for reference.

To determine how a change in fixed costs affects the profit-maximizing quantity, we typically analyze the cost and revenue functions. Without these functions or the corresponding data, it is not possible to provide an exact numerical answer.

However, I can explain the general concept. Fixed costs are expenses that do not change with the level of output or production. Examples of fixed costs in Manuel's case might include the permit cost, rent for the hot dog stand, or insurance premiums.

When fixed costs decrease, it reduces the overall cost of production for each level of output. This means that Manuel can achieve higher profits or reduce losses for any given level of sales. Consequently, the profit-maximizing quantity may change as a result.

If we assume that the decrease in the price of the permit is the only change in fixed costs and all other factors remain constant, the new profit hill can be expected to shift upward. This is because the reduction in fixed costs increases the potential for higher profits at each level of output.

Without more specific information about Manuel's profit and cost functions, it is not possible to determine the exact profit-maximizing levels before and after the decrease in the price of the permit.

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The length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.

To determine the length of segment OP on the unit circle, we need to use trigonometry. Let's break down the problem step by step:

Definition: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system.

Initial Ray: The initial ray is a line segment that starts from the origin (0, 0) and extends to a point on the unit circle. It forms an angle with the positive x-axis.

Rotation: We are rotating the initial ray counterclockwise by θ degrees. This means we are essentially finding a new point on the unit circle based on the angle θ.

Trigonometric Functions: The trigonometric functions sine (sin) and cosine (cos) are particularly useful for calculating the coordinates of points on the unit circle.

sin(θ) gives the y-coordinate of a point on the unit circle.

cos(θ) gives the x-coordinate of a point on the unit circle.

Coordinates of Point P: Since we are rotating the initial ray counterclockwise by θ degrees, the coordinates of point P on the unit circle can be obtained as follows:

x-coordinate of P: cos(θ)

y-coordinate of P: sin(θ)

Distance from the Origin (Length of Segment OP):

Using the coordinates of point P, we can calculate the distance between the origin (0, 0) and point P using the distance formula.

The distance formula states that for two points (x1, y1) and (x2, y2), the distance between them is given by:

d = √((x2 - x1)² + (y2 - y1)²)

In this case, point P has coordinates (cos(θ), sin(θ)), and the origin is (0, 0). Thus, the distance (length of segment OP) is:

d = √((cos(θ) - 0)² + (sin(θ) - 0)²)

= √(cos²(θ) + sin²(θ))

= √(1) [Using the trigonometric identity: sin²(θ) + cos²(θ) = 1]

= 1

Therefore, the length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.

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The new coordinates of the quadrilateral ABCD after the dilation is:

A' = (-6, 6)

B' = (-8, -4)

C' = (4, -8)

D' = (0, 0)

We have,

To find the coordinates of the new quadrilateral after a dilation with a scale factor of 2 centered at the origin, we multiply the x and y coordinates of each point by the scale factor.

Given the original coordinates of the quadrilateral ABCD:

A = (-3, 3)

B = (-4, -2)

C = (2, -4)

D = (0, 0)

To dilate the coordinates with a scale factor of 2, we multiply each coordinate by 2:

A' = 2 x (-3, 3) = (-6, 6)

B' = 2 x (-4, -2) = (-8, -4)

C' = 2 x (2, -4) = (4, -8)

D' = 2 x (0, 0) = (0, 0)

Thus,

The new coordinates of the quadrilateral ABCD after the dilation is:

A' = (-6, 6)

B' = (-8, -4)

C' = (4, -8)

D' = (0, 0)

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The quotient of[tex](6x/(3x+15)) / ((x^2+2x)/ (x^2+7x+10))[/tex] is 2x.

To simplify the expression[tex](6x/(3x+15)) / ((x^2+2x)/ (x^2+7x+10)),[/tex] we can use the rule for dividing fractions, which states that dividing by a fraction is the same as multiplying by its reciprocal.

Let's simplify the expression :

Simplify the numerator and denominator of the first fraction.

The numerator is 6x, and the denominator is (3x+15).

We can factor out a common factor of 3 from the denominator, which gives us 3(x+5).

So the first fraction simplifies to 6x / 3(x+5).

Simplify the numerator and denominator of the second fraction.

The numerator is (x^2+2x), and the denominator is [tex](x^2+7x+10).[/tex]

We can factor both the numerator and denominator:

Numerator: x(x+2)

Denominator: (x+5)(x+2)

Canceling out the common factor of (x+2), we are left with x / (x+5).

Multiply the two simplified fractions.

Multiplying the fractions, we get:

[tex](6x / 3(x+5)) \times (x / (x+5))[/tex]

Simplify the resulting expression.

Canceling out the common factor of (x+5) in the numerator and denominator, we are left with:

2x / 1

So the quotient simplifies to 2x.

Therefore, the quotient of [tex](6x/(3x+15)) / ((x^2+2x)/ (x^2+7x+10))[/tex] is 2x.

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Answer:

The three costs that contribute to the total cost of a mortgage are the down payment, mortgage tax and monthly interest.

Step-by-step explanation:

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ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.

What is Angle- Angle - Side theorem?

The angle-angle-side theorem, or AAS, tells us that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

If we consider triangle FSH and triangle FSI, we will observe the following;

angle HFS = angle IFSangle FSH = angle FSIlength HS = length SI

So based on the angle-angle-side theorem, or AAS, we can see that triangle FSH is congruent to triangle FSI.

Thus, our answer will be; ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.

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Answer:

Step-by-step explanation:

Apologies, but your questions seem to have errors and incomplete information.

For the first question about Jessica and Melissa sharing 12 pieces of dried pears, it is unclear how much Jessica ate as the fraction is missing. Similarly, information about how many pears Melissa ate is also missing. To answer the question accurately, I need the missing information.

For the second question about the children playing in the park, you mentioned that one-third of the children returned home, but you didn't provide the total number of children initially in the park. Without that information, I cannot determine how many children stayed in the park.

Please provide complete and accurate information for a precise answer.

The first five terms of the following generalized Fibonacci sequence are -19, 14, -5, 9, 4

Finding the first five terms of the following generalized Fibonacci sequence

From the question, we have the following parameters that can be used in our computation:

The generalized Fibonacci sequence

In the sequence, we can see that the last two terms are added to get the new term

Also, we have

a(1) = -19

a(2) = 14

Using the above as a guide, we have the following:

a(3) = -19 + 14 = -5

a(4) = -5 + 14 = 9

a(5) = 9 - 5 = 4

Hence, the first five terms of the following generalized Fibonacci sequence are -19, 14, -5, 9, 4

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The rental car is getting approximately 37.56 miles per gallon.

To convert kilometers per liter (km/l) to miles per gallon (mpg), you can use the following conversion factor:

1 km/l ≈ 2.82475 mpg

Therefore, if your rental car is getting 13.3 km/l of gasoline, the equivalent value in miles per gallon would be:

13.3 km/l x 2.82475 mpg ≈ 37.56 mpg

So, your rental car is getting approximately 37.56 miles per gallon.

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Answer:

$1250

Step-by-step explanation:

A = P + Prt

1337.5 = P + P(0.04)(21/12)

1337.5 = 1.07P

P = 1250

Answer: $1250

Collecting the like terms in the expression 5z - 7 - 13z + 35 gives -8z + 28

How to collect like terms on the expression

From the question, we have the following parameters that can be used in our computation:

5z - 7 - 13z + 35

Consider an expression given as

Ax + Bx

Where A and B are constants and x is a variable

The terms Ax and Bx are like terms

using the above as a guide, we have the following:

5z - 7 - 13z + 35 = 5z - 13z - 7 + 35

Evaluate the like terms

5z - 7 - 13z + 35 = -8z + 28

Hence, collecting the like terms gives -8z + 28

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The volume of the cylinder is 98π cubic centimeters.

To find the volume of a cylinder, we use the formula:

Volume = π × [tex]r^2[/tex] × h

Given:

Radius (r) = 7 cm

Height (h) = 2 cm

Substituting the values into the formula, we have:

Volume = π × (7 [tex]cm)^2[/tex] × 2 cm

Calculating the values inside the parentheses:

Volume = π × 49 [tex]cm^2[/tex] × 2 cm

Multiplying the values:

Volume = 98π [tex]cm^3[/tex]

Therefore, the volume of the cylinder is 98π cubic centimeters.

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The quadrilateral PQRS is a parallelogram since opposite sides PQ and SR have the same lengths and slopes.

To determine the correctness of the statements and calculate the lengths and slopes of the given quadrilateral PQRS, let's perform a coordinate proof.

Length and slope of PQ:

The coordinates of points P and Q are P(0, 2) and Q(3, -4), respectively.

Length of PQ: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(3 - 0)² + (-4 - 2)²]

= √[9 + 36]

= √45

= 3√5 (approx.)

Slope of PQ: (y₂ - y₁) / (x₂ - x₁)

= (-4 - 2) / (3 - 0)

= -6 / 3

= -2

Therefore, the length of PQ is approximately 3√5, and its slope is -2.

Length and slope of SR:

The coordinates of points S and R are S(-2, 1) and R(1, -5), respectively.

Length of SR: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(1 - (-2))² + (-5 - 1)²]

= √[9 + 36]

= √45

= 3√5 (approx.)

Slope of SR: (y₂ - y₁) / (x₂ - x₁)

= (-5 - 1) / (1 - (-2))

= -6 / 3

= -2

Therefore, the length of SR is approximately 3√5, and its slope is -2.

Length and slope of SP:

The coordinates of points S and P are S(-2, 1) and P(0, 2), respectively.

Length of SP: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(0 - (-2))² + (2 - 1)²]

= √[4 + 1]

= √5

Slope of SP: (y₂ - y₁) / (x₂ - x₁)

= (2 - 1) / (0 - (-2))

= 1 / 2

Therefore, the length of SP is √5, and its slope is 1/2.

Length and slope of HQ:

The coordinates of points H and Q are H(0, 2) and Q(3, -4), respectively.

Length of HQ: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(3 - 0)² + (-4 - 2)²]

= √[9 + 36]

= √45

= 3√5 (approx.)

Slope of HQ: (y₂ - y₁) / (x₂ - x₁)

= (-4 - 2) / (3 - 0)

= -6 / 3

= -2

Therefore, the length of HQ is approximately 3√5, and its slope is -2.

Based on the calculations, we can conclude that both statements are correct:

The quadrilateral PQRS is a parallelogram since opposite sides PQ and SR have the same lengths and slopes.

The quadrilateral PQRS is not a rectangle since the lengths of the adjacent sides PQ and SP are not equal.

Summary:

Length of PQ is approximately 3√5, and its slope is -2.

Length of SR is approximately 3√5, and its slope is -2.

Length of SP is √5, and its slope is 1/2.

Length of HQ is approximately 3√5, and its slope is -2.

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The expression that is equivalent to 6/q is 6q/q²

How to determine what the expression is equivalent to

From the question, we have the following parameters that can be used in our computation:

6/q = ?/q²

Multiply both sides of the equation by q²

So, we have

q² * 6/q = ?/q² * q²

Evaluate the products on both sides of the equation

? = 6q

Hence, the expression that is equivalent to 6/q is 6q/q²

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Answer:

84

Step-by-step explanation:

a² + b² = c²

4² + b² = 10²

16 + b² = 100

b² = 100 - 16

b² = √84

Answer:

Hi

Please mark brainliest ❣️

Thanks

Step-by-step explanation:

Using Pythagorean theorem

hyp² = adj² + opp²

10² = 4² +b²

100-16 = b²

84= b²

√84 = b²

.°. b = 9.17

or b = √84

The number that belongs in the green box is 32.8

How to find the number that belongs in the green box

From the question, we have the following parameters that can be used in our computation:

The triangle

The third angle in the triangle is calculated as

Third angle = 180 - 37 - 64

Evaluate

Third angle = 79

If the number that belongs in the green box is x, then we have

x/sin(79) = 30/sin(64)

This gives

x = sin(79) * 30/sin(64)

Evaluate

x = 32.8

Hence, the number that belongs in the green box is 32.8

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When you know the volume of a prism and some dimensions, you can solve for an unknown dimension.

How to solve for unknown dimension ?

When armed with knowledge about the volume of a prism alongside certain known dimensions, the possibility emerges to determine an elusive dimension within the prism. By leveraging the given information, the missing dimension can be unearthed, unraveling the intricacies of the prism's complete set of measurements.

This calculation empowers us to gain a comprehensive understanding of the geometric structure, further enriching our grasp of its spatial characteristics. The interplay between the volume and the known dimensions acts as a gateway to unlocking the enigma of the unknown dimension.

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The reason that is NOT typically used in a proof is two angles being congruent. Option C.

Among the options A, B, C, and D, the reason that is NOT typically used in a proof is option C: two angles being congruent.

Congruence is a fundamental concept in geometry that refers to the equality of shape and size. When two angles are congruent, it means they have the same measure. While congruence is an important concept used in geometric proofs, it is typically not used as a standalone reason in a proof.

In geometric proofs, various properties, theorems, and postulates are used to establish relationships and make deductions. Let's briefly discuss the other options:

A. Definition of supplementary angles: This is a valid reason that is commonly used in proofs involving supplementary angles. The definition states that if the sum of two angles is 180 degrees, they are considered supplementary.

B. Substitution property: The substitution property is a logical property used in algebraic proofs. It allows us to replace one expression with another expression that is equal to it. This property is often employed when simplifying equations or substituting known values.

D. Parallel lines: The concept of parallel lines is frequently utilized in geometric proofs, particularly in proving angles and angle relationships. Properties such as alternate interior angles, corresponding angles, and vertical angles are established based on the parallel lines. Option C is correct.

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