Drag the tiles to the correct boxes to complete the pairs.
Match each polynomial function with one of its factors.
f(x) = x3 − 3x2 − 13x + 15
f(x) = x4 + 3x3 − 8x2 + 5x − 25
f(x) = x3 − 2x2 − x + 2
f(x) = -x3 + 13x − 12
x − 2
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x + 3
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x + 4
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x + 5
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The two triangles are congruent because they have identical side lengths, implying that all corresponding angles and side lengths are equal.

Congruent triangles are triangles that have the same shape and size, with corresponding angles and side lengths being equal.

The image shows two triangles with given side lengths where the corresponding side lengths are each equal to each other.

Therefore, we can conclude that the two triangles are congruent since they have the exact same side lengths, which indicates that all corresponding angles and side lengths are equal.

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Step-by-step explanation:

Plug in 8 for x, and -7 for y in both systems.

[tex]8a- 7b = 10[/tex]

[tex]8b - 7a = - 5[/tex]

Now, we use the elimination method to isolate a or b.

Let multiply the top equation by 8 and the bottom one by 7.

[tex]64a - 56b= 80[/tex]

[tex]56b - 49a = - 35[/tex]

Eliminate the b variable by adding the two systems.

[tex]15 a= 45[/tex]

[tex]a = 3[/tex]

Now, plug 3 for a into either the systems to find b.

[tex]24 - 7b = 10[/tex]

[tex] - 7b = - 14[/tex]

[tex]b = 2[/tex]

A) Average Velocity = Displacement / Time Interval = -9 / (4 - 1) = -3 meters per second.

B) Acceleration(4) = 96/4^4 - 8/4^3 = 1/8 meters per second squared.

C) The body changes direction when the velocity changes sign.

a. To find the body's displacement, we need to find the difference between the initial and final positions. The position function is given as s [tex]= f(t) = 16/t^2 - 4/t.[/tex]

[tex]For t = 1, the position is:s(1) = 16/1^2 - 4/1 = 12For t = 4, the position is:s(4) = 16/4^2 - 4/4 = 3[/tex]

The displacement is given by the difference in position:

Displacement = s(4) - s(1) = 3 - 12 = -9 meters

To find the average velocity, we divide the displacement by the time interval:

Average Velocity = Displacement / Time Interval = -9 / (4 - 1) = -3 meters per second

b. The speed of the body at the endpoints of the interval can be found by taking the absolute value of the derivative of the position function. The acceleration can be found by taking the derivative of the velocity function.

At t = 1:

[tex]Velocity = f'(t) = d(s)/dt = d/dt(16/t^2 - 4/t) = -32/t^3 + 4/t^2[/tex]

[tex]Speed at t = 1: |Velocity(1)| = |-32/1^3 + 4/1^2| = 28[/tex] meters per second

At t = 4:

[tex]Velocity = f'(t) = d(s)/dt = d/dt(16/t^2 - 4/t) = -32/t^3 + 4/t^2[/tex]

[tex]Speed at t = 4: |Velocity(4)| = |-32/4^3 + 4/4^2|[/tex]= 4 meters per second

To find the acceleration, we take the derivative of the velocity function:

[tex]Acceleration = f''(t) = d^2(s)/dt^2 = d/dt(-32/t^3 + 4/t^2) = 96/t^4 - 8/t^3[/tex]

At t = 1:

[tex]Acceleration(1) = 96/1^4 - 8/1^3[/tex]= 88 meters per second squared

At t = 4:

Acceleration(4) = 96/4^4 - 8/4^3 = 1/8 meters per second squared

c. The body changes direction when the velocity changes sign. From the velocity function, we can see that the velocity will change sign when the denominator of the expression becomes zero, as that would result in a zero in the denominator, making the velocity undefined.

For the given position function, the velocity will change sign when t = 0 or t = 2. However, these values are not within the interval 1 ≤ t ≤ 4. Therefore, the body does not change direction during the interval 1 ≤ t ≤ 4.

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The solution to the system of equations 9x - 16y =54 and 5x + 17y =30 is x = 6 and y = 0.

Given the system of equations in the question:

9x - 16y = 54 -------equ1

5x + 17y = 30 -------equ2

Using the elimination method:
First, multiply equation (1) by 17 and equation (2) by 16 to make the coefficients of y in both equations equal:

17 × (9x - 16y) = 17 × 54

16 × (5x + 17y) = 16 × 30

We have:

153x - 272y = 918 ....... equ3

80x + 272y = 480 .......equ4

Add Equation 3 and Equation 4:

(153x - 272y) + (80x + 272y) = 918 + 480

153x - 272y + 80x + 272y = 918 + 480

233x = 1398

x = 1398/233

x = 6

Now, pug x = 6 into equation 2 and solve for y:
5x + 17y = 30

5(6) + 17y = 30

30 + 17y = 30

17y = 30 - 30

17y = 0

y = 0

Therefore, the value of x is 6 and y is 0.

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The 3! is used to calculate the number of permutations of three objects, and is included in the Poisson probability formula to account for the number of ways in which three events can occur in a specific order over a given period.

In Poisson probability calculation, 3! (3 factorial) is used to find the number of ways in which three independent events can occur in a specific order over a given period. In other words, 3! refers to the number of permutations of 3 objects.

The formula used in Poisson distribution is as follows: [tex]P (X = x) = (e^{-\lambda } \lambda^x)/x![/tex], where λ is the mean rate of the event occurring over a specific time interval, and x is the number of events that occur within that time interval.

The x! term is included in the formula because it represents the number of ways in which x events can occur in a specific order over the given time period. The exponents of λ are used to represent the probability of an event occurring in the interval.

In the case of 3! we are calculating the number of ways that three independent events can occur in a specific order over a given period. This can be expressed as 3 x 2 x 1 = 6, which is why 3! becomes 6 in the Poisson probability calculation.

The term 3! (6) is included in the formula to account for the number of ways in which three events can occur in a specific order over a given period.

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The y-intercept of the line with a slope of 2 and passing through the point (3, 5) is -1.

The y-intercept of a line is the point where the line intersects the y-axis. In the slope-intercept form of a linear equation, y = mx + b, the y-intercept is represented by the value of b.

To determine the y-intercept, we need to have information about the line, such as its slope and at least one point it passes through.

If we are given the slope, m, and a point (x₁, y₁) on the line, we can use the point-slope form of the equation, y - y₁ = m(x - x₁), to find the equation of the line and determine the y-intercept.

For example, let's say we are given the slope m = 2 and a point (3, 5) on the line. We can substitute these values into the point-slope form:

y - 5 = 2(x - 3)

Expanding the equation:

y - 5 = 2x - 6

Next, we can rearrange the equation to isolate y:

y = 2x - 6 + 5

y = 2x - 1

From this equation, we can see that the y-intercept is -1. The point where the line crosses the y-axis is (0, -1), and the value of b in the slope-intercept form is -1.

Therefore, the y-intercept of the line with a slope of 2 and passing through the point (3, 5) is -1.

In general, to determine the y-intercept of a line, we need to know the slope and at least one point it passes through. With this information, we can use the point-slope form of the equation and solve for the y-intercept.

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The arbitrage profit is $1,750,801.62.Borrow $1,000,000 in the U.S. at an annual interest rate of 8%.

Convert the borrowed amount to sterling pounds at the spot exchange rate.

Invest the borrowed sterling pounds in Britain at an annual interest rate of 15%.

Enter into a forward contract to exchange the future value of the investment back to U.S. dollars at the forward exchange rate.

To determine the arbitrage profit, we need to compare the potential gains from two different investment strategies: investing in the U.S. and investing in Britain.

i. Compute arbitrage profit:

1. Investing in the U.S.:

If we invest $1,000,000 in the U.S. at an annual interest rate of 8%, after 180 days (half a year), the investment will grow to:

$1,000,000 * (1 + 0.08/2) = $1,040,000

2. Investing in Britain:

To invest in Britain, we convert the $1,000,000 to sterling pounds using the spot exchange rate:

$1,000,000 * 1.618 = £1,618,000

Then, we invest the £1,618,000 in Britain at an annual interest rate of 15%:

£1,618,000 * (1 + 0.15/2) = £1,737,850

Now, we need to convert the future value of the investment back to U.S. dollars using the forward exchange rate:

£1,737,850 * 1.6052 = $2,790,801.62

Arbitrage profit = Future value of investment in Britain - Future value of investment in the U.S.

= $2,790,801.62 - $1,040,000

= $1,750,801.62

Therefore, the arbitrage profit is $1,750,801.62.

ii. Possible moves to eliminate arbitrage profit in the short run:

To eliminate the arbitrage profit, investors could take advantage of the discrepancy in interest rates and exchange rates by performing the following actions:

- Borrow $1,000,000 in the U.S. at an annual interest rate of 8%.

- Convert the borrowed amount to sterling pounds at the spot exchange rate.

- Invest the borrowed sterling pounds in Britain at an annual interest rate of 15%.

- Enter into a forward contract to exchange the future value of the investment back to U.S. dollars at the forward exchange rate.

By executing these actions, the arbitrage profit would be eliminated as the gains from the investment in Britain would be offset by the interest payments on the borrowed amount.

iii. The situation where the arbitrage profit is nil is referred to as "Arbitrage equilibrium" or "Arbitrage-free market." In this state, there are no opportunities for risk-free profits through exploiting discrepancies in prices, interest rates, or exchange rates. The market prices and rates are in balance, reflecting all available information and preventing arbitrage opportunities.

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Step-by-step explanation:

Area of equalaterial triangle is 1/2bh

Here the base is 7.

The height is 4.2

[tex] \frac{1}{2} (7)(4.2) = 14.7[/tex]

The kinetic energy of this car is equal to 77160.49 Joules.

In Mathematics, the kinetic energy of a physical object can be calculated by using the following equation (formula):

K.E = 1/2 × mv²

Where:

Note: 50 km/h to m/s = 50 × 1000/3600 = 13.89 m/s

By substituting the given parameters into the formula for the kinetic energy of a physical object, we have the following:

Kinetic energy, K.E = 1/2 × 800 × 13.89²

Kinetic energy, K.E = 400 × 192.90

Kinetic energy, K.E = 77160.49 Joules.

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Complete Question:

An 860 kg car moves at 50 km/h, what will its kinetic energy be?

The domain of the function y = √(x + 7) + 5 is all real numbers (all real numbers). Option D.

To determine the domain of the function y = √(x + 7) + 5, we need to identify the values of x for which the function is defined.

In this case, the function includes two operations: addition and square root. To ensure the function is defined, we need to consider the restrictions imposed by each operation.

The square root function (√) is defined only for non-negative real numbers. In other words, the expression inside the square root (√(x + 7)) must be greater than or equal to zero for the function to be defined.

Setting x + 7 ≥ 0, we can solve for x:

x ≥ -7

Therefore, the expression inside the square root (√(x + 7)) is defined for x values greater than or equal to -7.

Additionally, there are no restrictions on the addition operation (+5). It can be applied to all real numbers.

Combining both restrictions, we find that the function y = √(x + 7) + 5 is defined for all real numbers x, since there are no limitations or exclusions. Option D is correct.

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∠LMN is an obtuse angle.

==========================================

Explanation

Let's go through the answer choices to see which are true, and which are false.

Answer:

12 apples

Step-by-step explanation:

Apples eaten by friend = 5

Apples eaten by me = 7

Toral number of apples eaten together

= 5 + 7

= 12

________

hope this helps!

Answer:

12 apples

Step-by-step explanation:

If 1 person eats 5 apples, and another person eats 7, we should add the amount of apples together to get the total.

5+7

=12

So, 12 apples were ate altogether.

Hope this helps! :)

Using the principle of inclusion-exclusion, we can see that the total number of numbers with the given restrictions is:

586,071

o find the number of 7-digit numbers where the digits 1, 2, and 3 each appear at least once, we can use the principle of inclusion-exclusion.

Step 1: Find the total number of 7-digit numbers without any restrictions.

Since each digit can be chosen independently from 0 to 9 (excluding leading zeros), there are 9 options for the first digit (1-9) and 10 options for each subsequent digit (0-9). Therefore, the total number of 7-digit numbers without any restrictions is 9 * 10⁶.

Step 2: Subtract the number of 7-digit numbers that do not contain 1, 2, or 3.

The number of 7-digit numbers that do not contain 1, 2, or 3 is equal to the total number of 7-digit numbers with digits from 0 to 9 (excluding 1, 2, and 3). So for each digit position, there are 7 options (0, 4, 5, 6, 7, 8, 9). Thus, the total number of 7-digit numbers without 1, 2, or 3 is 7⁷.

Step 3: Add back the number of 7-digit numbers that do not contain two of the digits 1, 2, or 3.

The number of 7-digit numbers that do not contain two of the digits 1, 2, or 3 is equal to the total number of 7-digit numbers with digits from 0 to 9 (excluding two of 1, 2, and 3). For each digit position, there are 6 options (0, 4, 5, 6, 7, 8, 9). Thus, the total number of 7-digit numbers without two of 1, 2, or 3 is 6⁷.

Step 4: Subtract the number of 7-digit numbers that do not contain all three digits 1, 2, and 3.

The number of 7-digit numbers that do not contain all three digits 1, 2, and 3 is equal to the total number of 7-digit numbers with digits from 0 to 9 (excluding all three of 1, 2, and 3). For each digit position, there are 4 options (0, 4, 5, 6, 7, 8, 9). Thus, the total number of 7-digit numbers without all three of 1, 2, and 3 is 4⁷.

Now we can apply the principle of inclusion-exclusion to find the desired count:

Total number of 7-digit numbers = 9 * 10⁶ - 7⁷ + 6⁷ - 4⁷

Calculating this expression, we find:

Total number of 7-digit numbers = 586,071

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The experimental probability of members of Chris's class ending up with heads is about 3/5 or 0.6 (60%).

The experimental probability is one that can be calculated by:

dividing the number of favorable outcomes (the number of heads) by the total number of outcomes (the total number of students).

So, the number of heads recorded was 18, and the total number of students was 30.

Experimental probability = Number of heads / Total number of students

= 18 / 30

So one can simplify the fraction by:

Experimental probability = 3 / 5

Therefore, the experimental probability of members of Chris's class ending up with heads is 3/5 .

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The Ratio of 1.24 to 0.08 can be expressed as 15.5, 31:2, or 31/2.

To find the ratio of 1.24 to 0.08, we divide 1.24 by 0.08.

1.24 ÷ 0.08 = 15.5

Therefore, the ratio of 1.24 to 0.08 is 15.5.

We can also express this ratio as a fraction. To do this, we write 15.5 as a fraction over 1:

15.5/1

To simplify the fraction, we can multiply both the numerator and the denominator by 10 to get rid of the decimal:

(15.5 × 10)/(1 × 10) = 155/10

Further simplifying the fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 5:

(155 ÷ 5)/(10 ÷ 5) = 31/2

So, another way to express the ratio 1.24:0.08 is 31:2.

In conclusion, the ratio of 1.24 to 0.08 can be expressed as 15.5, 31:2, or 31/2.

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The probability of x being equal to 2 is 0.13

The probability of x being more than 2 is 0.35

The probability of x ≤ 2 is 0.65

To compute the probabilities using the given probability distribution, we can refer to the provided values for each value of x.

From the table, the probability of x being equal to 2 is 0.13.

The probability of x being more than 2 can be computed by adding the probabilities of x = 3 and x = 4:

P(x > 2) = P(x = 3) + P(x = 4)

P(x > 2) = 0.24 + 0.11

P(x > 2) = 0.35

To compute P(x ≤ 2), add the probabilities of x = 1 and x = 2:

P(x ≤ 2) = P(x = 1) + P(x = 2)

P(x ≤ 2) = 0.52 + 0.13

P(x ≤ 2) = 0.65

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a) The group of student for whom all places are suitable is 15.

b) The presentation of the information in a Venn Diagram about the choices of locations for an educational trip by students of Class 10 studying at Shree Sharada Secondary School is attached.

c) The total number of students studying in Class 10 in Sharada Secondary School is 90.

d) The ratio of students who said that Pokhara was the only suitable place and Lumbini was the only suitable place is 4:5.

A Venn Diagram is a graphical representation of groups of data to show their contrasts or differences and their similarities.

Students of Class 10

Shree Sharada Secondary Schoo

Educational Trip Locations

Pokhara, P = 40

Lumbini, L = 30

Ilam, I = 45

All three places (P, L, and I) = 15

No Opinion = 5

The number of P only = 25 (40 - 15)

The number of L only = 15 (30 - 15)

The number of I only = 30 (45 - 15)

The number of P, L, and I = 15

The number expressing no opinion = 5

b) The Venn Diagram is attached.

c) The total number of students in Class 10 = 90 (25 + 15 + 30 + 15 + 5)

Ratios of Lumbibi as a suitable place to Pokhara only:

The number of P only = 25

The number of L only = 15

Initial ratio = 15:25 = 3:5

Addition of 5 to Lumbini:

The number of P only = 25

The number of L only = 15

d) Final ratio = 20:25 = 4:5

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Answer:

Step-by-step explanation:

Kim's commission was $8140, and her gross pay was $10560 for last month.

Washington Plastic Products pays Kim $2420 monthly salary plus 11% commission each month.

If her sales were $74000 last month, her commission and gross pay can be calculated as follows:

Kim's commission = 11% of $74000

= $8140Kim's gross pay

= $2420 + $8140 = $10560

Therefore, Kim's commission was $8140 and her gross pay was $10560 last month.

In the given problem, Kim's monthly salary is $2420, and she earns an 11% commission on her sales.

Assuming her sales last month were $74000, her commission and gross pay for the month can be found by multiplying her sales by the commission rate and adding her monthly salary.

Commission = 11% of $74000 = $8140Gross Pay = $2420 + $8140 = $10560

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Bob can deduct a total of $2,210 on his Schedule C related to these expenses.

To determine the total amount Bob can deduct on his Schedule C related to these expenses, consider the specific categories that can be deducted. Based on the information provided, categorize the expenses as follows:

Course fees for continuing legal education: $1,400

Airfare and hotel room: $500

Meals provided by the hotel's restaurant: $300

Snacks from the newsstand: $10

To calculate the total amount that Bob can deduct, add up these expenses:

$1,400 + $500 + $300 + $10 = $2,210

Therefore, Bob can deduct a total of $2,210 on his Schedule C related to these expenses.

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The term that best describes the figure shown is (a) line LM

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

The figure

Where, we have

Points L and M

On the figure, we have two arrows that extend indefinetely

By definition of lines, lines extend indefinetely

This means that the term that best describes the figure shown is (a) line LM

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The product of 2 and -5 can be shown on a number line as a movement in the negative direction. To represent this on a number line, we start at zero and move 2 units to the right (in the positive direction) to the point labeled +2. Then, since the product is negative, we move 5 units to the left (in the negative direction) from the point +2. This brings us to the point labeled -3.

Start at zero on the number line.

Move 2 units to the right (in the positive direction) to the point labeled +2.

Since the product is negative, we need to move in the opposite direction (left) from the point +2.

Move 5 units to the left (in the negative direction) from the point +2.

The point we reach after moving 5 units to the left from the point +2 is labeled -3.

Therefore, on the number line, the product of 2 and -5 is represented by the point labeled -3. This indicates that starting from zero and moving 2 units to the right and then 5 units to the left results in a position of -3.

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The probable question could be:

How is the product of 2 and -5 shown on a number line?

The fraction of the area of the tray occupied by the blondies is 1/9.

To find the fraction of the area of the tray occupied by blondies, we need to determine the area occupied by the blondies and compare it to the total area of the tray.

Let's calculate the area of each individual blondie:

The blondies are squares with 3-inch sides, so the area of each blondie is 3 inches × 3 inches = 9 square inches.

Now, let's calculate the area occupied by the blondies in each row:

Since there are 4 rows of blondies and each row contains 4 blondies, the total number of blondies is 4 rows × 4 blondies per row = 16 blondies.

So, the total area occupied by the blondies is 16 blondies × 9 square inches per blondie = 144 square inches.

Next, let's determine the area occupied by the brownies:

The brownies are squares with 6-inch sides, so the area of each brownie is 6 inches × 6 inches = 36 square inches.

Since there are also 4 rows of brownies and each row contains 4 brownies, the total number of brownies is 4 rows × 4 brownies per row = 16 brownies.

Therefore, the total area occupied by the brownies is 16 brownies × 36 square inches per brownie = 576 square inches.

Now, let's calculate the total area of the tray:

Given that the tray is completely full and has an area of 648 square inches, we can subtract the area occupied by the brownies from the total area to find the remaining area occupied by the blondies:

Total area of the tray - Area occupied by the brownies = Area occupied by the blondies

648 square inches - 576 square inches = 72 square inches.

So, the fraction of the area of the tray occupied by the blondies is:

Area occupied by the blondies / Total area of the tray = 72 square inches / 648 square inches.

To simplify this fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 72 in this case:

72 square inches / 648 square inches = 1/9.

Therefore, the blondies' percentage of the tray's surface area is 1/9.

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The sentence that contains correct punctuation is "Tammy is a good worker; she gets everything done on time."

Option A contains correct punctuation. It has a semicolon which separates two independent clauses and also makes a connection between them. This is the correct use of punctuation. It is also known as a semicolon conjunction.

Option B is incorrect as there is no semicolon between "colors" and "red." There should be a semicolon instead of a comma to create a connection between the two clauses.
Option C is incorrect as there is no need for a semicolon after "going home" because there is no second independent clause following it.
Option D is incorrect as there should be a semicolon between "everything" and "including" since there are two distinct groups of items being listed.
In conclusion, option A is the correct answer because it follows the rules of proper punctuation by using a semicolon correctly to join two independent clauses.

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The solution of the compound inequality is as follows:

x < -4 or x ≥ 3.

Compound inequality is an inequality that combines two simple inequalities. In other words, a compound inequality contains at least two inequalities that are separated by either "and" or "or".

Therefore,  let's find the compound inequality graphed on the number line as follows:

Hence,

if the symbol is (≥ or ≤) then you fill in the dot, like the top two examples in the graph below.

if the symbol is (> or <) then you do not fill in the dot .

Therefore, the compound inequality is as follows:

x < -4 or x ≥ 3.

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Answer:

Step-by-step explanation:

The expression 2x - 11x - 6 can be simplified as follows:

2x - 11x - 6 = -9x - 6

So, the equivalent expression is -9x - 6.

The answer is:

-9x - 6

Work/explanation:

To simplify this expression, we combine the like terms :

[tex]\huge\text{2x - 11x - 6} \\ \\ \\ \text{-9x - 6}[/tex]

There's nothing that we can do for this expression. It's been simplified as much as possible.

Answer:

y = - 5x - 1 and y = [tex]\frac{1}{3}[/tex] x + 8

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

14

y = - 5x + 4 ← is in slope- intercept form

with slope m = - 5

• Parallel lines have equal slopes , then

y = - 5x + c ← is the partial equation

to find c substitute P (- 1, 4 ) into the partial equation

4 = - 5(- 1) + c = 5 + c ( subtract 5 from both sides )

- 1 = c

y = - 5x - 1 ← equation of parallel line

15

y + 3x = 13 ( subtract 3x from both sides )

y = - 3x + 13 ← in slope- intercept form

with slope m = - 3

given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-3}[/tex] = [tex]\frac{1}{3}[/tex] , then

y = [tex]\frac{1}{3}[/tex] x + c ← is the partial equation

to find c substitute P (6, 10 ) into the partial equation

10 = [tex]\frac{1}{3}[/tex] (6) + c = 2 + c ( subtract 2 from both sides )

8 = c

y = [tex]\frac{1}{3}[/tex] x + 8 ← equation of perpendicular line

Answer:

Answer:

Answer is A. Find the functioning zeros and vertical asymptotes

This statement follows from the fact that m∠4 + m∠6 = 180°. The statement that completes the proof is C. ∠4 and ∠7 are supplementary.

To complete the proof, let's analyze the given statements and reasons:

m ∥ n (Given) - This statement establishes that lines m and n are parallel.

m∠6 = m∠7 (Vertical angles theorem) - When two lines intersect, the vertical angles formed are congruent.

? (Same-side interior angles theorem) - This step is missing in the given proof. To prove that ∠4 and ∠6 are supplementary, we need to use the same-side interior angles theorem.

The same-side interior angles theorem states that when two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal are supplementary. In this case, lines m and n are parallel, and the transversal is the line that intersects them.

To complete the proof, we can include the missing statement:

∠4 and ∠6 are interior angles on the same side of the transversal (Same-side interior angles theorem).

Now, we can continue with the remaining steps of the proof:

m∠4 + m∠7 = 180° (Definition of supplementary angles) - Supplementary angles add up to 180 degrees.

m∠4 + m∠6 = 180° (Substitution property of equality) - We substitute ∠7 with ∠6 since they are congruent.

∠4 and ∠6 are supplementary (Definition of supplementary angles) - This statement follows from the fact that m∠4 + m∠6 = 180°.

Therefore, the statement that completes the proof is:

C. ∠4 and ∠7 are supplementary.

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