Please help !!!! Given m∥n, find the value of x.

(3x+5) (x-25) = 180 degrees

Answer:

Step-by-step explanation:

To find the number of students who studied in the library or cafeteria (including both), we need to add the number of students who studied in the library and the number of students who studied in the cafeteria, but we need to subtract the number of students who studied in both the library and cafeteria to avoid counting them twice.

So, the number of students who studied in library or cafeteria is:

170 + 135 - 76 = 229

Therefore, 229 students studied in the library or cafeteria (including both).

Answer:

r = -2

Step-by-step explanation:

5r + 20/10 = 3r -6/3

5r + 2 = 3r - 2

2r + 2 = -2

2r = -4

r = -2

Answer: r = -2

Step-by-step explanati

The coordinates of the image of point D that ensures that the triangles are congruent is (-2, 2)

Given the triangle ABC and the incomplete triangle DEF

From the triangle transformation of points B and C, we can see that

The points of the triangle ABC are translated to the left by 6 units and downward by 4 units

Mathematically, this can be represented as

(x, y) = (x - 6, y - 4)

Given that

A = (4, 6)

So, we have

D = (4 - 6, 6 - 4)

Evaluate the difference

D = (-2, 2)

Hence, the position is (-2, 2)

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The equation of the line with slope 2 that goes through the point (6,1) in slope-intercept form is y = 2x - 11. This means that the y-intercept of the line is -11, and the slope of the line is 2, which means that for every increase of 1 in x, the line will increase by 2 in y.

To find the equation of a line with a given slope and a point on the line, we can use the point-slope form of a linear equation:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point on the line.

In this case, the slope is given as 2 and the point (6,1) is on the line. Plugging these values into the equation, we get:

y - 1 = 2(x - 6)

Expanding the right side, we get:

y - 1 = 2x - 12

Adding 1 to both sides, we get:

y = 2x - 11

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The cοst οf painting the triangle bοrder at the given rate is Rs. [tex]108\sqrt{(2)[/tex].

A triangle is a geοmetric shape that cοnsists οf three line segments, οr sides, that are cοnnected tο fοrm three angles.

Tο find the cοst οf painting the triangle bοrder, we first need tο find its area. Let's call the third side οf the triangle "x".

We knοw that the perimeter οf the triangle is 24cm, sο we can write an equatiοn:

6cm + 8cm + x = 24cm

Simplifying this, we get:

x = 10cm

Nοw we can use Herοn's fοrmula tο find the area οf the triangle:

s = (6cm + 8cm + 10cm)/2 = 12cm

Area [tex]= \sqrt{(s(s-6cm)(s-8cm)(s-10cm))[/tex]

[tex]= \sqrt{(12cm6cm4cm*2cm)[/tex]

[tex]= 2\sqrt{(72cm^2)[/tex]

[tex]= 12\sqrt{(2) cm^2[/tex]

Finally, we can calculate the cοst οf painting the bοrder at a rate οf Rs. 9 per square cm:

Cοst = (Area) x (Rate)

[tex]= (12\sqrt{(2)} cm^2) x (Rs. 9/cm^2)[/tex]

[tex]= Rs. 108\sqrt{(2)[/tex]

Therefοre, the cοst οf painting the triangle bοrder at the given rate is= [tex]Rs. 108\sqrt{(2)[/tex]

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The area of the bigger solid is  67.24 cm².

Assume the smaller shape has a length of 2x and a width of 2y. The larger shape would then have 5x length and 5y width.

Because the area of a rectangle is the product of its length and width, the area of the smaller shape is:

Area of smaller shape = 2x * 2y = 4xy

We know that the similar shapes have an area of 78 cm2. As a result, we can write:

4xy + larger area = 78

(larger shape area) / (smaller shape area) = (linear scale factor)²

Substituting the values from the problem yields:

(larger shape area) / (4xy) = (5/2)2 = 25/4

When we multiply both sides by 4xy, we get:

larger shape area = (25/4) * 4xy = 25xy

Now we can plug the expression we discovered for the area of the smaller shape into the equation we found earlier:

4xy + 25xy = 78

29xy = 78

xy = 78/29

Substituting this xy value into the expression we discovered for the area of the larger shape yields:

larger shape area = 25xy = 25 * (78/29) = 67.24 cm2 (rounded to two decimal places)

As a result, the larger shape has an area of approximately 67.24 cm2.

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The ratio of frequencies between G# and D# is: G# / D# = 415.3 / 293.7 ≈ 1.414

To find the missing frequencies for G# and D# using the ratio 1.0595, we need to multiply the frequency of the previous note by 1.0595. Starting from A440, we can use this ratio to calculate the frequencies of G# and D#:

G#: 440 x 1.0595^8 ≈ 830.6 Hz

D#: 440 x 1.0595^6 ≈ 622.3 Hz

The ratio of frequencies between G# and D# is:

830.6 / 622.3 ≈ 1.334

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In 2005, even in cities with low population densities, housing costs were likely at least 61% of the national average.

We know that the relative housing cost for a US city is defined to be the ratio average housing cost for the city /​national average housing cost, expressed as a percent also the scatterplot above shows the relative housing cost and the population density for several large US cities in the year 2005, therefore,

the equation is y = 0.0125x + 61

Therefore, with this we know that in 2005, even in cities with low population densities, housing costs were likely at least 61% of the national average.

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The total interest to be paid over the 5-year period is $403.22.

In a loan with a declining balance, interest is added to the outstanding balance at the start of each period, such as each month or each quarter. The outstanding balance, which decreases as the borrower makes repayments, is used to determine the interest rate. As a result, the borrower gradually pays less interest and more of each payment is used to lowering the main balance due. The lowering balance approach is frequently applied to mortgages, personal loans, and auto loans.

Given that, the quarterly installments are $1899.75.

For the entire year for a total of 5 years the total amount is:

Total amount = 4 * 5 * $1899.75 = $37995

For reducing balance loan the interest is calculated as:

For the first quarter, the interest to be paid is:

interest = (total amount - 0) * (0.14 / 4)

interest = (37995 - 0) * (0.14 / 4) = $94.50

Similarly for the remaining quarters we have:

Second quarter = $87.95

Third quarter: $81.05

Fourth quarter: $73.76

Fifth quarter: $65.96

The total interest to be paid over the 5-year period is:

$94.50 + $87.95 + $81.05 + $73.76 + $65.96 = $403.22

Hence, the total interest to be paid over the 5-year period is $403.22.

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If the top of Bria's bookcase has a length-to-width ratio of 3:1, the width of the top should be about 10 inches to provide an area of 300 square inches for her soap carving collection.

The top of the bookcase is a rectangle, the area of the top can be expressed as the product of its length and width:

Area of the top = length × width

Area of the top = 300 square inches

length × width = 300 square inches

width = 300 square inches / length

Assuming that the top of the bookcase is with a length-to-width ratio of

3 : 1

length = 3 × width

substituting the value gives

width = 300 / (3 × width)

Simplifying, we have:

width² = 100 square inches (the polynomial)

Taking the square root of both sides, we get:

width = √100 ≈ 10 inches

Then the length will be = 30 inches

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Answer:

[tex]b) \quad(-1,0,2)[/tex]

Step-by-step explanation:

The domain is the set of all input values for a function. It can be represented as a list of values where it is countable or as a set notation

Here there are only 3 ordered pairs. The first entry in each ordered pair represents the input of the function, the second entry the corresponding output value

Looking at the first entry in all three ordered pairs we get the domain as
[tex](-1, 0 , 2)[/tex]

The cumulative distribution function (CDF) for given random variable fx(x)  is given by F(x) = 1 - e^[(-a²)(x²/2)]        x > 0,

                           F(x) = 0                                x ≤ 0.

The cumulative distribution function (CDF) F(x) for a Rayleigh random variable X is defined as,

F(x) = P(X ≤ x)

To find the CDF of X, we integrate the PDF of X over the interval [0, x],

F(x) = ∫₀ˣ a²x e^[(-a²)(x²/2)] dx

Using the substitution u = (-a²x²/2),

Simplify the integral as follows,

F(x) = ∫₀ˣ a²x e^[(-a²)(x²/2)] dx

= ∫₀^((-a²x²)/2) -e^u du (where u = (-a²x²/2) and x = √(2u/a²))

= [e^u]₀^((-a²x²)/2)

= 1 - e^[(-a²)(x²/2)]

Therefore, the CDF of X  for the Rayleigh random variable X has PDF fx (x)   is equal to,

F(x) = 1 - e^[(-a²)(x²/2)]        x > 0,

F(x) = 0                                x ≤ 0.

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The above question is incomplete, the complete question is:

For a constant parameter a > 0, a Rayleigh random variable X has PDF

fx (x)        =    a²xe^[(-a²)(x²/2)]      x > 0

                     0                              otherwise.

What is the CDF of X?

Answer:

m₆ = 9375

Step-by-step explanation:

Given sequence is

[tex]m,_i = -3(-5)^{i - 1}[/tex]

To find the 6th term, all you have to do is substitute i = 6 and compute

For i = 6 we get

[tex]m_6 = -3(-5)^{6 - 1}\\= -3(-5)^5\\\\= -3(-3125) \\\\= 9375[/tex]

A negative number raised to an odd number is negative that is why (-5)⁵ is negative

The point estimate for the population mean age at first marriage is 21.89, the true population mean age at first marriage falls between 21.815 and 21.965 years with a small margin of error due to a large sample size. A 99% confidence interval is (21.836, 21.944).

The point estimate at first marriage is 21.89.

We can interpret the 99% confidence interval as follows: we are 99% confident that the true population mean age at first marriage falls between 21.815 and 21.965 years.

To compute a 95% confidence interval, we can use the formula:

Margin of error = z*(SE)

where z is the z-score corresponding to the desired confidence level (1.96 for 95% confidence), and SE is the standard error of the mean, which is equal to the standard deviation divided by the square root of the sample size.

Thus, for the given data:

Margin of error = 1.96*(4.787/sqrt(26920)) = 0.054

The 95% confidence interval can be computed as:

21.89 ± 0.054

which gives us a range of (21.836, 21.944).

The margin of error for this confidence interval is small because the sample size is very large (n=26920). As the sample size increases, the standard error of the mean decreases, which in turn reduces the margin of error.

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_____The given question is incomplete, the complete qustion is given below:

a. Use the summary to determine the point estimate of the population mean and margin of error for the confidence interval

b. interpret the confidence interval

c. verify the results by computing a 95% confidence interval with the information provided

d. why is the margin of error for this confidence interval so small? A study asked respondents, "If ever married, how old were you when you first married? The results are summarized in the technology excerpt that follows. Complete parts (a) through (d) below. One-Sample T: AGEWED Variable N Mean StDev SE Mean 99.0% CI AGEWED 26920 21.890 4.787 0.029 (21.815, 21.965) L attention and maintarhaan Hansen

In response to the stated question, we may state that As a result, the 66th derivative of f(x) = 4 sin(x) is 4 sin(x) (x).

In mathematics, the derivative of a function with real variables measures how sensitively the function's value varies in reaction to changes in its parameters. Derivatives are the fundamental tools of calculus. Differentiation (the rate of change of a function with respect to a variable in mathematics) (in mathematics, the rate of change of a function with respect to a variable). The use of derivatives is essential in the solution of calculus and differential equation problems. The definition of "derivative" or "taking a derivative" in calculus is finding the "slope" of a certain function. Because it is frequently the slope of a straight line, it should be enclosed in quotation marks. Derivatives are rate of change metrics that apply to almost any function.

Using the chain rule and the derivative of the sine function repeatedly yields the 66th derivative of the function [tex]f(x) = 4 sin (x).[/tex]

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x), and this pattern repeats itself every two derivatives.

As a result, the first derivative of f(x) is:

[tex]f'(x) = 4 cos (x)[/tex]

The second derivative is as follows:

[tex]f"(x) = -4 sin (x)[/tex]

The third derivative is as follows:

[tex]f"'(x) = -4 cos (x)[/tex]

The fourth derivative is as follows:

[tex]f""(x) = 4 sin (x)[/tex]

And so forth.

[tex]f^{(66)(x)} = 4 sin (x)[/tex]

Because the pattern repeats every four derivatives, the 66th derivative is the same as the second, sixth, tenth, fourteenth, and so on.

As a result, the 66th derivative of f(x) = 4 sin(x) is 4 sin(x) (x).

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Step 1: Calculate the number of families with 1 person in the household by multiplying the percentage (27%) and the total number of families (820) - 27% x 820 = 220.4.

Step 2: Calculate the number of families with 2 people in the household by multiplying the percentage (33%) and the total number of families (820) - 33% x 820 = 271.6.

Step 3: Calculate the number of families with 3 people in the household by multiplying the percentage (24%) and the total number of families (820) - 24% x 820 = 196.8.

Step 4: Calculate the number of families with 4 people in the household by multiplying the percentage (10%) and the total number of families (820) - 10% x 820 = 82.

Step 5: Calculate the number of families with 5 people in the household by multiplying the percentage (4%) and the total number of families (820) - 4% x 820 = 32.8.

Step 6: Calculate the number of families with 6 or more people in the household by multiplying the percentage (2%) and the total number of families (820) - 2% x 820 = 16.4.

Step 7: Add the numbers calculated in Steps 1 - 6 to get the total number of families with at least 3 people in the household - 220.4 + 271.6 + 196.8

Answer:

The answer is G = (4,3)

Step-by-step explanation:

G = (4,3)

Mathematicians divide the world of numbers into seven subsets, namely natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers.

Natural numbers: These are the counting numbers, starting from 1 and continuing indefinitely. They can also be denoted as "N" or "ℕ".

Whole numbers: These are the natural numbers, as well as zero. They can also be denoted as "W" or "ℤ⁺⁰".

Integers: These are the whole numbers, as well as their negative counterparts. They can also be denoted as "Z" or "ℤ".

Rational numbers: These are numbers that can be expressed as a fraction, where the numerator and denominator are integers (and the denominator is not zero). They can also be denoted as "Q" or "ℚ".

Irrational numbers: These are numbers that cannot be expressed as a fraction of two integers. They are often represented by decimal expansions that neither terminate nor repeat. Examples include π and the square root of 2.

Real numbers: These are all the rational and irrational numbers, taken together. They can also be thought of as numbers that can be located on a number line. They are often denoted as "R" or "ℝ".

Complex numbers: These are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (defined as the square root of -1). Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers. They are often denoted as "C" or "ℂ".

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Answer:

C

Step-by-step explanation:

The graph is of the given equation is represented in the figure below.

The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic.

By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude indicates that the price per pound of deli meat has changed by 3.5 units.

Given linear equation represents the cost measured in dollars, as a function of the amount of deli meat, measured in pounds:

[tex]y = 3.5x[/tex]

The graph is of that linear equation as plotted below diagram.

By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude denotes a change in the price per pound of deli meat of 3.5 units.

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The graph is of the given equation is represented in the figure below linear line: Y = 3.5x

The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic.

By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude indicates that the price per pound of deli meat has changed by 3.5 units.
Given linear equation represents the cost measured in dollars, as a function of the amount of deli meat, measured in pounds:
Linear line: Y = 3.5x

The graph is of that linear equation as plotted below diagram.
By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude denotes a change in the price per pound of deli meat of 3.5 units.

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The volume of the new rectangular prism is 160 in³ after it has been dilated with a scale factor of 2.

In this case, the scale factor is 2, which means that the dimensions of the original figure will be multiplied by 2 to get the dimensions of the new figure.

Volume of rectangular prism = length x width x height

20 = l x w x h

Next, we need to find the new dimensions of the rectangular prism after it has been dilated by a scale factor of 2. We can do this by multiplying each dimension of the original rectangular prism by 2.

New length = 2 x l

New width = 2 x w

New height = 2 x h

Now we can find the volume of the new rectangular prism by using the same formula as before, but with the new dimensions:

Volume of new rectangular prism = (2 x l) x (2 x w) x (2 x h)

Simplifying this expression, we get:

Volume of new rectangular prism = 8 x (l x w x h)

We know that l x w x h is equal to the volume of the original rectangular prism, which is 20 in³. So we can substitute this value into the expression to get:

Volume of new rectangular prism = 8 x 20 in³

Volume of new rectangular prism = 160 in³

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The mathematical model that represents the general price movement of mortgages is:     ΔPM = f(Δk, ΔPrin, ΔC)

Where ΔPM is the change in the price of the mortgage, Δk is the change in the required rate of return, ΔPrin is the change in the principal payment, and ΔC is the change in the interest payment.

This means that the change in the price of a mortgage is a function of the change in the required rate of return, the change in the principal payment, and the change in the interest payment.

The securities that are issued to financial institutions to increase liquidity in the secondary mortgage market by financing the origination of mortgages are:

Financial institutions may also receive private-label pass-through securities, although these are not guaranteed by government entities like Ginnie Mae, Freddie Mac, or Fannie Mae.

They are not backed by the government, but by private institutions, which can make them riskier.

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Here, y can be expressed as a linear combination of a1, a2, and a3 is y = (-1/8) a1 + (9/4) a2 - (5/2) a3.

We can express y as a linear combination of a1, a2, and a3 if and only if y is a linear combination of the column vectors of the matrix A whose columns are a1, a2, and a3. We can write this as:

y = c1 a1 + c2 a2 + c3 a3

where c1, c2, and c3 are constants to be determined. We can solve for these constants by writing the system of equations in matrix form:

A [c1; c2; c3] = y

where [c1; c2; c3] is a column vector of the constants c1, c2, and c3. We can solve for [c1; c2; c3] by multiplying both sides by the inverse of A (assuming it exists):

[c1; c2; c3] = A^(-1) y

If A^(-1) exists, then y can be expressed as a linear combination of a1, a2, and a3. Otherwise, y cannot be expressed as a linear combination of a1, a2, and a3.

For y = [1 2 4 0], we have:

A = [-1 3 0 1; 3 1 2 0; 1 1 1 1]

We can compute the inverse of A using row reduction:

[A | I] = [-1 3 0 1 | 1 0 0;

3 1 2 0 | 0 1 0;

1 1 1 1 | 0 0 1]

[R2 - 3R1, R3 - R1] = [-1 3 0 1 | 1 0 0;

0 -8 2 -3 | -3 1 0;

0 -2 1 0 | -1 0 1]

[R2 / (-8), R3 + 2R2] = [1/8 -3/8 0 3/8 | 3/8 -1/8 0;

0 1 0 -1/4 | 3/4 -1/4 0;

0 0 1 -1/2 | 1/2 -1/2 1]

Therefore, A^(-1) = [1/8 -3/8 0 3/8;

0 1 0 -1/4;

0 0 1 -1/2;

0 0 0 0]

We can now compute [c1; c2; c3]:

[c1; c2; c3] = A^(-1) y = [1/8 -3/8 0 3/8;

0 1 0 -1/4;

0 0 1 -1/2;

0 0 0 0] [1; 2; 4; 0] = [-1/8; 9/4; -5/2; 0]

Therefore, y as a linear combination of a1, a2, and a3:

y = (-1/8) a1 + (9/4) a2 - (5/2) a3

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_____The given question is incomplete, the complete question is given below:

Exercise 1.3.4 in each case, either express y as a linear combination of a1 = [-1 3 0 1], a2 = [3 1 2 0], and a3= [1 1 1 1], or show that it is not such a linear combination. here: y = [1 2 4 0]

The likelihood that one or more of the dice will land on an even number is 1296.

The likelihood of an event is quantified by its probability, which is a number. It is stated as a number between 0 and 1, or in percentage form, as a range between 0% and 100%. The likelihood of an event increasing with probability of occurrence.

Four 6-sided dice are rolled what is the probability that at least two dice show least 2 die the same.

For 2 of the same: 5×5×642) =900

For 3 of the same: 5×643) =120

For 4 of the same: 644) =6

Combined: 900+120+6=1026

Total possibilities: 64=1296

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The probability that at least one of the dice lands on an even number is 15/16 or approximately 0.938.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

We can solve this problem by finding the probability that all four dice land on odd numbers and then subtracting this probability from 1 to get the probability that at least one of the dice lands on an even number.

The probability that one dice lands on an odd number is 3/6 = 1/2, and the probability that all four dice land on odd numbers is:

(1/2) × (1/2) × (1/2) × (1/2) = 1/16

Therefore, the probability that at least one of the dice lands on an even number is:

1 - 1/16 = 15/16

So the probability that at least one of the dice lands on an even number is 15/16 or approximately 0.938.

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Answer:

£22

Step-by-step explanation:

50% of 88=88/100 ×50=44

44÷2=25%=22

75% of £88 is deducted, so that 88-66=£22

The correct order of the steps to test the general hypothesis is given as B, A, E, C, D, F.

A hypothesis is basically a uncertain fact that is backed by some logical and solid information. While using the hypothesis, it s important that we use the correct method to verify the credibility of the hypothesis in the system.

So, as per the given option, the correct sequence should be, B, A, E, C, D, F.

We first set up the null hypothesis and the alternative hypothesis, and then we choose the degree of significance. The next phases in hypothesis testing are choosing the test statistics and creating the decision-making rule.

When we get to a conclusion based on the hypothesis after carefully calculating and computing the hypothesis' performance, the procedure is said to be finished.

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Complete question - place the following steps in order. what is the correct order for the general procedure for hypothesis testing?

A. Select the level of significance

B. setup null and alternative hypothesis

C. Establish the decision rule

D. performance computation

E. select the test statistics

F. Draw conclusions

In linear equation, 28  pieces of cake did Mrs Devi buy.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.  

                                   Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

Cost of M cake = $3.5

For every 3 M cakes she bought 7 B cakes.

Let 1 unit be 3 M cakes and 7 B cakes. 1 unit costs= 3(3.5) + 7(2.5)= $28

She bought $112 / $28 = 4 units of cakes.

Which means 12 M cakes and 28 B cakes.

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Answer:

B. 2.1

Step-by-step explanation:

If you draw a line from C to intersect AB perpendicularly at point D so we have 2 right triangles ACD and BCD.

For △ACD, AC is hypotenuse so sinA = CD/AC

=> CD = 5 x sin(20) = 5 x 0.342 = 1.71

then we have AB = AD + BD

Pythagorean theorem: c^2 = a^2 + b^2

for △ACD, 5^2 = 1.71^2 + AD^2

AD^2 = 5^2 - 1.71^2 = 22.0759

AD = 4.70

BD = AB - AD = 6 - 4.70 = 1.30

for △BCD, BC is hypotenuse

BC^2 = BD^2 + CD^2 = 1.30^2 + 1.71^2 = 4.61

BC = √4.61 = 2.1

1. The functiοn [tex]f(x) = 3x2^x+4-1[/tex]undergοes the fοllοwing transfοrmatiοns

A vertical translatiοn dοwnward by 1 unit (the [tex]"-1[/tex]" at the end)

An upward vertical stretch by a factοr οf 3 (the "3" cοefficient in frοnt)

An expοnential grοwth with base 2 (the expοnent "x" in the term [tex]"2^x"[/tex])

A hοrizοntal shift tο the left by 4 units (the "-4" in the expοnent οf [tex]"2^x"[/tex])

2. The functiοn [tex]f(x) = -1/2x5^x-2+6[/tex] undergοes the fοllοwing transfοrmatiοns:

3. The functiοn [tex]g(x) = 1/5log(x+5)+3[/tex] undergοes the fοllοwing transfοrmatiοns:

4. The functiοn [tex]g(x) = -4log(x)-2[/tex] undergοes the fοllοwing transfοrmatiοns:

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The value of the angles in the triangle are:

∠A = 60°, ∠B = 80° and ∠C = 40°

We  are given that BD = DC

Thus, ∠DBC = ∠BCD ---- 1  (angle in isosceles triangle)

We also have ∠BDC = 100°

In ΔBDC

∠BDC + ∠DBC + ∠BCD = 180°  (sum of angles of triangle is 180°)

Using 1:

∠BDC + 2∠DBC = 180°  

100° + 2∠DBC = 180°  

2∠DBC = 180 - 100

2∠DBC = 80

∠DBC = 80/2

∠DBC = 40°

∠DBC = ∠BCD = ∠2 = 40°

Thus, ∠C = 40°

We are given that  m∠1 = m∠2

Thus, ∠1 = ∠2 = 40°

Now,  ∠BDC +  ∠BDA = 180° (Linear pair)

100° + ∠BDA = 180°

∠BDA = 180 - 100

∠BDA = 80°

In ΔABD

∠ABD + ∠BDA +  ∠BAD = 180° (sum of angles of triangle is 180°)

∠1 + ∠BDA +  ∠BAD = 180°

40° +  80° +  ∠BAD = 180°

120° +  ∠BAD = 180°

∠BAD = 60°

So,  ∠A = 60°

∠B =  ∠1  + ∠2 = 40° + 40° = 80°

Therefore,  ∠A = 60°, ∠B = 80° and ∠C = 40°

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Complete Question

m∠1=m∠2

D∈

AC

, BD = DC

m∠BDC = 100°

Find: m∠A, m∠B, m∠C