30 60 90 triangle. Hypotenuse 40. Find legs X and y

Answer:62 miles/hour

Step-by-step explanation:372 miles / 6 hours = 62 miles/hour 62 * 15 = 930 A driver travels 372 miles in 6 hours. At that rate, the driver will travel 930 miles in 15 hours

Answer: 62 mph

Step-by-step explanation:

372/6=62

:)

Answering the question, we may state that According to the graph, from function largest y-intercept to smallest y-intercept, we have:

[tex]f(x) = 5 f(x) = 2 f(x) + 1 f(x) = 1/2 f(x) = 1/5 (x)[/tex]

Mathematicians investigate the relationships between numbers, equations, and related structures, as well as the locations of forms and possible placements for these items. A set of inputs and their corresponding outputs are referred to as a "function" in this context. If each input results in a single, unique output, the relationship between the inputs and outputs is known as a function. Each function has its own domain, codomain, or scope. A common way to denote functions is with the letter f. (x). is an x for entry. One-to-one capabilities, so multiple capabilities, in capabilities, and on functions are the four main categories of accessible functions.

According to the graph, from largest y-intercept to smallest y-intercept, we have:

[tex]f(x) = 5 f(x) = 2 f(x) + 1 f(x) = 1/2 f(x) = 1/5 (x)[/tex]

As a result, the sequence is:

[tex]f(x) = 5^(x) (highest y-intercept) (highest y-intercept)[/tex]

[tex]f(x) = 2^(x) + 1 f(x) = 1/2^ (x)[/tex]

[tex]f(x) = 1/5^(x) (lowest y-intercept) (lowest y-intercept)[/tex]

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1. The values of x, y, and z are x = 15.5, y = 9.54, and z = 0. 2. The values of x and y are x = 1.4375 and y = 8. 3. The values of x and y are x = 6 and y = 52.5. 4. X can have any value and the triangles will still be similar

1. We are given that ΔPRS is congruent to ΔCFH.

From ΔPRS, we know that:

∠P = 180 - 28 - ∠R

∠P = 152 - 13y

From ΔCFH, we know that CH is the hypotenuse and CF is one of the legs. So, using the Pythagorean Theorem, we have:

CH^2 = CF^2 + FH^2

39^2 = 24^2 + FH^2

FH^2 = 39^2 - 24^2

FH = sqrt(39^2 - 24^2) = 30

Since ΔPRS is congruent to ΔCFH, their corresponding sides are equal. Therefore:

PS = CH = 39

2x - 7 = CF = 24

Solving for x and y:

2x - 7 = 24

2x = 31

x = 15.5

39 = 2x - 7

46 = 2x

x = 23

∠P = 152 - 13y

28 = 152 - 13y

124 = 13y

y = 9.54

Solving for z:

PS = 2x - 7

39 = 2(15.5) - 7

39 = 31

z = 0

Therefore, the values of x, y, and z are x = 15.5, y = 9.54, and z = 0.

2. We are given that ΔABC is similar to ΔDEF. Therefore, the corresponding sides are proportional:

AB/DE = BC/EF = AC/DF

Substituting the given values:

8/(y-6) = 19/(4x-1) = 14/DF

We can solve for x and y using any two of the three ratios.

Let's first solve for x and y using the first two ratios:

8/(y-6) = 19/(4x-1)

Cross-multiplying, we get:

8(4x-1) = 19(y-6)

Expanding the brackets, we get:

32x - 8 = 19y - 114

32x - 19y = -106

Now let's use the third ratio:

14/DF = 8/(y-6)

Cross-multiplying, we get:

14(y-6) = 8DF

Simplifying, we get:

y = (4/7)DF + 6

Substituting this into the equation we got earlier:

32x - 19y = -106

32x - 19[(4/7)DF + 6] = -106

32x - (76/7)DF - 114 = -106

32x - (76/7)DF = 8

Multiplying both sides by 7, we get:

224x - 76DF = 56

Using the equation we got from the third ratio:

14(y-6) = 8DF

14y - 84 = 8DF

14y = 8DF + 84

y = (4/7)DF + 6

Substituting this into the equation we just got:

14[(4/7)DF + 6] = 8DF + 84

8DF + 84 = (56/7)DF + 84

8DF = (56/7)DF

DF = 7

Substituting DF = 7 into the third ratio:

14/DF = 8/(y-6)

14/7 = 8/(y-6)

2 = y-6

y = 8

Now we can substitute y = 8 into the equation we got earlier:

32x - 19y = -106

32x - 19(8) = -106

32x - 152 = -106

32x = 46

x = 1.4375

Therefore, the values of x and y are x = 1.4375 and y = 8.

3. Since ΔZMK ≈ ΔAPY, we know that the corresponding angles are congruent:

m∠M = m∠A

m∠K = m∠Y

Therefore, we can write two equations:

m∠M = 2y + 7

m∠K = 41°

Also, we know that:

m∠M + m∠K + (13x - 37)° = 180°

Substituting the values we have:

112° + 41° + (13x - 37)° = 180°

13x + 116 = 180

13x = 64

x = 4.9231

Substituting x into the third equation:

112° + 41° + (13x - 37)° = 180°

13x + 116 = 180

13(4.9231) + 116 + m∠K = 180

m∠K = 41°

Substituting m∠K = 41° into the second equation:

m∠K = m∠Y

13x - 37 = 41

13x = 78

x = 6

Substituting x into the first equation:

m∠M = 2y + 7

112 = 2y + 7

105 = 2y

y = 52.5

Therefore, the values of x and y are x = 6 and y = 52.5.

4. Since ΔBTS ≈ ΔGHD, we know that the corresponding angles are congruent:

m∠S = m∠H

m∠B = m∠G

Therefore, we can write two equations:

m∠S = 7y + 5

m∠B = m∠G = 21°

Also, we know that:

m∠B + m∠T + m∠S = 180°

Substituting the values we have:

21° + m∠T + 56° = 180°

m∠T = 103°

Now we can use the fact that the sum of the angles in a triangle is 180° to find m∠G:

m∠B + m∠T + m∠G = 180°

21° + 103° + m∠G = 180°

m∠G = 56°

Since we have a pair of similar triangles, we can use their side lengths to set up a proportion:

BS/BT = GD/GH

Substituting the given values:

25/31 = (4x-11)/GH

Solving for GH:

GH = (31/25)(4x-11)

Now we can use the fact that the sum of the angles in a triangle is 180° to find m∠H:

m∠G + m∠H + m∠D = 180°

56° + m∠H + 90° = 180°

m∠H = 34°

Substituting the values we have:

m∠S = 7y + 5

56 = 7y + 5

51 = 7y

y = 7.2857

Substituting y into the first equation:

m∠S = 7y + 5

m∠S = 7(7.2857) + 5

m∠S = 59

Now we can use the fact that the sum of the angles in a triangle is 180° to find m∠T:

m∠B + m∠T + m∠S = 180°

21° + m∠T + 59° = 180°

m∠T = 100°

Now we can use the fact that we have a pair of similar triangles to find x:

BS/BT = GD/GH

25/31 = (4x-11)/GH

25/31 = (4x-11)/((31/25)(4x-11))

Simplifying:

25/31 = 25/31

Therefore, x can have any value and the triangles will still be similar.

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Answer: a. Here is a sketch of the situation described above:

 80 +                       .           July (x = 7)

      |                      .

      |                     .

      |                     .

 60 +           .               .

      |                .

      |                    .

      |                        .

 40 +          .                   .     .

      |                              .

      |                               .

      |                                .

 20 +   .         . . . . . . . . . . . . . .

      |                                      .

      |                                          .

 0  +_______________________________________________

     1   2   3   4   5   6   7   8   9   10  11  12

                          January                 December

b. One possible trigonometric equation that models the temperature throughout the year is:

T(x) = (36cos((2π/12)(x-7))) + 41

where T(x) represents the average temperature in degrees Fahrenheit for month x (with January corresponding to x=1), and the constant term of 41 is added to shift the curve up to match the lowest average temperature recorded.

c. To find the average monthly temperature for the month of March, we simply plug in x=3 into the equation above:

T(3) = (36cos((2π/12)(3-7))) + 41

= (36*cos(-π/3)) + 41

≈ 51.4°F

So the average monthly temperature for the month of March is approximately 51.4 degrees Fahrenheit.

d. To find the period of time during which the average temperature is less than 41°F, we need to solve the inequality:

T(x) < 41

Substituting the equation for T(x) from part b, we get:

(36cos((2π/12)(x-7))) + 41 < 41

Simplifying this inequality, we get:

cos((2π/12)*(x-7)) < 0

We can solve this inequality by finding the values of x for which the cosine function is negative. The cosine function is negative in the second and third quadrants of the unit circle, so we have:

(2π/12)*(x-7) ∈ (π, 2π) ∪ (3π, 4π)

Simplifying this expression, we get:

π/6 < x-7 < π/2 or 5π/6 < x-7 < 2π/3

Adding 7 to both sides of each inequality, we get:

7 + π/6 < x < 7 + π/2 or 7 + 5π/6 < x < 7 + 2π/3

Simplifying these expressions, we get:

7.524 < x < 8.571 or 11.286 < x < 11.857

Therefore, the average temperature is less than 41°F during the period of time from approximately November 24th to December 19th, and from approximately February 15th to March 20th.

Step-by-step explanation:

Answer:

(4, 4) in point form

x = 4, y = 4 in equation form

Explanation:

...

In the given problem, we need to find the area of the trapezoid in which the height is 8 yards. The area of the given trapezoid is 92 square yards.

If the length of a trapezoid's parallel sides and the distance (height) between them are known, the area of the shape may be determined.

A = (a+b)h/2 is the formula for a trapezoid's surface area.

where "a" and "b" are the lengths of the base of the trapezoid and "h" represents the height of the figure.

We have been given the values,

The length of the bases is 10 and 13 yards and,

the height of the trapezoid is 8 yards.

So, according to the formula for determining the area,

Area of a trapezoid,

"A" = {(10 + 13)8} / 2

⇒ A = {184}/2

⇒ A = 92 square yards.

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The required vertices of [tex]$\Delta A'B'C'$[/tex] are A'(-10, 15), B'(-15, -10), and C'(5, -5).

The transformation that defines AA'B'C' can be described as a dilation with center at the origin and scale factor of 5/2.

To find the coordinates of A', B', and C', we can use the following formulas:

[tex]$\begin{align*}A'(x,y) &= \left(\frac{5}{2}\right)x, \left(\frac{5}{2}\right)y \B'(x,y) &= \left(\frac{5}{2}\right)x, \left(\frac{5}{2}\right)y \C'(x,y) &= \left(\frac{5}{2}\right)x, \left(\frac{5}{2}\right)y \\end{align*}$[/tex]

Using the coordinates of A(-4,6), B(-6, -4), and C(2,-2), we can calculate the coordinates of A', B', and C' as follows:

For point A(-4,6), we have:

[tex]$A'(x,y) = \left(\frac{5}{2}\right) x, \left(\frac{5}{2}\right) y = \left(\frac{5}{2}\right) (-4), \left(\frac{5}{2}\right) (6) = (-10, 15)$[/tex]

Therefore, the coordinates of A' are (-10, 15).

For point B(-6,4), we have:

[tex]$B'(x,y) = \left(\frac{5}{2}\right) x, \left(\frac{5}{2}\right) y = \left(\frac{5}{2}\right) (-6), \left(\frac{5}{2}\right) (4) = (-15, 10)$[/tex]

Therefore, the coordinates of B' are (-15, 10).

For point C(2,2), we have:

[tex]$C'(x,y) = \left(\frac{5}{2}\right) x, \left(\frac{5}{2}\right) y = \left(\frac{5}{2}\right) (2), \left(\frac{5}{2}\right) (-2) = (5, -5)$[/tex]

the coordinates of C' are (5, -5).

Therefore, the vertices of [tex]$\Delta A'B'C'$[/tex] are A'(-10, 15), B'(-15, -10), and C'(5, -5).

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The function values are:
a) g(102) = 207
b) g(1-42) = -79
c) g(1-72) = -139
d) g(182) = 369
e) g(1a+22) =  2a + 49
f) g(1a2+2) = 2a2 + 9

The problem is asking to evaluate the function g(x) at specific values of x. To find g(102), for example, we substitute 102 for x in the expression for g(x) and simplify:

g(102) = 2(102) + 5 = 207

Similarly, for g(1-42), we substitute -42 for x:

g(1-42) = 2(-42) + 5 = -79

g(1-72) = 2(1-72) + 5 = -139

g(182) = 2(182) + 5 = 369

For g(1a + 22), we substitute "a+22" for x:

g(1a+22) = 2(a+22) + 5 = 2a + 49

And for g(1a²+2), we substitute "a²+2" for x:

g(1a²+2) = 2(a²+2) + 5 = 2a² + 9

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Where is your question so we may help

Answer:

Step-by-step explanation:

mark me brainliest so i can solve

Check the picture below.

The value of the average residual for a car’s price include the following: C. 3860.7.

In Mathematics, a coefficient of determination (r² or r-squared) can be defined as a number between zero (0) and one (1) that is typically used for measuring the extent (how well) to which a statistical model predicts an outcome.

In Mathematics, a residual value is a difference between the measured (given or observed) value from a residual plot and the predicted value from a residual plot.

Based on the computer output (see attachment), we can logically deduce that the coefficient of determination (r²) and average residual for a car's price are as follows;

r² = 68% = 68/100 = 0.68.

Average residual for a car's price, S = 3860.7.

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The GCF of the polynomial a^(5)b^(7)-a^(3)b^(2)+a^(2)b^(6)-a^(2)b^(2) is a^(2)b^(2), and the factored form of the polynomial is a^(2)b^(2)(a^(3)b^(5)-a+b^(4)-1).

The GCF, or greatest common factor, is the largest factor that all terms in a polynomial have in common. In this case, we need to find the GCF of the polynomial a^(5)b^(7)-a^(3)b^(2)+a^(2)b^(6)-a^(2)b^(2).

First, we need to look at the exponents of each term to determine the GCF. The smallest exponent for a is 2, and the smallest exponent for b is 2. Therefore, the GCF for this polynomial is a^(2)b^(2).

Next, we need to factor out the GCF from each term in the polynomial. This is done by dividing each term by the GCF and then multiplying the GCF by the resulting polynomial.

So, the factored form of the polynomial is:

a^(2)b^(2)(a^(3)b^(5)-a+b^(4)-1)

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Answer:

a) 1024 - 14280x + 720x² - 240x³

b) 117616

Step-by-step explanation:

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Answer:

2010-36=1974

She was born in 1974.

Answer:1974

Step-by-step explanation:

2010-36=1974

3 - 1 = 5, which is true

a. To solve for the first variable, we need to isolate it on one side of the equation. We can do this by subtracting 2y from both sides of the first equation: 2x + y = 5 becomes 2x + y - 2y = 5 - 2y, which simplifies to 2x = 5 - 2y. Now we can divide both sides by 2 to solve for the first variable x: x = (5 - 2y)/2.

b. Now we can use the value of x we just found to solve for the second variable y in the second equation: 2y = 2x - 8. Substituting the value of x in for 2x gives us 2y = (5 - 2y) - 8, which simplifies to 3y = -3. Now, we can divide both sides by 3 to solve for the second variable y: y = -3/3 or simply y = -1.

c. To find the numerical value of the first variable x, substitute the value of y we just found (i.e. y = -1) into the equation we found in Step a. This gives us x = (5 - 2(-1))/2, which simplifies to x = 3/2 or x = 1.5.

d. To check your solution, substitute the numerical values you found for x and y into the original equations. For the first equation, 2x + y = 5, we have 2(1.5) + (-1) = 5. Simplifying this gives us 3 - 1 = 5, which is true, so the solution is correct!

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Answer:

Je suis désolé, mais je ne sais pas à quoi fait référence l'exo 135. Pouvez-vous me donner plus de détails sur ce que vous cherchez à comprendre ou résoudre ? Je suis heureux de vous aider si je peux comprendre la question.

Converting 33 pints into quarts is 16.5 Quarts.

The pint (symbol: pt) is a unit of volume or capacity in both the imperial and United States customary measurement systems.

The quart (abbreviation qt.) is an English unit of volume equal to a quarter gallon. It is divided into two pints or four cups.

To calculate 33 Pints to the corresponding value in Quarts,

We multiply the quantity in Pints by 0.5 (conversion factor).

In this case we should multiply 33 Pints by 0.5 to get the equivalent result in Quarts:

33 Pints x 0.5 = 16.5 Quarts

Hence, 33 Pints is equivalent to 16.5 Quarts.

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The complete factorization of [tex]-3x^2+6x+9[/tex] is -3(x - 3)(x + 1).

A mathematical expression, equation, or polynomial is factorized, sometimes referred to as factored, when it is broken down into factors or simpler expressions.

A technique for factoring a number or a polynomial is called factorisation. The polynomials are divided into the sums of their component parts. As an illustration, x2 + 2x can be factored as x(x + 2), where x and x+2 are the factors that can be multiplied to obtain the original polynomial.

To factor completely [tex]-3x^2+6x+9[/tex], we first need to factor out the greatest common factor, which is -3:

[tex]-3(x^2 - 2x - 3)[/tex]

Now we can factor the quadratic expression inside the parentheses:

-3(x - 3)(x + 1)

Hence, the complete factorization of [tex]-3x^2+6x+9[/tex] is -3(x - 3)(x + 1).

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The number of each type of coin they have is 42 quarters and 12 nickels.

To find out how many of each type of coin Kevin and Randy Muise have, we can use a system of equations. Let's call the number of quarters "q" and the number of nickels "n". We can create two equations based on the information given:

q + n = 54 (the total number of coins)

0.25q + 0.05n = 11.10 (the total value of the coins)

Now we can use the substitution method to solve for one of the variables. Let's solve for "n" in the first equation:

n = 54 - q

Now we can substitute this value of "n" into the second equation:

0.25q + 0.05(54 - q) = 11.10

Simplifying and solving for "q":

0.25q + 2.7 - 0.05q = 11.10

0.20q = 8.40

q = 42

Now we can plug this value of "q" back into the first equation to find "n":

n = 54 - 42

n = 12

So Kevin and Randy Muise have 42 quarters and 12 nickels in their jar.

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The equation of a perpendicular line is y = -x + 2

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

y - 3 = 1(x - 2)

We can use the point-slope form of a linear equation to write the equation of the perpendicular line:

y - y1 = m(x - x1)

By comparison, we have

m1 = 1

For perpendicular lines. we have

m = -1/m1

So, we have

m = -1

An example of an equation wit a slope of -1 is

y = -x + 2

Hence, the equaton is y = -x + 2

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1. ∆ABC and ACD are not necessarily similar

2. ∆ABC and ADE are similar by SAS similarity

3. . ∆ABC and AGF are similar by SAS similarity

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.

1. ABC and ACD are not similar because there is only one corresponding Similar sides

2. ABC and ADE are similar because there are two corresponding sides and an equal angle A'

3. ABC and ACD are similar because is equal angle A and two corresponding sides.

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Answer:

We can simplify this expression as follows:

5.4 x 10^-8 / 1.5 x 10^4 = (5.4/1.5) x (10^-8 / 10^4) = 3.6 x 10^-12

Therefore, the result in scientific notation is 3.6 x 10^-12.

The following operations with functions:

1. (g-f)(1) = 7

2. (h-g)(t) = -1

3. (g -f(a) = -38 - a

4. g(3) - h(3) = -16

5. (h+g)(10) = -6

6. (f-g)(4) = 1

Complete each operation with functions.
1. (g-f)(1) = g(1) - f(1) = (2-1) - (-2-4) = 1 + 6 = 7
2. (h-g)(t) = h(t) - g(t) = (2t+1) - (2t+2) = -1
3. (g-f)(a) = g(a) - f(a) = (-30-3) - (a+5) = -33 - a - 5 = -38 - a
4. g(3) - h(3) = (2(3)-5) - (4(3)+5) = (6-5) - (12+5) = 1 - 17 = -16

5. (h+g)(10) = h(10) + g(10) = (3(10)+3) + (-4(10)+1) = (30+3) + (-40+1) = 33 - 39 = -6
6. (f-g)(4) = f(4) - g(4) = (4(4)-3) - (4+2(4)) = (16-3) - (4+8) = 13 - 12 = 1

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Olivia will take time of 4 hour to drive from Guilford to Bath.

Let the speed of Dave be 'x' km/h

Then,

Olivia's speed = ( x + 15 )km/h

Time = 3 hours.

Distance =  180 kilometres

Using relations:

Speed  = distance /time

x + 15 = 180/3

x + 15 = 60

x = 60 - 15

x = 45 km/hr.

Time taken by Olivia to drive from Guilford to Bath.

45 = 180/t

t = 180 / 45

t = 4 hours.

Thus,  it take Olivia 4 hour to drive from Guilford to Bath.

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If a+b=4, and a²+b²=12, then what is a⁴+b⁴ is (B) 136.

To find the value of a⁴ + b⁴, we can use the identity (a² + b²)² = a⁴ + 2a²b² + b⁴. We are given that a² + b² = 12, so we can plug that value into the identity to get:

(12)² = a⁴ + 2a²b² + b⁴

144 =  a⁴ + 2a²b² + b⁴

We can also use the identity (a + b)² = a² + 2ab + b² to find the value of 2a²b². We are given that a + b = 4, so we can plug that value into the identity to get:

(4)² = a² + 2ab + b²

16 = a² + 2ab + b²

Subtracting a^2 + b^2 from both sides gives us:

16 - (a² + b²) = 2ab

16 - 12 = 2ab

4 = 2ab

2 = ab

So we can plug the value of 2ab back into the first identity to get:

144 = a4⁴ + 2(2)² + b⁴

144 = a^⁴ + 8 + b⁴

Subtracting 8 from both sides gives us:

136 = a⁴ + b⁴

So the value of a⁴ + b⁴ is 136, which is option (B).

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8 is an abelian group.

The maximum possible order of an element in the group ????8 is 8. No, ????8 is not a cyclic group, as the only cyclic group of order 8 is a group with one element. However, ????8 is an abelian group. An abelian group is a group in which the result of the group operation is independent of the order of its operands.

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Answer: 19

Step-by-step explanation: The formula for the area of a quadrilateral is length x width.

Here, we have the width missing. So our equation would be,

12x=228.

dividing both sides by 12 gives us x=228/12.

solving this, we get x=19.

So, your answer will be 19.

Answer:

The width of the rectangle is 19 feet

Step-by-step explanation:

The formula for the area of a rectangle is length times width which can be expressed as

[tex]A=lw[/tex]

We can rearrange the equation and solve for the width.

Divide both sides of the equation by [tex]l[/tex].

[tex]\frac{A}{l}=w[/tex]

Now we have an equation to evaluate the width.

Numerical Evaluation

Substituting our values into the equation yields

[tex]\frac{228}{12}=w[/tex]

[tex]w=19[/tex]

The value of P for the given isosceles trapezoid is 21.

An isosceles trapezoid is a four-sided figure with two parallel sides (called bases) of different lengths, and two non-parallel sides of equal length.

The non-parallel sides are also called legs. The two parallel sides are connected by two diagonal lines that intersect each other at a midpoint, forming two congruent triangles.

The following properties are characteristic of an isosceles trapezoid:

For this case, if m∠I = 106⁰, then m∠J = 106⁰

So the value of P is calculated as follows;

m∠J = 5p + 1 = 106

5p + 1 = 106

5p = 106 - 1

5p = 105

p = 105 / 5

p = 21

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Answer:

(10,14)

Step-by-step explanation:

if you look there's a pattern

ABDC is a rectangle, we can conclude that AD is congruent to BC.

What is Triangle ?

Triangle can be defined in which it consists of three sides, three angles and sum of three angles is always 180 degrees.

In the given diagram, we have a parallelogram ABCD. To prove that AD is congruent to BC, we need to show that ABDC is a rectangle.

Here's the proof:

Since ABCD is a parallelogram, we know that:

AB is parallel to CD

BC is parallel to AD

Also, we have:

∠A + ∠B = 180° (opposite angles of a parallelogram)

∠D + ∠C = 180° (opposite angles of a parallelogram)

From the diagram, we can see that:

∠A + ∠D = 180° (adjacent angles of a parallelogram)

∠B + ∠C = 180° (adjacent angles of a parallelogram)

Adding the last two equations, we get:

∠A + ∠D + ∠B + ∠C = 360°

But we know that the sum of the angles in a rectangle is 360°. Therefore, if we can prove that ABDC is a rectangle, we can conclude that AD is congruent to BC.

To show that ABDC is a rectangle, we need to prove that:

AB is perpendicular to BC

BC is perpendicular to CD

CD is perpendicular to AD

AD is perpendicular to AB

Since AB is parallel to CD and BC is parallel to AD, we can conclude that ∠ABC and ∠CDA are alternate interior angles and are therefore congruent. Similarly, ∠ABD and ∠DCB are alternate interior angles and are congruent.

Now, we can prove that ABDC is a rectangle by showing that all its angles are right angles. We can do this by proving that:

∠ABC + ∠ABD = 90° (interior angles of a triangle)

∠CDA + ∠DCB = 90° (interior angles of a triangle)

Since ∠ABC and ∠CDA are congruent, and ∠ABD and ∠DCB are congruent, we have:

∠ABC + ∠ABD = ∠CDA + ∠DCB

Substituting the values of these angles, we get:

2∠ABC = 2∠CDA

∠ABC = ∠CDA

Therefore, ∠ABC and ∠CDA are both 45 degrees. Similarly, we can show that ∠ABD and ∠DCB are both 45 degrees. Hence, all angles of ABDC are 90 degrees, and we have proven that ABDC is a rectangle.

Since , ABDC is a rectangle, we can conclude that AD is congruent to BC.

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