Can the two triangles be proven congruent by SAS explain?

The ratio of the length of each transversal will be  [tex]\frac{\overline {CB}}{ \overline {AB}} = \frac{\overline {EF}}{\overline{DE}} =1[/tex]

The ratio of the lines can be compared using Three Parallel Lines Theorem.

What is Three Parallel Lines Theorem?

When two transversals are intersected by three parallel lines, the transversals are proportionally divided.

According to the three parallel lines theorem, the segments created by each transversal and the three parallel lines are proportionate.

The findings are:

Reasons:

The question is a four part question

Let the equations of the parallel lines be as follows;

Line, x; y = x

Line, y; y = x + 1

Line z; y = x + 2

The points at which transversal 1 intersect the lines x, y, and z, are;

A(0.4, 0.4), B(0.6, 1.6), and C(0.8, 2.8)

The length of segment [tex]{\overline {AB}}[/tex]  = √((0.6 - 0.4)² + (1.6 - 0.4)²) = 0.2·√(37)

The length of segment [tex]{\overline {CB}}[/tex] = √((0.8 - 0.6)² + (2.8 - 1.6)²) = 0.2·√(37)

The ratio of the lengths of the segment formed by transversal 1 is therefore;

         [tex]\sqrt{x} \ ratio\ of\ the\ length \ of segments =\frac{\overline {CB}}{ \overline {AB}} = \frac{2.\sqrt{37} }{2.\sqrt{37} } =1[/tex]

The points at which transversal 2 intersect the lines x, y, and z, are;

D(1.1, 3.1), E(1.3, 2.3), and F(1.5, 1.5)

The length of segment  [tex]{\overline {DE}}[/tex]= √((1.3 - 1.1)² + (2.3 - 3.1)²) = 0.2·√(17)

The length of segment [tex]{\overline {EF}}[/tex] = √((1.5 - 1.3)² + (1.5 - 2.3)²) = 0.2·√(17)

The ratio of the lengths of the segment formed by transversal 2 is therefore;

[tex]\sqrt{x} \ ratio\ of\ the\ length \ of segments =\frac{\overline {EF}}{ \overline {DE}} = \frac{0.2.\sqrt{17} }{0.2.\sqrt{17} } =1[/tex]

Therefore;

            [tex]\frac{\overline {CB}}{ \overline {AB}} = \frac{\overline {EF}}{\overline{DE}} =1[/tex]

Which gives;

The the comparison can also be made with the triangle proportionality theorem.

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The used postulate is explained below.

What is congruent triangles?

Two triangles are stated to be congruent if the 3 facets and the 3 angles of each the angles are same in any orientation.

When angles and facet blanketed in among those angles of 1 triangle are identical as any other one, each those triangle are Congruent.

This standards of congruency is is aware of as ASA criterion as a facet blanketed among angles are all identical.

Two triangles are congruent, if angles and the facet blanketed among them in a single triangle is same to the 2 angles and the facet blanketed among them of the alternative triangle.

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The value of x is in the parallelogram is 10

In a parallelogram, the opposite sides are parallel to each other.

The opposite sides of a parallelogram are equal in length.

The measurement of opposite angles of a parallelogram is equal.

Like all other quadrilaterals, the sum of all the angles of a parallelogram is 360°.

The adjacent or the adjoining angles of a parallelogram add up to 180°.

The adjacent or the adjoining angles of a parallelogram add up to 180°.

9x +8 and 8x + 2 are adjacent angles of the parallelogram

Mathematically, 9x + 8 + 8x + 2 = 180

17x + 10 = 180

17x + = 180 - 10

17x = 170

x = 170/17

x = 10

In conclusion the value of x is 10

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The equation that best represents the area of the forest each year is A. y = 2,000(0.94)^x.

An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario.

Since the certain forest covers an area of 2,000 square kilometers and each year, this area decreases by 6%.

The equation will be:

= 2000(1 - 6%)^x

= 2000(0.94)^x

In conclusion, the correct option is A.

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The rate of change in the given interval would be 1.25 ft/s.

What is the rate of change?

The slope of a graphed function is determined using the average rate of change formula. Divide the change in y-values by the change in x-values to find the average rate of change.

By observation, we can take the points from the graph in between the given interval.

The points would be (3, 2.5) and (5, 4)

Then the rate of change = y2 - y1/ x2 - x1

                                         = 4 - 2.5/5 - 3

                                         = 2.5/2

                                         = 1.25 ft/s.

Hence, the rate of change in the given interval would be 1.25 ft/s.

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a) The system has infinitely many solutions, which are:

(1 - y, y, -3).

b) The system has no solution.

To obtain the number of solutions of each system, the equations are build from the augmented matrix presented.

For item a, the system is given as follows:

Both the left side of the matrix (3 x 3) and the right side (3 x 1), have the same rank, that is, the same number of non-zero rows, hence the system has infinitely many solutions, which are:

For item b, the system is given as follows:

The last statement is false, hence the system has no solutions. It can also be seem by the different ranks of each matrix, with the 3 x 3 matrix having a rank of 2 and the 3 x 1 matrix having a rank of 3.

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First, 108/9=12

This means that are 12 groups of people.

If there are 8 students in each group, 12 x 8 = 96.

96 should be your answer.

The claim that all isosceles triangles are equivalent to one another is untrue.

There are two equal sides and two equal angles in an isosceles triangle.

The term is derived from the Greek words iso (same) and Skelos (leg).

A scalene triangle is one in which none of its sides are equal, as opposed to an equilateral triangle, which has all of its sides equal.

A right triangle with two equal-length legs is called an isosceles right triangle.

The corresponding angles would also be congruent because the lengths of the two legs of the right triangle are equal.

As a result, the two acute angles and two legs of an isosceles right triangle are congruent.

Therefore, the claim that all isosceles triangles are equivalent to one another is untrue.

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Answer: If two parallel lines are cut by a transversal, then corresponding angles are congruent.

Step-by-step explanation:

Answer:3; the ride costs $3 per mile

Step-by-step explanation:

Rate of Change of the function is 3. then the ride costs $3 per mile.

The average Rate of Change of the function f(x) can be calculated as;

[tex]f(x) = \dfrac{f(b) - f(a)}{b-a}[/tex]

We are given  the total cost of a ride-sharing trip, y, is a function of the distance traveled,

Distance (mi), Total Cost ($). y

2                       0

5                       1  

8                        2

11                       3

14                       4

Therefore,

Rate of Change of the function

(5 - 2) / (1- 0)

= 3 / 1

Hence, we can conclude that the ride costs $3 per mile.

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The value of the variables x and y are,

11) x = 13,  y = 18.4

12) x = 6.9,  y = 8.

What is right angle triangle?

A right triangle is a triangle with one angle at a right angle, meaning that two of its sides are perpendicular. It is also known as a right-angled triangle, an orthogonal triangle, or more commonly as a rectangled triangle. Trigonometry's foundation is the right triangle's relationship between its sides and other angles.

11)Let the given triangle is a right angle triangle.

So we can use trigonometric ratio,

[tex]tan(45) = \frac{x}{13} \\x = 13(tan(45))\\x = 13[/tex]

Using this to find the value of y.

[tex]sin(45) = \frac{x}{y} \\sin(45) = \frac{13}{y} \\y = \frac{13}{sin(45)} \\y = 18.4[/tex]

So, the values of x and y are x = 13 and y = 18.4.

12) The given triangle is a right angle triangle.

So we can use trigonometric ratio,

[tex]tan(30) = \frac{4}{y}\\ y = \frac{4}{tan(30)} \\y = 6.9[/tex]

Now to find the value of x.

[tex]sin(30) = \frac{4}{x} \\x = \frac{4}{sin(30)} \\x = 8[/tex]

Hence, the values of x and y are, x = 6.9 and y = 8.

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Answer:

To find a polynomial, you must identify the terms, coefficients, and degrees. The degree of a polynomial is the highest exponent of the variable in the equation. The coefficients of a polynomial are the values that are multiplied by the variables. The terms of a polynomial are the pieces of the equation, including the coefficients and the variables. After identifying the terms, coefficients, and degree of the polynomial, you can write it in standard form.

Step-by-step explanation:

The lowest form of 6/42 is 1/7 and,

the lowest form of 14/28 is 1/2.

What is the lowest form of fraction?

When there is no common factor between the fraction's numerator (top) and denominator (bottom), the fraction is said to be in its simplest form. This is also known as its Lowest form.

In fraction 6/42, both denominator and numerator have common factor as 6, So we divide the denominator and numerator by 6.

Hence, we get 1/7.

Similarly, In fraction 14/28, both denominator and numerator have common factor as 14, So we divide the denominator and numerator by 14.

Hence, we get 1/2.

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Answer:

-2

Step-by-step explanation:

a is -3

d is -1

you subtract the two

-3 - (-1)

when you have two negatives they cancel each other out and become positive

-3 + 1

-3 + 1 = -2

that is your answer

The multiples of a number are numbers found by multiplying the original number by any whole number. Prime multiples are number which multiply each other and form a prime

What are Prime Numbers ?

A prime number is a number that is only divisible by 1 and itself. So, prime numbers have only two factors, 1 and the number itself.

This means these numbers cannot be divided by any number other than 1 and the number itself without leaving a remainder.

Numbers that have more than 2 factors are known as composite numbers.

For example :

The factors of 3 are 1 and 3.

Therefore, the prime multiples of 3 are 1 and  3.

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A mathematical proof is an inferential justification for a mathematical claim that demonstrates how the premises logically support the conclusion.

Result for an image A proof formula:

There are numerous techniques to demonstrate something; we'll talk about three of them: direct proof, proof by contradiction, and proof by induction. We'll discuss each of these proofs, including what they are and when and how to use them.

Proof: If n is an integer, then. We shall demonstrate the converse statement, "if n is divisible by 4, then n is divisible by 2," in order to demonstrate that "if n is not divisible by 2, then n is not divisible by 4." Let's say n can be divided by 4.

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The area of the polygon EFG with given vertices is, 7.5 units.

What is area of triangle?

The area of a triangle is the area that it completely encloses in a two-dimensional plane. A triangle is a closed shape that, as we all know, has three sides and three vertices. As a result, the total area occupied by a triangle's three sides is known as its area. The general formula for calculating a triangle's area is the product of the triangle's base and height divided by half.

Let the given vertices are,

E(3, 1), F(3, -2), G(-2, -2)

There are three vertices, so the polygon is a triangles.

Since,

Area of triangle = 1/2 x base x height.

If we draw the given vertices on the coordinate plane then the base is FG and the height of the triangle is EF.

Now to find the lengths of side FG and EF.

By using the distance formula,

[tex]d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\EF =\sqrt{(3-3)^2+(-2-1)^2} = \sqrt{9} = 3\\ FG = \sqrt{(-2-3)^2 + (-2-(-2))^2} = \sqrt{25+0} = 5[/tex]

⇒ Area = 1/2 x 5 x 3

             = 1/2 x 15

    Area = 7.5

Hence, the area of the polygon EFG is 7.5 units.

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There are 125 of the smallest cubes that will fill the largest cube using the size of their respective volume

The volume of cube is calculated by using a × a × a = a³ where "a" is the edge.

The volume of the smallest cube = 1.5inches × 1.5inches × 1.5inches

The volume of the smallest cube = 3.375inches³

The volume of the largest cube = 7.5inches × 7.5inches × 7.5inches

The volume of the largest cube = 421.875inches³

Thus, the number of the smallest cubes it would take to fill the largest cube is derived by

421.875inches³/3.375inches³ = 125.

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The inverse of the function f(x) = [tex]2x^2 - 4[/tex] is

[tex]f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}[/tex]

First option is correct

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here, f(x) = [tex]2x^2 - 4[/tex]

Let f(x) = y

[tex]2x^2 - 4[/tex] = y

[tex]2x^2 = y + 4\\x^2 = \frac{y + 4}{2}\\x = \pm \sqrt{\frac{y + 4}{2}}[/tex]

[tex]f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}[/tex]

First option is correct

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Isosceles triangle problem can be solved by given steps

what is isosceles triangle ?

An isosceles triangle in geometry is a triangle with two sides that are the same length. Sometimes it is stated as having exactly two sides that are the same length, and other times it is specified as having at least two sides that are the same length, the latter form containing the equilateral triangle as a particular case.

explanation:

In isosceles triangle

1. Two sides of the triangle are congruent

2. The angle opposite those sides are congruent

x + 72 +72 = 180

             x  = 180 -144

             x = 36

Hence the isosceles triangle problem can be solved by above steps and explanation

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The range of a graph can be found using the values depicted on the y-axis.

According to the question,

We have the following information:

We have a graph and we need to know how do we find the range of that graph.

We know that the domain of a graph is the collection or in other words set of input values depicted on x -axis and the range of a graph is the set of possible output values portrayed on y-axis of the graph.

In noting down the range of a graph, we first list down the down value and then the up value or in other words we move from down to up in order to find the range of a graph.

Hence, the range of a graph can be found using the values depicted on the y-axis.

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By using the law of exponents, it has been proved that

[tex]a^{-n} = \frac{1}{a^n}[/tex]

What are exponent?

Exponent tells us how many times a number is multiplied by itself.

For example : In [tex]2^4 = 2 \times 2 \times .... \times 2[/tex]

Here, 2 is multiplied by itself 4 times.

If [tex]a^m = a \times a \times a \times ..... \times a[/tex] (m times),

a is the base and m is the index.

The laws of exponents are

[tex]a^m \times a^n = a^{m+n}\\\\a^ m \div a^n = a^{m - n}\\\\a^ 0 =1\\\\a^{-m} = \frac{1}{a^m}\\\\(ab)^m = a^m \times b^m\\\\(\frac{a}{b})^m = \frac{a^m}{b^m}\\\\(a^m)^n = a^{mn}[/tex]

We know this law

[tex]a^p \times a^q = a^{p + q}[/tex]

Putting  p = n and q = -n

[tex]a^n \times a^{-n} = a^{n + (-n)} = a^{n - n} = a^0 = 1[/tex]

So,

[tex]a^n \times a^{-n} = 1\\a^{-n} = \frac{1}{a^n}\\a^{-n} = \frac{1^n}{a^n}\\a^{-n} = (\frac{1}{a})^n[/tex]

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Answer:

B) x + 1

Step-by-step explanation:

Consecutive integers mean continuous integers ( 1, 2 , 3 ,....). We will get the next number by adding 1 to the previous number.

Eg: 5 , 6 , 7 ,.....

6 = 5 + 1

7 = 6 + 1

So, if first number is x, then the next number will be x + 1

Answer:

Step-by-step explanation:

SLOPE OF THE GIVEN LINE

6x + 18y = 36

x + 3y = 6

3y = -x + 6

3/3 y = -1/3 x + 6/3

y = -1/3 x + 2

Slope = -1/3

slope of a perpendicular line  = 3

POINT SLOPE FORM

y-4 = 3(x-(-5)

y - 4 = 3(x+5)

SLOPE INTERCEPT FORM

y-4 = 3x + 15

y = 3x + 15 + 4

y = 3x + 19

STANDARD FORM

3x - y + 15 + 4 = 0

3x - y + 19 = 0

Equation of line in point slope form is y-4= 3 (x+5), y=3x+19 is equation in slope intercept form and equation in standard form is 3x-y+19=0.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

6x+18y=36

18y=-6x+36

Divide both sides by 18

y=-6/18x+36/18

y=-1/3x+2

Slope of given line is -1/3, so slope of perpendicular line is 3.

The equation of line in point slope form with slope 3 and goes through the point (-5,4) is given below

y-4= 3 (x+5)

To find the equation in slope intercept form we need to find y intercept

4=3(-5)+b

4=-15+b

19=b

y=3x+19 is equation in slope intercept form.

The equation in standard form is 3x-y+19=0

Hence,  equation of line in point slope form is y-4= 3 (x+5), y=3x+19 is equation in slope intercept form and equation in standard form is 3x-y+19=0.

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The correct answer is the 5 ways to find if two triangles are congruent are SSS, SAS, ASA, AAS and HL.

Two triangles are congruent if and only if they have the same three sides and angles.

SSS (side, side, side): SSS, which stands for "side, side, side," denotes that there are two triangles with identical angles on all three sides. The triangles are congruent if their third sides are the same as those of another triangle.

SAS (side, angle, side): "Side, Angle, Side" stands for two triangles whose two sides and one included angle are known to be equal. The triangles are congruent if two sides and the included angle of one triangle equals the corresponding sides and angle of another triangle.

ASA (angle, side, angle): ASA, which stands for "angle, side, angle," denotes the existence of two triangles whose included side and two known angles are equal. The triangles are congruent if two angles and the included side of one triangle match the corresponding angles and side of another triangle.

AAS (angle, angle, side): AAS, which stands for "angle, angle, side," denotes the equality of the non-included side and two angles in two triangles. The triangles are congruent if two of one triangle's angles and its excluded side are equal to the corresponding angles and sides of another triangle.

HL (hypotenuse, leg): The two triangles are congruent if one right-angled triangle's hypotenuse and one of its legs are equal to its corresponding hypotenuse and leg in another right-angled triangle.

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What is division?

Division is the process by which value of single unit can be calculated from the value of multiple unit.

The number to be divided is known as dividend, the number by which the dividend is divided is the divisor, the result obtained is the quotient and the remaining part is the remainder.

There is a well known formula for division

Divisor x Quotient + Remainder = Dividend.

This is a problem sum of division

Cost of 15 flowers = $200

Cost of 1 flower = $200 [tex]\div[/tex] 15 = $13.33

Cost of 5 flowers = $70

Cost of 1 flower = $70 [tex]\div[/tex] 5 = $14

So unit price of 15 flower arrangement of $200 is lower than unit price of 5 flower arrangement of $70

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The exponential decay formula can take one of three forms:

f (x) = abˣ , f (x) = a (1 – r)ˣ , P = P₀[tex]e^{-kt}[/tex]

Where,

a (or) P₀ = Initial amount ,b = decay factor, e = Euler’s constant, r = Rate of decay (for exponential decay), k = constant of proportionality

What is Exponential Decay ?

Exponential decay occurs in mathematical functions when the pace by which changes are occurring are decreasing and must thus reach a limitation, which is the horizontal asymptote of an exponential function.

The amount drops gradually, followed by a quick reduction in the speed of change and increases over time. The exponential decay formula is used to determine the decrease in growth. The exponential decay formula can take one of three forms:

f (x) = abx

f (x) = a (1 – r)x

P = P0 e-k t

Where,

a (or) P0 = Initial amount

b = decay factor

e = Euler’s constant

r = Rate of decay (for exponential decay)

k = constant of proportionality

x (or) t = time intervals (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem).

In exponential decay, the quantity decreases very rapidly at first, and then more slowly. The rate of change decreases over time. The rate of decay becomes slower as time passes. Hence, the exponential decay graph is denoted as

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B: 1000 + 25 + Water deposit (50) + Phone connection (40) + Elect. connection (60)

1000 + 25 + 200 = 1225

Answer: [tex](-\infty, \infty)[/tex]

Step-by-step explanation:

Absolute value is always non-negative.