Question

# Two angles of a regular polygon are given to be Find the value of and measure of each angle.

Hint:

### A polygon whose length of all sides is equal with equal angles at each vertex is called regular polygon.

## The correct answer is: 87

### Explanation:

- We have been given the two sides of a regular polygon that is - (2𝑥 + 27)° 𝑎𝑛𝑑 (3𝑥 − 3)°
- We have to find the value of x and measure of each angle.

Step 1 of 1:

We know that a regulat polygon is equiangular

So,

2x + 27 = 3x - 3

x = 27 + 3

x = 30

And the value of each angle will be

= 2x + 27

= 2(30) + 27

= 87

### Related Questions to study

### Solve the equation. Write a reason for each step.

8(−x − 6) = −50 − 10x

Ans:- x = 1

Explanation :-

Given ,8(-x − 6) = -50-10x.

By left distributive property - 8x − 48 = - 50 -10x

Adding 48 on both sides by additive property of equality both sides remain equal.

- 8x − 48 + 48 = - 50 -10x + 48

- 8x = -10x - 2

Adding 10x on both sides by additive property of equality both sides remain equal.

- 8x +10x = -10x - 2 +10x

2x = - 2

Dividing 2 by division property of equality both sides remains equal.

x = -1

∴ x = -1

### Solve the equation. Write a reason for each step.

8(−x − 6) = −50 − 10x

Ans:- x = 1

Explanation :-

Given ,8(-x − 6) = -50-10x.

By left distributive property - 8x − 48 = - 50 -10x

Adding 48 on both sides by additive property of equality both sides remain equal.

- 8x − 48 + 48 = - 50 -10x + 48

- 8x = -10x - 2

Adding 10x on both sides by additive property of equality both sides remain equal.

- 8x +10x = -10x - 2 +10x

2x = - 2

Dividing 2 by division property of equality both sides remains equal.

x = -1

∴ x = -1

### Find the measure of each angle of an equilateral triangle using base angle theorem.

Hint:

- the base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.
- An equilateral triangle is a triangle with all the three sides of equal length.

- We have to find the measure of each angle of an equilateral triangle using base angle theorem.

Let a triangle be ABC

Here,

AB = AC

Using base angle theorem

And,

So,

Therefore,

Step 2 of 2:

We know that the sum of all angles of a triangle is 180

^{0}.

Now,

So,

### Find the measure of each angle of an equilateral triangle using base angle theorem.

Hint:

- the base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.
- An equilateral triangle is a triangle with all the three sides of equal length.

- We have to find the measure of each angle of an equilateral triangle using base angle theorem.

Let a triangle be ABC

Here,

AB = AC

Using base angle theorem

And,

So,

Therefore,

Step 2 of 2:

We know that the sum of all angles of a triangle is 180

^{0}.

Now,

So,

### The length of each side of a regular pentagon is . Find the value of if its perimeter is .

Hint:

- A regular pentagon is a polygon that has 5 sides all of same length and all the angles of the same measure.

- We have been given in the question the length of each side of a regular pentagon which is (x+5) cm
- We have also been given the perimeter that is 50 cm.
- We have to find the value of x.

We have given a perimeter of a regular pentaogn 50.

A pentagon has sides.

The length of the side is x+5

So,

5(x + 5) = 50

x + 5 = 10

x = 5

Hence, Option A is correct.

### The length of each side of a regular pentagon is . Find the value of if its perimeter is .

Hint:

- A regular pentagon is a polygon that has 5 sides all of same length and all the angles of the same measure.

- We have been given in the question the length of each side of a regular pentagon which is (x+5) cm
- We have also been given the perimeter that is 50 cm.
- We have to find the value of x.

We have given a perimeter of a regular pentaogn 50.

A pentagon has sides.

The length of the side is x+5

So,

5(x + 5) = 50

x + 5 = 10

x = 5

Hence, Option A is correct.

### Name the property of equality the statement illustrates.

Every segment is congruent to itself.

Ans :- Option A

Explanation :-

The reflexive property states that any real number, a, is equal to itself. That is, a = a.

Similarly the segment is congruent to itself .

∴Option A

### Name the property of equality the statement illustrates.

Every segment is congruent to itself.

Ans :- Option A

Explanation :-

The reflexive property states that any real number, a, is equal to itself. That is, a = a.

Similarly the segment is congruent to itself .

∴Option A

Hint:

- the base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.

- We have been given a diagram of a triangle in the question named ABC we have also been given 𝑚∠𝐵 = 55°.
- We have to find out 𝑚∠A.

We have given figure

Here, AB = AC

It means the given triangle is an isosceles triangle.

Now,

By base angle theorem .

And it is given

So,

Step 2 of 2:

We know that the sum of angle of a triangle is 180

^{0}

Hint:

- We have been given a diagram of a triangle in the question named ABC we have also been given 𝑚∠𝐵 = 55°.
- We have to find out 𝑚∠A.

We have given figure

Here, AB = AC

It means the given triangle is an isosceles triangle.

Now,

By base angle theorem .

And it is given

So,

Step 2 of 2:

We know that the sum of angle of a triangle is 180

^{0}

### If f(x) satisfies the relation 2f(x) +f(1-x) = x^{2} for all real x , then f(x) is

We have given that

2f(x) +f(1-x) = x

^{2 }- - -- - - - - -(i)

We have to find the value of f(x)

By replacing x by (1-x) in equation (i)we get,

2f(1-x) + f(x) = (1-x)

^{2}

2f(1-x) + f(x) = 1 + x

^{2}– 2x - - - - - -(ii)

Multiplying the equation(i) by 2 we get,

4f(x) + 2f(1-x) = 2x

^{2}- - - - - - (iii)

Subtracting equation (ii) from (iii)

3f(x) = x

^{2}+ 2x -1

So,

Therefore option (b) is correct.

### If f(x) satisfies the relation 2f(x) +f(1-x) = x^{2} for all real x , then f(x) is

We have given that

2f(x) +f(1-x) = x

^{2 }- - -- - - - - -(i)

We have to find the value of f(x)

By replacing x by (1-x) in equation (i)we get,

2f(1-x) + f(x) = (1-x)

^{2}

2f(1-x) + f(x) = 1 + x

^{2}– 2x - - - - - -(ii)

Multiplying the equation(i) by 2 we get,

4f(x) + 2f(1-x) = 2x

^{2}- - - - - - (iii)

Subtracting equation (ii) from (iii)

3f(x) = x

^{2}+ 2x -1

So,

Therefore option (b) is correct.

### If f:R->R be a function whose inverse is (𝑥+5)/3 , then what is the value of f(x)

f^{-1}(x) = (x+5)/3

For solving this let us take

y = f^{-1}(x)

y = (x+5)/3

Further solving we get,

x = 3y – 5

f(y) = 3y – 5

Therefore,

f(x) = 3x – 5

### If f:R->R be a function whose inverse is (𝑥+5)/3 , then what is the value of f(x)

f^{-1}(x) = (x+5)/3

For solving this let us take

y = f^{-1}(x)

y = (x+5)/3

Further solving we get,

x = 3y – 5

f(y) = 3y – 5

Therefore,

f(x) = 3x – 5

Hint:

- The base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.

- We have been given in the question a diagram of a triangle named ABC and 𝑚∠𝐴 = 60°.
- We have to find the 𝑚∠𝐴 𝑎𝑛𝑑 𝑚∠𝐶.

In the given figure, AB = AC.

So, ABC is an isosceles triangle.

So, According to base-angle theorem, the angles opposite the congruent sides are congruent.

So,

Step 2 of 2:

Now we know that the sum of angle of triangle is equal to 180

^{0}.

So,

Since,

So,

Therefore,

Hint:

- The base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.

- We have been given in the question a diagram of a triangle named ABC and 𝑚∠𝐴 = 60°.
- We have to find the 𝑚∠𝐴 𝑎𝑛𝑑 𝑚∠𝐶.

In the given figure, AB = AC.

So, ABC is an isosceles triangle.

So, According to base-angle theorem, the angles opposite the congruent sides are congruent.

So,

Step 2 of 2:

Now we know that the sum of angle of triangle is equal to 180

^{0}.

So,

Since,

So,

Therefore,

### Let A= {x, y, z} and B= { p, q, r, s}, What is the number of distinct relations from B to A ?

A= {x, y, z}

B= { p, q, r, s},

For finding the district relations from B to A we have to take the cartesian product of B and A

B×A = {p, q, r, s} × {x, y, z}

= {(p, x) , (p, y) , (p, z) , (q, x) , (q, y), (q, z) , (r, x) , (r, y), (r, z) , (s, x), (s, y), (s, z)}

Therefore there are 12 distinct relations .

### Let A= {x, y, z} and B= { p, q, r, s}, What is the number of distinct relations from B to A ?

A= {x, y, z}

B= { p, q, r, s},

For finding the district relations from B to A we have to take the cartesian product of B and A

B×A = {p, q, r, s} × {x, y, z}

= {(p, x) , (p, y) , (p, z) , (q, x) , (q, y), (q, z) , (r, x) , (r, y), (r, z) , (s, x), (s, y), (s, z)}

Therefore there are 12 distinct relations .

### Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and f(g)

f(x) = x + 1

g(x) = 2x – 3

We have to find the value of

i) f(x) + g(x)

ii) f(x) – g(x)

iii) f(g(x))

Therefore,

i) f(x) + g(x) = x + 1 + 2x – 3

= 3x – 2

ii f(x) – g(x) = x + 1 – (2x – 3)

= x + 1 – 2x + 3

= 4 – x

iii) f(g(x)) = f(2x -3)

= (2x – 3) + 1

= 2x – 2

Therefore, f+g = 3x – 2

f – g = 4 – x

f(g) = 2x – 2

### Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and f(g)

f(x) = x + 1

g(x) = 2x – 3

We have to find the value of

i) f(x) + g(x)

ii) f(x) – g(x)

iii) f(g(x))

Therefore,

i) f(x) + g(x) = x + 1 + 2x – 3

= 3x – 2

ii f(x) – g(x) = x + 1 – (2x – 3)

= x + 1 – 2x + 3

= 4 – x

iii) f(g(x)) = f(2x -3)

= (2x – 3) + 1

= 2x – 2

Therefore, f+g = 3x – 2

f – g = 4 – x

f(g) = 2x – 2

### Name the property of equality the statement illustrates.

If ∠P ≅ ∠Q, then ∠Q ≅ ∠P.

The symmetric property states that for any real numbers, a and b, if a = b then b = a. Similarly with angles If ∠P ≅ ∠Q, then ∠Q ≅ ∠P.

∴Option B

### Name the property of equality the statement illustrates.

If ∠P ≅ ∠Q, then ∠Q ≅ ∠P.

The symmetric property states that for any real numbers, a and b, if a = b then b = a. Similarly with angles If ∠P ≅ ∠Q, then ∠Q ≅ ∠P.

∴Option B

### Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

f = {(1,1), (2,3), (0,–1), (–1, –3)}

And also we have given that

f(x) = ax + b

We have to find the value of a and b .

First of all if the f is a function then its points will satisfy f(x) = ax + b

f(1) = 1

f(2) = 3

f(0) = -1

f(-1) = -3

i) (1,1)

f(1) = a (1) + b

1 = a + b

ii) (2,3)

f(2) = a (2) + b

3 = 2a + b

Subtract equation (i) from (ii)

2a – a + b – b = 3 – 1

a = 2

Putting this value in equation (i)

1 = 2 + b

b = 1 – 2

b = -1

Therefore, value of a = 2 and b = -1 .

### Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

f = {(1,1), (2,3), (0,–1), (–1, –3)}

And also we have given that

f(x) = ax + b

We have to find the value of a and b .

First of all if the f is a function then its points will satisfy f(x) = ax + b

f(1) = 1

f(2) = 3

f(0) = -1

f(-1) = -3

i) (1,1)

f(1) = a (1) + b

1 = a + b

ii) (2,3)

f(2) = a (2) + b

3 = 2a + b

Subtract equation (i) from (ii)

2a – a + b – b = 3 – 1

a = 2

Putting this value in equation (i)

1 = 2 + b

b = 1 – 2

b = -1

Therefore, value of a = 2 and b = -1 .

### Planet Wiener receives $2.25 for every hotdog sold. They spend $105 for 25 packages of hot dogs and 10 packages of buns. Think of the linear function that demonstrates the profit based on the number of hotdogs sold

Rate of Change:___________

Initial Value:______________

Independent Variable:______

Dependent Variable:________

EQ of Line:________________

SOL – Let Dependent Variable : x given by no of hotdog sold

Independent Variable : y given by profit earned

It is given that Planet Wiener receives $2.25 for every hotdog sold.

Rate of change : 2.25

We know that Slope of the line = Rate of Change

So, m = 2.25 ---- (1)

Planet Wiener spends $105 for 25 packages of hot dogs and 10 packages of buns. (See it as an investment made before earning any profit. So, these $105 will actually be considered in ‘-’)

Initial Value : - 105

This initial value is obtained when no of hotdogs sold, x = 0 i.e., it will act as y – intercept, c = - 105 ---- (2)

Using slope intercept form equation of a line is given by

y = mx + c where m is slope and c is y – intercept.

Equation of line : y = 2.25x – 105 ( From (1) and (2) )

### Planet Wiener receives $2.25 for every hotdog sold. They spend $105 for 25 packages of hot dogs and 10 packages of buns. Think of the linear function that demonstrates the profit based on the number of hotdogs sold

Rate of Change:___________

Initial Value:______________

Independent Variable:______

Dependent Variable:________

EQ of Line:________________

SOL – Let Dependent Variable : x given by no of hotdog sold

Independent Variable : y given by profit earned

It is given that Planet Wiener receives $2.25 for every hotdog sold.

Rate of change : 2.25

We know that Slope of the line = Rate of Change

So, m = 2.25 ---- (1)

Planet Wiener spends $105 for 25 packages of hot dogs and 10 packages of buns. (See it as an investment made before earning any profit. So, these $105 will actually be considered in ‘-’)

Initial Value : - 105

This initial value is obtained when no of hotdogs sold, x = 0 i.e., it will act as y – intercept, c = - 105 ---- (2)

Using slope intercept form equation of a line is given by

y = mx + c where m is slope and c is y – intercept.

Equation of line : y = 2.25x – 105 ( From (1) and (2) )

### Sketch a graph modelling a function for the following situation:

A dog is sleeping when he hears the cat “meow” in the next room. He quickly runs to the next room where he slowly walks around looking for the cat. When he doesn’t find the cat, he sits down and goes back to sleep. Sketch a graph of a function of the dog’s speed in terms of time.

SOL – Let dog’s speed = y

Time taken = x

Acc. to the question, dog was sleeping when time, x = 0 , speed, y = 0

Then, he quickly runs to the next room i.e. speed is increasing over time. Since he moved quickly, implies that rate of change is very high.

Slope is positive and line is very steep

Then, he slowly walks around looking for the cat. i.e. there is a sudden decrease in the speed.

A line parallel to y – axis will be drawn to a point where speed is less.

And then he walks around looking for the cat i.e., speed is same over time

A line parallel to x – axis is drawn

Further, he sits down and goes back to sleep i.e. speed is 0

Since speed suddenly drops to 0, so line x = 0 which is parallel to y – axis and then line y = 0 which is parallel to x – axis and has slope = 0

### Sketch a graph modelling a function for the following situation:

A dog is sleeping when he hears the cat “meow” in the next room. He quickly runs to the next room where he slowly walks around looking for the cat. When he doesn’t find the cat, he sits down and goes back to sleep. Sketch a graph of a function of the dog’s speed in terms of time.

SOL – Let dog’s speed = y

Time taken = x

Acc. to the question, dog was sleeping when time, x = 0 , speed, y = 0

Then, he quickly runs to the next room i.e. speed is increasing over time. Since he moved quickly, implies that rate of change is very high.

Slope is positive and line is very steep

Then, he slowly walks around looking for the cat. i.e. there is a sudden decrease in the speed.

A line parallel to y – axis will be drawn to a point where speed is less.

And then he walks around looking for the cat i.e., speed is same over time

A line parallel to x – axis is drawn

Further, he sits down and goes back to sleep i.e. speed is 0

Since speed suddenly drops to 0, so line x = 0 which is parallel to y – axis and then line y = 0 which is parallel to x – axis and has slope = 0

### In the given diagram, m∠1 ≅ m∠3

Prove that ∠POR ≅ ∠QOS.

Hint :- given m∠1 ≅ m∠3 , adding common angle 2 to both sides proves the statement.

Explanation :-

Given , m∠1 ≅ m∠3

Adding angle 2 to both sides by the addition property of equality both sides remain equal.

m∠1 + m∠2 ≅ m∠2 + m∠3

We know that ∠1 +∠2 = ∠ POR and ∠2 + ∠3 = ∠QOS

Substituting the angle POR and QOS ,

∠POR ≅ ∠QOS

Hence proved

### In the given diagram, m∠1 ≅ m∠3

Prove that ∠POR ≅ ∠QOS.

Hint :- given m∠1 ≅ m∠3 , adding common angle 2 to both sides proves the statement.

Explanation :-

Given , m∠1 ≅ m∠3

Adding angle 2 to both sides by the addition property of equality both sides remain equal.

m∠1 + m∠2 ≅ m∠2 + m∠3

We know that ∠1 +∠2 = ∠ POR and ∠2 + ∠3 = ∠QOS

Substituting the angle POR and QOS ,

∠POR ≅ ∠QOS

Hence proved