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

Hint :- using the additive property and subtraction property ,division property on both sides .solve for x.
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

Solution:
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.

Explanation:

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

Step 1 of 1:
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 180⁰.
Now,




So,

Solution:
Hint:

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

Explanation:

• 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.

Step 1 of 1:
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.

Hint :- The reflexive property states that any real number, a, is equal to itself i.e a = a.
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

Solution:
Hint:

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

Explanation:

• 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.

Step 1 of 1:
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 180⁰

Solution:-
We have given that
2f(x) +f(1-x) = x²     - - -- - - - - -(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 + x² – 2x         - - - - - -(ii)
Multiplying the equation(i) by 2 we get,
4f(x) + 2f(1-x) = 2x²                - - - - - - (iii)
Subtracting equation (ii) from (iii)
3f(x) = x² + 2x -1
So,

Therefore option (b) is correct.

We have given that inverse of the function 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

Solution:
Hint:

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

Explanation:

• 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.

Step 1 of 1:
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 180⁰.
So,

Since,
So,





Therefore,

We have given the two sets A and B
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 .

We have given the function

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

Ans :- Option B
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

We have given a function from Z to Z

 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 .

HINT – $105 is an investment made before earning any profit.
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) )

HINT – Figure out slope of the graph with respect to situations given.
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

Aim :- To prove ∠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