The difference between the smallest and second smallest angles of a quadrilateral is equal to the difference between the biggest and second biggest angles. If the biggest angle is twice the smallest angle, what is the measure of the smallest angle?
(Solution pls... Tnx)
let the angles in order of size be
x , a , b, 2x
"The difference between the smallest and second smallest angles" ---> a-x
"the difference between the biggest and second biggest angles" ---> 2x-b
2x - b = a - x
3x = a+b
also we know
x + a + b + 2x = 360
x + 3x + 2x = 360
6x = 360
x = 60
The smallest ist 60° and the largest is 120°
We were lucky that a+b could be expressed in terms of x
To solve this problem, let's assign variables to the angles of the quadrilateral.
Let's say:
- The smallest angle is represented by 'x'
- The second smallest angle is represented by 'y'
- The second biggest angle is represented by 'z'
- The biggest angle is represented by '2x' (since it is twice the size of the smallest angle)
According to the problem, the difference between the smallest and second smallest angles is equal to the difference between the biggest and second biggest angles.
So, we can write the following equation:
x - y = 2x - z
Now, let's simplify the equation:
z - x = x - y (Rearrange the terms)
Since we are dealing with angles, we know that the sum of the angles in a quadrilateral is always 360 degrees.
Therefore, we can write the equation:
x + y + z + 2x = 360
Combining like terms, we have:
3x + y + z = 360
Now we have a system of two equations:
(1) z - x = x - y
(2) 3x + y + z = 360
To solve this system, we can use substitution or elimination. Here, we'll use substitution.
From equation (1), we can isolate 'z' as follows:
z = 2x - (x - y)
z = 2x - x + y
z = x + y
Now let's substitute this value into equation (2):
3x + y + (x + y) = 360
Combine like terms:
4x + 2y = 360
Next, we'll isolate 'y' in terms of 'x' by isolating 'y' in the equation above:
2y = 360 - 4x
y = (360 - 4x) / 2
y = 180 - 2x
Now, we'll substitute this expression for 'y' into equation (1):
x - (180 - 2x) = 2x - (x - (180 - 2x))
Simplify:
x - 180 + 2x = 2x - x + 180 - 2x
Combine like terms:
3x - 180 = 180 - x
Move 'x' terms to one side and the constant terms to the other:
4x = 360
Divide both sides by 4:
x = 90
Therefore, the measure of the smallest angle is 90 degrees.