# The angles of a triangle are in inverse ratio to the numbers 2/5, 1/2, 2/3. Which are the angles ?

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let a b c be the angles of the triangle:

==> a + b + c= 180........(1)

a = 1/(2/5) *x ==> a= (5/2) x

b= 1/(1/2) * x ==> b = 2*x

c= 1/(2/3)*x ==> c = (3/2) *x

Now substitute in (1):

==> a + b + c = 180

==> (5/2)x + 2x + (3/2)x = 180

==> x*(5/2 + 2 + 3/2) = 180

==> x*(6) = 180

==> x= 180/6 = 30

**==> a = 5/2 *30 = 75 degrees**

**==> b= 2 *30 = 60 degrees **

**==> c = 3/2 *30 = 45 degrees**

**TO CHECK:**

**A + B + C = 75 + 60 + 45 = 180 .**

We'll note the measures of the angles of the triangle as x,y,and z.

The sum of the measures of the angles of a triangle is:

x+y+z = 180 degrees

The angles are in inverse proportion to the numbers 2/5, 1/2, 2/3.

x/[1/(2/5)] = y/[1/(1/2)] = z/[1/(2/3)]

We'll re-write the proportions:

x/(5/2) = y/2 = z/(3/2)

We'll apply the proportion rule:

(x+y+z)/(5/2 + 2 + 3/2)=x/(5/2) = y/2 = z/(3/2)

But x+y+z = 180

180/6 = 2x/5

30 = 2x/5

5*30 = 2x

We'll divide by 2:

x = 5*15

**x = 75 degrees**

y/2 = 30

**y = 60 degrees**

75 + 60 + z = 180

z = 180 - 75 - 60

z = 180 - 135

**z = 45 degrees **

We assume angles A, B, C of the triangle are in the inverse ratio to numbers (2/5), (1/2) and 2/3.

Therefore A = x/(2/5) = 5x/2

B = x/(1/2) = 2x

C = x/(2/3) = 3x/2, where x is the constant of (inverse)proportion.

Since A+B+C = 180 in a triangle , 5x/2+2x+3x/2 = 180 deg. Or

6x = 180 deg

x = 180/6 = 30 deg.

Therefore,

A = 5x/2 = 5*30/2 = 75 deg

B = 2x = 2*30 =60 deg

C = 3x/2 = 3*30/2 = 45 deg.

Tally: A+B+C = (75+60+45) deg = 180 deg.