the sum of the two angles of a triangle is equal to the third angle and the difference of the two angles is equal to two-thirds the third angle. find the measure of each angle

smallest angle --- x

middle sized -----y
largest ---------x+y

x + y + x+y = 180
x+y = 90 ***

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

sub into ***
5y + y = 90
y = 15
x = 75

smallest = 15
middle sized = 75
largest = 90

Let's assume the three angles of the triangle are represented by A, B, and C.

According to the information given:
1) The sum of the two angles is equal to the third angle. So we have the equation A + B = C.

2) The difference of the two angles is equal to two-thirds the third angle. This gives us the equation A - B = (2/3)C.

Now we have a system of two equations with two variables. We can solve it to find the measures of each angle.

To do that, let's start by eliminating one variable. We can eliminate A by adding the two equations together:

(A + B) + (A - B) = C + (2/3)C
2A = (5/3)C

Now we can solve for A by dividing both sides of the equation by 2:

A = (5/6)C

We can substitute this value of A in equation 1 to find B:

(5/6)C + B = C
B = C - (5/6)C
B = (1/6)C

So we have A = (5/6)C, B = (1/6)C, and C = C.

To find the measure of each angle, we need to assign a numerical value to one of the angles. Let's assume C = 180 degrees (since it is a triangle).

Now we can substitute C = 180 into the equations:

A = (5/6)C = (5/6) * 180 = 150 degrees
B = (1/6)C = (1/6) * 180 = 30 degrees

Therefore, the measures of the angles are:
A = 150 degrees
B = 30 degrees
C = 180 degrees

To find the measure of each angle, let's represent them as variables.

Let the first angle be 'x', the second angle be 'y', and the third angle be 'z'.

1. The sum of the two angles of a triangle is equal to the third angle:
So we can write an equation: x + y = z. Equation 1.

2. The difference of the two angles is equal to two-thirds the third angle:
We can write another equation: x - y = (2/3)z. Equation 2.

Now, we have a system of two equations with two variables. We can solve these equations simultaneously to find the values of x, y, and z.

Let's solve it using the method of substitution:

From Equation 1, we'll solve for y in terms of x:
y = z - x.

Now, substitute this value of y in Equation 2:
x - (z - x) = (2/3)z.
Simplify this equation: 2x - z = (2/3)z.
Multiply both sides by 3 to eliminate the fraction: 6x - 3z = 2z.
Move the variables to one side: 6x - 2z = 3z.

Next, let's solve for x:
Add 2z to both sides: 6x = 5z.
Divide both sides by 6: x = (5/6)z.

Now that we have the value of x in terms of z, we can substitute it into Equation 1 to find y:

y = z - x
y = z - (5/6)z
y = (6/6)z - (5/6)z
y = (1/6)z

We have the values of x and y: x = (5/6)z and y = (1/6)z.

To find the value of z, we can use the fact that the sum of the angles in a triangle is 180 degrees:
x + y + z = 180.

Substitute the values of x and y:
(5/6)z + (1/6)z + z = 180.
(6/6)z + (1/6)z = 180.
(7/6)z = 180.

Multiply both sides by 6/7 to solve for z:
z = (180 * 6)/7 = 154.29.

Now that we have z, we can find x and y:
x = (5/6)z = (5/6) * 154.29 = 128.57.
y = (1/6)z = (1/6) * 154.29 = 25.71.

Therefore, the measure of each angle is:
x ≈ 128.57 degrees.
y ≈ 25.71 degrees.
z ≈ 154.29 degrees.