Let's solve the given problem step by step using the information provided.
Let's assign variables to the measures of the angles:
Measure of the first angle: X
Measure of the second angle: Y
Measure of the third angle: Z
According to the problem, the sum of the measures of the angles is 180. Therefore, we can write the equation:
X + Y + Z = 180
The problem also states that the sum of the measures of the second and third angles is two times the measure of the first angle. We can express this as an equation:
Y + Z = 2X
Additionally, it is given that the third angle is 20 more than the second angle. This can be expressed as an equation:
Z = Y + 20
Now, we have a system of three equations:
Equation 1: X + Y + Z = 180
Equation 2: Y + Z = 2X
Equation 3: Z = Y + 20
To solve this system, we can substitute Equation 3 into Equation 2 to eliminate Z:
Y + (Y + 20) = 2X
2Y + 20 = 2X
2Y = 2X - 20
Y = X - 10
Now, we have an expression for Y in terms of X. We can substitute this expression into Equation 1 to eliminate Y:
X + (X - 10) + Z = 180
2X - 10 + Z = 180
Z = 190 - 2X
So, we have expressions for Y and Z in terms of X:
Y = X - 10
Z = 190 - 2X
Now, we can substitute these expressions into the original equation for the sum of the measures of the angles to solve for X:
X + (X - 10) + (190 - 2X) = 180
X + X - 10 + 190 - 2X = 180
2X - 2X + X = 180 - 190 + 10
X = 0
Substituting X = 0 back into the expressions for Y and Z, we get:
Y = 0 - 10 = -10
Z = 190 - 2(0) = 190
Therefore, the measures of the angles are:
First angle (X): 0 degrees
Second angle (Y): -10 degrees
Third angle (Z): 190 degrees
Note: It is unusual to have negative angles in a geometric context. Please double-check the problem statement to ensure there are no errors or inconsistencies in the given information.