the measures of the2nd angle of a triangle is 20degree more than the complements of the first triangle and the 3rd angle is 4 degree less than half the supplement of the 1st.find the measure of each angle

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160°

To find the measure of each angle in the triangle, we'll need to break down the problem and apply some mathematical reasoning.

Let's assign variables to the unknown angles. Let:
- First angle = x
- Second angle = y
- Third angle = z

According to the given information:

1) "The measures of the second angle of a triangle is 20 degrees more than the complement of the first angle."
The complement of an angle is the angle that, when added to the first angle, gives a total of 90 degrees. So, the complement of the first angle is (90 - x).
Hence, we can write the equation for the second angle as: y = (90 - x) + 20.

2) "The third angle is 4 degrees less than half the supplement of the first angle."
The supplement of an angle is the angle that, when added to the first angle, gives a total of 180 degrees. So, the supplement of the first angle is (180 - x).
Half of the supplement of the first angle is (180 - x) / 2.
Hence, we can write the equation for the third angle as: z = (180 - x) / 2 - 4.

Since we know that the sum of the angles in a triangle is always 180 degrees, we can set up the equation:

x + y + z = 180

Now, substitute the expressions we derived earlier into the equation:

x + (90 - x) + 20 + (180 - x) / 2 - 4 = 180

Simplify the equation:

x + 90 - x + 20 + (180 - x) / 2 - 4 = 180
110 - x + (180 - x) / 2 - 4 = 180
110 - x + 90 - x - 2 = 180
(200 - 3x) / 2 - 2 = 180
(200 - 3x) / 2 = 182
200 - 3x = 364
-3x = 164
x = -164 / 3
x ≈ -54.67 (ignore this as we can't have negative angles)

Since angles cannot be negative, we can disregard the negative solution. Therefore, x ≈ 54.67.

Now, substitute this value back into the equations for y and z:

y = (90 - x) + 20
y = (90 - 54.67) + 20
y ≈ 55.33 + 20
y ≈ 75.33

z = (180 - x) / 2 - 4
z = (180 - 54.67) / 2 - 4
z ≈ 125.33 / 2 - 4
z ≈ 62.67 - 4
z ≈ 58.67

Therefore, the measures of each angle are approximately:
x ≈ 54.67 degrees
y ≈ 75.33 degrees
z ≈ 58.67 degrees