The second angle of a triangle is 4 times as large as the first. The third angle is more than the sum of the other two angles. Find the measure of all three angles.

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Let x = first angle, the 4x = second, but how much more is the third angle? (x+4x+?)

They will add to 180º.

To solve this problem, let's assign variables to each angle for easier calculations.

Let's call the first angle A, the second angle B, and the third angle C.

Based on the information given:
1. The second angle is 4 times as large as the first angle. So, B = 4A.
2. The third angle is more than the sum of the other two angles. So, C > A + B.

We can now use these equations to find the measure of each angle.

Since the sum of the angles in a triangle is always 180 degrees, we can write an equation using angle A, B, and C:

A + B + C = 180

Substituting B = 4A into this equation:

A + 4A + C = 180

Simplifying:

5A + C = 180 ----(1)

Now we can use the information that C > A + B:

Substituting B = 4A into this inequality:

C > A + 4A

Simplifying:

C > 5A ----(2)

From equations (1) and (2), we can set up a system of equations to solve for A and C:

5A + C = 180
C > 5A

To find a specific solution, we need one more equation. Let's use the fact that the sum of the angles in a triangle is 180 degrees:

A + B + C = 180

Substituting B = 4A into this equation:

A + 4A + C = 180

Simplifying:

5A + C = 180 ----(3)

Now we have a system of equations (equations 1, 2, and 3) to solve for A and C.

Solving these equations will give us the values of A and C, and we can find B since B = 4A.