The angles of a triangle are `(x-40^@),(x-20^@),(x/2-10^@)` .Find the value of x & then find the angles of the triangle.

Answer

To solve the problem, we need to find the value of \( x \) and then determine the angles of the triangle given as \( (x - 40^\circ), (x - 20^\circ), \) and \( \left(\frac{x}{2} - 10^\circ\right) \).

### Step 1: Set up the equation
We know that the sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can set up the equation:

\[ (x - 40^\circ) + (x - 20^\circ) + \left(\frac{x}{2} - 10^\circ\right) = 180^\circ \]

### Step 2: Combine like terms
Now, let's combine the terms on the left side of the equation:

\[ x - 40^\circ + x - 20^\circ + \frac{x}{2} - 10^\circ = 180^\circ \]

This simplifies to:

\[ 2x + \frac{x}{2} - 70^\circ = 180^\circ \]

### Step 3: Eliminate the constant
Next, we can add \( 70^\circ \) to both sides of the equation:

\[ 2x + \frac{x}{2} = 180^\circ + 70^\circ \]

This simplifies to:

\[ 2x + \frac{x}{2} = 250^\circ \]

### Step 4: Find a common denominator
To combine \( 2x \) and \( \frac{x}{2} \), we convert \( 2x \) to have a common denominator of 2:

\[ \frac{4x}{2} + \frac{x}{2} = 250^\circ \]

This simplifies to:

\[ \frac{5x}{2} = 250^\circ \]

### Step 5: Solve for \( x \)
Now, we can multiply both sides by 2 to eliminate the fraction:

\[ 5x = 500^\circ \]

Next, divide both sides by 5:

\[ x = 100^\circ \]

### Step 6: Find the angles of the triangle
Now that we have \( x \), we can find the angles of the triangle:

1. First angle:
\[ x - 40^\circ = 100^\circ - 40^\circ = 60^\circ \]

2. Second angle:
\[ x - 20^\circ = 100^\circ - 20^\circ = 80^\circ \]

3. Third angle:
\[ \frac{x}{2} - 10^\circ = \frac{100^\circ}{2} - 10^\circ = 50^\circ - 10^\circ = 40^\circ \]

### Final Result
Thus, the value of \( x \) is \( 100^\circ \) and the angles of the triangle are \( 60^\circ, 80^\circ, \) and \( 40^\circ \).

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