The total number of non- similar triangles which can be formed such that all the angles of the triangle are integers is

Options

A. 900
B. 1800
C. 2700
D. 15931

Answer

To find the total number of non-similar triangles that can be formed such that all the angles of the triangle are integers, we can follow these steps:

### Step 1: Understand the properties of triangle angles
The sum of the angles in a triangle is always 180 degrees. If we denote the angles of the triangle as \( x \), \( y \), and \( z \), we have:
\[ x + y + z = 180 \]
where \( x, y, z \) are all positive integers.

### Step 2: Set the conditions for the angles
We can assume without loss of generality that:
\[ x \leq y \leq z \]
This implies that:
\[ x \geq 1 \quad \text{and} \quad z \leq 180 \]

### Step 3: Express the angles in terms of one variable
Let’s express \( y \) and \( z \) in terms of \( x \) and a variable \( t \):
- Let \( y = x + t \) (where \( t \) is a non-negative integer)
- Then, substituting into the angle sum equation:
\[ x + (x + t) + z = 180 \]
This simplifies to:
\[ z = 180 - 2x - t \]

### Step 4: Determine the limits for \( t \)
From the triangle inequality, we know that all angles must be positive:
1. \( z > 0 \) implies \( 180 - 2x - t > 0 \)
\[ t < 180 - 2x \]
2. Since \( t \) is non-negative, we have:
\[ 0 \leq t < 180 - 2x \]

### Step 5: Find the range for \( x \)
Since \( x \) must also be a positive integer and \( z \) must be less than or equal to 180, we can find the maximum value of \( x \):
- The maximum value of \( x \) occurs when \( z \) is minimized:
\[ x < 90 \quad \text{(since } z \text{ must be at least } 1\text{)} \]
Thus, \( x \) can take values from 1 to 89.

### Step 6: Count the number of triangles for each \( x \)
For each integer value of \( x \) from 1 to 89, we can find the number of valid \( t \) values:
- The number of valid \( t \) values for a given \( x \) is:
\[ t_{\text{max}} = 180 - 2x - 1 \]
Thus, the number of possible \( t \) values is:
\[ t_{\text{count}} = (180 - 2x) - 1 \]

### Step 7: Calculate the total number of non-similar triangles
Now we sum the number of valid \( t \) values over all possible \( x \):
\[ \text{Total} = \sum_{x=1}^{89} (180 - 2x - 1) = \sum_{x=1}^{89} (179 - 2x) \]

### Step 8: Evaluate the summation
This summation can be simplified:
\[ = \sum_{x=1}^{89} 179 - \sum_{x=1}^{89} 2x \]
Using the formula for the sum of the first \( n \) integers:
\[ \sum_{x=1}^{n} x = \frac{n(n + 1)}{2} \]
We find:
\[ \sum_{x=1}^{89} x = \frac{89 \times 90}{2} = 4005 \]
Thus,
\[ \text{Total} = 179 \times 89 - 2 \times 4005 \]
Calculating this gives:
\[ = 15931 - 8010 = 7921 \]

### Final Answer
The total number of non-similar triangles that can be formed such that all the angles of the triangle are integers is **7921**.