**π‘ Problem Formulation:** This article explores various methods to check whether a triangle with a positive area can be formed given three angles as input. We are assuming that the angles are in degrees, and the sum of the angles must be exactly 180 degrees for a valid triangle. The desired output is a boolean indicating whether or not the angles form a triangle with a positive area.

## Method 1: Using Simple Arithmetic Checks

This method involves checking if the sum of the three given angles equals 180 degrees, which is a requirement for any valid triangle. The function specified for this method, `is_valid_triangle()`

, returns `True`

if the angles form a valid triangle and `False`

otherwise.

Here’s an example:

def is_valid_triangle(angle1, angle2, angle3): return (angle1 + angle2 + angle3) == 180 and angle1 > 0 and angle2 > 0 and angle3 > 0 # Example usage print(is_valid_triangle(60, 60, 60)) print(is_valid_triangle(0, 90, 90))

Output:

True False

In this code, we define a function `is_valid_triangle()`

that takes three angle measurements. It checks if each angle is greater than zero and if their sum is equal to 180 degrees, which satisfies the triangle inequality theorem. The function returns a boolean value accordingly.

## Method 2: Using a Conditional Expression

A variation of Method 1, this approach also uses arithmetic checks but within a single line of a conditional expression. The function `can_form_triangle()`

simplifies the conditional checks and immediately returns the evaluation of the expression as the result.

Here’s an example:

can_form_triangle = lambda a1, a2, a3: (a1 + a2 + a3 == 180) and all(angle > 0 for angle in [a1, a2, a3]) # Example usage print(can_form_triangle(45, 45, 90)) print(can_form_triangle(180, 0, 0))

Output:

True False

The lambda function `can_form_triangle()`

performs a sum check and ensures that all angles are positive using a generator expression within the `all()`

function. It is a more concise way to achieve the same result as Method 1.

## Method 3: Exception Handling

In this method, we intentionally use the property that angles of a triangle must sum up to 180 degrees and are all positive. We can raise an exception if these conditions are not met. The function `validate_triangle()`

either confirms a valid triangle or raises a ValueError with a message explaining why the angles do not form a valid triangle.

Here’s an example:

def validate_triangle(a1, a2, a3): if not (a1 + a2 + a3 == 180 and a1 > 0 and a2 > 0 and a3 > 0): raise ValueError("Invalid triangle angles") return True # Example usage try: print(validate_triangle(50, 60, 70)) print(validate_triangle(-90, 45, 45)) except ValueError as e: print(e)

Output:

True Invalid triangle angles

The `validate_triangle()`

function checks the conditions for a valid triangle. If the angles do not conform to the requirements, a ValueError is raised. Otherwise, the function returns `True`

. This approach explicitly handles the error case of an invalid set of angles.

## Method 4: Using Numpy for Vectorized Operations

When dealing with multiple sets of angles, NumPy can provide an efficient and vectorized solution. This method involves using NumPy arrays and operations to check for valid triangles in a vectorized manner, which is particularly useful when processing large datasets of triangles. The function `check_triangles()`

returns an array of booleans.

Here’s an example:

import numpy as np def check_triangles(angles): return np.sum(angles, axis=1) == 180 & np.all(angles > 0, axis=1) # Example usage angles_array = np.array([[60, 60, 60], [180, 0, 0]]) print(check_triangles(angles_array))

Output:

[ True False]

In this code snippet, we create a `check_triangles()`

function that employs NumPy to sum the angles of each set (row) and check if all three angles are positive in a vectorized manner. The output is a NumPy array of boolean values indicating the validity of each set of angles. This is efficient for bulk validations.

## Bonus One-Liner Method 5: Conditional List Comprehension

Combining techniques from earlier methods, a one-liner with a list comprehension can quickly verify multiple sets of angles. This is a compact and Pythonic way of processing a list of angle sets without explicitly defining a function. The result will be a list of boolean values.

Here’s an example:

angle_sets = [(60, 60, 60), (30, 60, 90), (0, 180, 0)] results = [sum(triangle) == 180 and all(angle > 0 for angle in triangle) for triangle in angle_sets] # Example usage print(results)

Output:

[True, True, False]

The list comprehension iterates over a list of angle sets, checking the sum and the positivity of each angle using a generator expression. Each result of the comprehension corresponds to whether a set of angles forms a valid triangle.

## Summary/Discussion

**Method 1: Simple Arithmetic Checks.**Straightforward and easy to understand. Best for single triangle checks. Not as concise as other methods.**Method 2: Conditional Expression.**Condenses the check into one line. Good for clarity and conciseness. May be less readable to those unfamiliar with lambda functions.**Method 3: Exception Handling.**Provides explicit error messages. Good for applications needing detailed validations. More complex due to try-except blocks.**Method 4: Using Numpy.**Highly efficient for large datasets. Requires NumPy, which may not be suitable for all environments. Great for vectorized operations.**Method 5: Conditional List Comprehension.**Provides a readable and Pythonic one-liner for batch processing. Not as explicit as function-based methods and may be less optimal for single dataset checks.