22. Triangle Angle Validator | Conditional Statements

📋 Problem Statement

Input all angles of a triangle and check whether it is a Valid or Not Valid triangle.

📐 Triangle Angle Sum Property

For three angles to form a valid triangle, they must satisfy two conditions:

Condition 1: The sum of all three angles must equal exactly 180°
Condition 2: Each individual angle must be greater than 0°

Mathematical representation:

a + b + c = 180
a > 0 AND b > 0 AND c > 0

Both conditions must be true simultaneously.

a°
/\
/ \
/ \
/ \
/________\
b° c°

Valid Triangle: a + b + c = 180° AND a > 0, b > 0, c > 0

📥 Input Format

Enter three integers as input

📤 Output Format

Print the output as "Valid" or "Not Valid"

💡 Examples

Example 1:

Input: 130 30 60

Output: Not Valid

Verification:
• a = 130°, b = 30°, c = 60°
• Check sum: 130 + 30 + 60 = 220° ✗
• Expected sum: 180°
• Check positive: All angles > 0 ✓
Result: Sum ≠ 180° → Not Valid
The angles add up to more than 180°, which violates the fundamental triangle property.

Example 2:

Input: 70 70 40

Output: Valid

Verification:
• a = 70°, b = 70°, c = 40°
• Check sum: 70 + 70 + 40 = 180° ✓
• Check positive: 70 > 0, 70 > 0, 40 > 0 ✓
Result: All conditions satisfied → Valid triangle
This forms an isosceles triangle with two equal angles of 70°.

Example 3:

Input: 60 60 60

Output: Valid

Verification:
• a = 60°, b = 60°, c = 60°
• Check sum: 60 + 60 + 60 = 180° ✓
• Check positive: All angles > 0 ✓
Result: Perfect equilateral triangle → Valid
All angles equal = equilateral triangle.

Example 4 (Edge Case):

Input: 0 90 90

Output: Not Valid

Verification:
• a = 0°, b = 90°, c = 90°
• Check sum: 0 + 90 + 90 = 180° ✓
• Check positive: a = 0 (NOT > 0) ✗
Result: One angle is 0° → Not Valid
Even though the sum is 180°, an angle of 0° creates a degenerate triangle (straight line).

Example 5 (Edge Case):

Input: -30 110 100

Output: Not Valid

Verification:
• a = -30°, b = 110°, c = 100°
• Check sum: -30 + 110 + 100 = 180° ✓
• Check positive: a = -30 (negative) ✗
Result: Negative angle → Not Valid
Negative angles don't exist in geometric triangles.

⚠️ Constraints

-10^9 <= Angles <= 10^9

✅ Solution

#include <iostream>

using namespace std;

int main() {
    int a, b, c;
    cin >> a >> b >> c;
    
    // Check both conditions: sum = 180 AND all angles positive
    if (a + b + c == 180 && a != 0 && b != 0 && c != 0)
        cout << "Valid";
    else
        cout << "Not Valid";
    
    return 0;
}

🔑 Key Concepts

Triangle Angle Sum Property: Fundamental geometric rule that angles sum to 180°
Compound AND Conditions: Multiple conditions must all be true simultaneously
Zero Check: Using != 0 to ensure positive angles (excludes negative and zero)
Edge Case Handling: Preventing degenerate triangles (0° or negative angles)
Single Condition Statement: Combining all checks into one if statement
Geometric Validation: Applying mathematical theorems in programming

💡 Understanding the Validation Logic

Why Check for Non-Zero?

Using a != 0 instead of a > 0 is intentional:

a != 0: Excludes both 0 and negative values
- 0° → creates a degenerate triangle (flat line)
- Negative angles → geometrically impossible
a > 0: More explicit "greater than zero" check
- Both approaches work for this problem
- != 0 is slightly more concise

Complete Validation:

// All four conditions combined with AND (&&)
if (
    a + b + c == 180  // Condition 1: Sum equals 180
    &&
    a != 0            // Condition 2: First angle valid
    &&
    b != 0            // Condition 3: Second angle valid
    &&
    c != 0            // Condition 4: Third angle valid
)

⚠️ Common Mistakes to Avoid

Only checking the sum: if (a + b + c == 180) would accept (0, 90, 90) as valid

// ❌ WRONG: Missing positive checks
if (a + b + c == 180)
    cout << "Valid";  // Accepts invalid cases!

Forgetting zero check: Zero angles create degenerate triangles (straight lines)
Not checking negative angles: Angles like -30° would be accepted if only checking sum
Using OR instead of AND: || would make the logic incorrect
Separate if statements: Not combining conditions can lead to wrong logic flow
Wrong equality check: Using = instead of == for comparison

🚀 Alternative Validation Approaches

Approach 1: Explicit Positive Check

if (a + b + c == 180 && a > 0 && b > 0 && c > 0)
    cout << "Valid";
else
    cout << "Not Valid";

Approach 2: Range Check (More Comprehensive)

// Check if each angle is in valid range (0, 180)
if (a + b + c == 180 && 
    a > 0 && a < 180 &&
    b > 0 && b < 180 &&
    c > 0 && c < 180)
    cout << "Valid";
else
    cout << "Not Valid";

Approach 3: Separate Validation Steps

bool sum_valid = (a + b + c == 180);
bool angles_positive = (a > 0 && b > 0 && c > 0);

if (sum_valid && angles_positive)
    cout << "Valid";
else
    cout << "Not Valid";

🎓 Extension: Triangle Classification by Angles

Once you know it's valid, you can classify the triangle by its angles:

Acute Triangle: All three angles < 90° (e.g., 60°, 60°, 60°)
Right Triangle: One angle = 90° (e.g., 90°, 45°, 45°)
Obtuse Triangle: One angle > 90° (e.g., 120°, 40°, 20°)
Equiangular Triangle: All angles equal = 60° each (special case of acute)

// Extended classification
if (a == 90 || b == 90 || c == 90)
    cout << "Right Triangle";
else if (a > 90 || b > 90 || c > 90)
    cout << "Obtuse Triangle";
else
    cout << "Acute Triangle";

🌍 Real-world Applications

Navigation Systems: GPS triangulation requires valid angle measurements
Computer Graphics: Rendering engines validate triangle meshes for 3D models
Surveying: Land measurement using theodolites and angle measurements
Architecture: Structural design validation for triangular supports
Robotics: Path planning and obstacle detection using triangulation
Game Development: Collision detection and physics calculations
Astronomy: Stellar parallax calculations using angular measurements
Engineering: Stress analysis in truss structures
Education Software: Automated geometry problem validation

📚 Mathematical Background

The Triangle Angle Sum Theorem

Euclidean Geometry: In flat (Euclidean) space, triangle angles always sum to exactly 180°
Historical Note: Known since ancient Greek mathematics (Euclid's Elements)
Proof Method: Can be proven using parallel lines and alternate interior angles
Universal Property: True for all triangles regardless of size or shape
Non-Euclidean Exception: In spherical geometry (curved surfaces), angles can sum to > 180°

Why Each Angle Must Be Positive:

Zero Angle: Creates a degenerate triangle (collapses to a line)
Negative Angle: Geometrically undefined in standard Euclidean geometry
Practical Limit: Each angle must be in range (0°, 180°) exclusive

🎯 What You'll Learn

Applying geometric theorems (angle sum property) in code
Writing compound conditional statements with multiple AND conditions
Understanding edge case validation (zero and negative values)
Distinguishing between valid and degenerate geometric shapes
Combining mathematical rules with logical operators
Input validation techniques for geometric problems
Clean, readable single-condition validation

💪 Challenge Yourself

Try solving these variations:

Level 1: Also output the type of triangle (Acute/Right/Obtuse)
Level 2: Check if angles form an equiangular triangle (all 60°)
Level 3: Given two angles, calculate and display the third angle
Level 4: Validate multiple triangles and count how many are valid
Level 5: Check if angles can form a valid triangle AND classify by both angles and hypothetical sides

Test Cases to Try:

Normal: (60, 60, 60), (90, 45, 45), (30, 70, 80)
Edge: (0, 90, 90), (180, 0, 0), (1, 1, 178)
Invalid: (100, 100, 100), (-30, 110, 100), (0, 0, 0)

Validate Triangle Using Angles

📋 Problem Statement

📐 Triangle Angle Sum Property

📥 Input Format

📤 Output Format

💡 Examples

⚠️ Constraints

✅ Solution

🔑 Key Concepts

💡 Understanding the Validation Logic

⚠️ Common Mistakes to Avoid

🚀 Alternative Validation Approaches

🎓 Extension: Triangle Classification by Angles

🌍 Real-world Applications

📚 Mathematical Background

🎯 What You'll Learn

💪 Challenge Yourself