💡 Problem Formulation: In geometry, two triangles are considered congruent if their corresponding sides and angles are equal. Our goal is to write a program to verify if a pair of triangles is congruent based upon given side lengths (and possibly angles). For instance, if Triangle A has sides of lengths 3, 4, 5 and Triangle B also has sides of similar lengths, the desired output should affirm congruency between the triangles.
Method 1: SSS (Side Side Side) Congruency Check
This method involves checking if all three sides of one triangle are equal to the corresponding sides of the other triangle. It assumes that the triangle’s sides are provided in sorted order (ascending or descending). If all three sides match, the triangles are congruent according to SSS postulate.
Here’s an example:
def is_congruent_sss(tri_a, tri_b):
return tri_a == tri_b
# Example usage
triangle_1 = [3, 4, 5]
triangle_2 = [5, 3, 4]
# Sort the sides
triangle_1.sort()
triangle_2.sort()
print(is_congruent_sss(triangle_1, triangle_2))
Output: True
This code snippet defines a function is_congruent_sss() which takes the sorted sides of two triangles as input and returns True if they are congruent according to the SSS postulate. It demonstrates the use of list sorting and simple equality comparison to verify congruency.
Method 2: SAS (Side Angle Side) Congruency Check
In this method, triangles are declared congruent if any two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of the other triangle. It is useful when angle measures are also known.
Here’s an example:
from math import radians, cos
def is_congruent_sas(side1_a, angle_a, side2_a, side1_b, angle_b, side2_b):
return side1_a == side1_b and side2_a == side2_b and cos(radians(angle_a)) == cos(radians(angle_b))
# Example usage
print(is_congruent_sas(3, 90, 4, 3, 90, 4))
Output: True
The function is_congruent_sas() compares two sides and the cosine of the included angle for both triangles, returning True if they are equivalent. It shows how trigonometric functions can be applied to check congruency following the SAS postulate.
Method 3: ASA (Angle Side Angle) Congruency Check
ASA congruence criterion checks if two angles and the intervening side are the same in both triangles. This method is applicable when the length of one side and measures of the adjacent angles are known for each triangle.
Here’s an example:
def is_congruent_asa(angle1_a, side_a, angle2_a, angle1_b, side_b, angle2_b):
return side_a == side_b and angle1_a == angle1_b and angle2_a == angle2_b
# Example usage
print(is_congruent_asa(45, 5.6569, 90, 45, 5.6569, 90))
Output: True
The is_congruent_asa() function verifies congruency based on the ASA postulate. If the provided side and adjacent angles match for both triangles, the function returns True. It highlights the importance of angle measurements for triangle congruency checks.
Method 4: AAS (Angle Angle Side) Congruency Check
AAS criteria states that two triangles are congruent if two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle. This method is particularly useful when two angles and any side are known.
Here’s an example:
def is_congruent_aas(angle1_a, angle2_a, side_a, angle1_b, angle2_b, side_b):
return side_a == side_b and angle1_a == angle1_b and angle2_a == angle2_b
# Example usage
print(is_congruent_aas(60, 60, 3, 60, 60, 3))
Output: True
The function is_congruent_aas() implements the AAS congruency check. By comparing two angles and a non-included side of the triangles, it properly determines if the triangles are congruent.
Bonus One-Liner Method 5: RHS (Right-angle Hypotenuse Side) Congruency Check
This is a specialized case of the SSS postulate for right-angled triangles. The hypotenuse and one other side are compared to establish congruency using a concise one-liner function.
Here’s an example:
is_congruent_rhs = lambda hyp_a, side_a, hyp_b, side_b: hyp_a == hyp_b and side_a == side_b # Example usage print(is_congruent_rhs(5, 3, 5, 3))
Output: True
Using a Python lambda function, is_congruent_rhs() provides a quick and elegant one-liner solution for the RHS congruency check specifically applicable to right-angled triangles.
Summary/Discussion
- Method 1: SSS Congruency Check. Straightforward for triangles of unknown orientation. Reliable if all sides are known. Limited when angle information is also needed.
- Method 2: SAS Congruency Check. Efficient when two sides and included angle can be measured. Requires a bit more math (trigonometry). Can be precise.
- Method 3: ASA Congruency Check. Useful for comparing triangles with known angles. Simple calculations but less common real-world application.
- Method 4: AAS Congruency Check. Handy when specific non-included side and angles are known. Like ASA, can be limited in application.
- Bonus Method 5: RHS Congruency Check. Ideal for right-angled triangles. Very specific use case. Elegant and simple to implement if conditions are met.