Congruent Triangles Calculator

Enter the corresponding side lengths and angle measures for two triangles to determine if they are congruent. Use the standard notation where side 'a' is opposite angle 'A', side 'b' opposite angle 'B', and side 'c' opposite angle 'C'.

Triangle 1

Side a: Side b: Side c: Angle A (°): Angle B (°): Angle C (°):

Triangle 2

Side a: Side b: Side c: Angle A (°): Angle B (°): Angle C (°):

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Understanding Congruent Triangles

In geometry, two triangles are said to be congruent if they have the same size and shape. This means that all three corresponding sides are equal in length, and all three corresponding angles are equal in measure. When two triangles are congruent, one can be perfectly superimposed on the other through a series of rigid transformations (translations, rotations, reflections).

Why is Congruence Important?

The concept of congruence is fundamental in geometry and has wide-ranging applications in various fields:

Architecture and Engineering: Ensuring structural stability and symmetry in designs.
Manufacturing: Producing identical parts for assembly.
Computer Graphics: Creating and manipulating identical objects in digital environments.
Proof in Geometry: Congruence postulates are essential tools for proving geometric theorems and relationships.

Congruence Postulates and Theorems

While comparing all six parts (three sides and three angles) of two triangles would definitively prove congruence, mathematicians have developed shortcuts known as congruence postulates (or theorems). These postulates state that if certain combinations of corresponding parts are equal, then the entire triangles must be congruent.

1. SSS (Side-Side-Side) Congruence Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Example:

Triangle 1: Sides a=5, b=7, c=8
Triangle 2: Sides a=5, b=7, c=8

These triangles are congruent by SSS.

2. SAS (Side-Angle-Side) Congruence Postulate

If two sides and the included angle (the angle between those two sides) of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Example:

Triangle 1: Side a=6, Angle B=40°, Side c=9
Triangle 2: Side a=6, Angle B=40°, Side c=9

These triangles are congruent by SAS.

3. ASA (Angle-Side-Angle) Congruence Postulate

If two angles and the included side (the side between those two angles) of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Example:

Triangle 1: Angle A=50°, Side b=10, Angle C=70°
Triangle 2: Angle A=50°, Side b=10, Angle C=70°

These triangles are congruent by ASA.

4. AAS (Angle-Angle-Side) Congruence Theorem

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

Example:

Triangle 1: Angle A=30°, Angle B=70°, Side a=12
Triangle 2: Angle A=30°, Angle B=70°, Side a=12

These triangles are congruent by AAS.

5. HL (Hypotenuse-Leg) Congruence Theorem (for Right Triangles only)

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

Example:

Triangle 1: Angle C=90°, Hypotenuse c=13, Leg a=5
Triangle 2: Angle C=90°, Hypotenuse c=13, Leg a=5

These triangles are congruent by HL.

How to Use the Calculator

Our Congruent Triangles Calculator allows you to input the corresponding side lengths (a, b, c) and angle measures (A, B, C in degrees) for two triangles. The calculator will then analyze the provided information and determine if the triangles are congruent based on the established postulates (SSS, SAS, ASA, AAS, HL). Remember to enter corresponding parts for accurate results:

Side 'a' is opposite Angle 'A'.
Side 'b' is opposite Angle 'B'.
Side 'c' is opposite Angle 'C'.

If you don't have a specific measurement, you can leave the field blank. The calculator will use the available information to check for congruence and will also attempt to calculate a missing angle if two angles are provided for a triangle.