Triangle Congruence Criteria

Two triangles are congruent if all their corresponding sides and angles are identical. In other words, if you overlay them, they will align perfectly.

To determine whether two triangles are congruent, we use specific congruence criteria. These allow us to verify congruence without measuring and comparing every single side and angle.

Here are the main congruence criteria:

SAS Criterion (Side-Angle-Side)

Two triangles are congruent if they have two equal sides and the included angle between them is also equal.

For example, consider two triangles ABC and DEF, where \( AB = DE \) and \( AC = DF \), and the included angle \( \widehat{A} = \widehat{D} \) is also congruent.

In this case, the two triangles are congruent (\(\triangle ABC \cong \triangle DEF\)).

If two houses have roof slopes of the same length and the same pitch angle, then their roofs must be identical. 

SSS Criterion (Side-Side-Side)

Two triangles are congruent if all three corresponding sides are equal.

For instance, if two triangles have matching sides, such that \( AB = DE \), \( BC = EF \), and \( AC = DF \), then they must be congruent (\(\triangle ABC \cong \triangle DEF\)).

Imagine two houses with identical roof slopes and a horizontal beam of the same length at the base. If all measurements match, the two roofs must be identical.

ASA Criterion (Angle-Side-Angle)

Two triangles are congruent if they share a common side and have two equal adjacent angles.

For example, if \( AB = DE \) and the adjacent angles \( \widehat{A} = \widehat{D} \) and \( \widehat{B} = \widehat{E} \) are also congruent, then the two triangles are congruent (\(\triangle ABC \cong \triangle DEF\)).

If two houses have the same base beam and the same angles between the roof slopes and the beam, then the slopes must be identical.

AAS Criterion (Angle-Angle-Side)

Two triangles are congruent if they have one equal side and two equal angles, regardless of their order.

For example, if \( AB = DE \) and two angles are congruent, \( \widehat{A} = \widehat{D} \) and \( \widehat{C} = \widehat{F} \), then the two triangles must be congruent (\(\triangle ABC \cong \triangle DEF\)).

If two houses have roof slopes of the same length and the same pitch angles, we can be sure that the structures are identical—even if we don’t know the length of the base.

Congruence Criteria for Right Triangles

For right triangles, there are two additional congruence criteria:

• Hypotenuse-Leg (HL) Criterion
  Two right triangles are congruent if they have the same hypotenuse and one congruent leg.

  For example, if two right triangles \( ABC \) and \( DEF \) have a common hypotenuse \( BC = EF \) and one congruent leg \( AC = DF \) (or \( AB = DE \) ), then the two triangles must be congruent (\(\triangle ABC \cong \triangle DEF\)).
  

• Leg-Leg (LL) Criterion
  Two right triangles are congruent if both of their legs are equal.

  For example, if two right triangles \( ABC \) and \( DEF \) have matching legs, such that \( AC = DF \) and \( AB = DE \), then the two triangles must be congruent (\(\triangle ABC \cong \triangle DEF\)).
  

In summary, congruence criteria provide a quick and reliable way to determine if two triangles are identical without measuring every individual element.

These principles are fundamental in geometry, as they play a crucial role in many proofs and constructions.