Question
what are the congruence criterions?
Answers
GPT-4o mini
In geometry, congruence criteria are rules that allow us to determine whether two triangles are congruent, meaning they have the same size and shape. The primary congruence criteria for triangles are:
1. **Side-Side-Side (SSS) Congruence Criterion**: If all three sides of one triangle are equal to all three sides of another triangle, then the two triangles are congruent.
2. **Side-Angle-Side (SAS) Congruence Criterion**: If two sides and the included angle (the angle between the two sides) of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
3. **Angle-Side-Angle (ASA) Congruence Criterion**: If two angles and the side between them (the included side) of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
4. **Angle-Angle-Side (AAS) Congruence Criterion**: If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.
5. **Hypotenuse-Leg (HL) Congruence Criterion** (specific to right triangles): If the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
These criteria provide methods to prove that two triangles are congruent based on their sides and angles without needing to measure all their parts directly.
1. **Side-Side-Side (SSS) Congruence Criterion**: If all three sides of one triangle are equal to all three sides of another triangle, then the two triangles are congruent.
2. **Side-Angle-Side (SAS) Congruence Criterion**: If two sides and the included angle (the angle between the two sides) of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
3. **Angle-Side-Angle (ASA) Congruence Criterion**: If two angles and the side between them (the included side) of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
4. **Angle-Angle-Side (AAS) Congruence Criterion**: If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.
5. **Hypotenuse-Leg (HL) Congruence Criterion** (specific to right triangles): If the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
These criteria provide methods to prove that two triangles are congruent based on their sides and angles without needing to measure all their parts directly.