Two triangles are congruent if they satisfy any of the following:
-- two sides and the included angle of one triangle equal to the corresponding parts of the other one
-- two angles and the included side of one triangle equal to the corresponding parts of the other one
-- all three sides of one triangle equal to the corresponding parts of the other one
-- they are right triangles, and hypotenuse and one leg of one triangle equal to the
corresponding parts of the other one
-- they are right triangles, and hypotenuse and one acute angle of one triangle equal
to the corresponding parts of the other one
sssThere are five methods for proving the congruence of triangles. In SSS, you prove that all three sides of two triangles are congruent to each other. In SAS, if two sides of the triangles and the angle between them are congruent, then the triangles are congruent. In ASA, if two angles of the triangles and the side between them are congruent, then the triangles are congruent. In AAS, if two angles and one of the non-included sides of two triangles are congruent, then the triangles are congruent. In HL, which only applies to right triangles, if the hypotenuse and one leg of the two triangles are congruent, then the triangles are congruent.
You left out one very important detail . . . the statement is true for a RIGHT triangle.
You can't use AAA to prove two triangles congruent because triangles can have the same measures of all its angles but be bigger or smaller, AAA could probably be used to prove two triangles are similar not congruent.
Excuse me, but two triangles that have A-A-S of one equal respectively to A-A-S of the other are not necessarily congruent. I would love to see that proof!
congruent
It is a theorem, not a postulate, since it is possible to prove it. If two angles and a side of one triangle are congruent to the corresponding angles and side of another triangle then the two triangles are congruent.
Yes, you can use either the ASA (Angle-Side-Angle) Postulate or the AAS (Angle-Angle-Side) Theorem to prove triangles congruent, as both are valid methods for establishing congruence. ASA requires two angles and the included side to be known, while AAS involves two angles and a non-included side. If you have the necessary information for either case, you can successfully prove the triangles are congruent.
The SAS theorem is used to prove that two triangles are congruent. If the triangles have a side-angle-side that are congruent (it must be in that order), then the two triangles can be proved congruent. Using this theorem can in the future help prove corresponding parts are congruent among other things.
To prove that two triangles are congruent, you can use the Side-Angle-Side (SAS) Postulate. This states that if two sides of one triangle are equal to two sides of another triangle, and the angle between those sides is also equal, then the triangles are congruent. Alternatively, the Angle-Side-Angle (ASA) Theorem can also be used if two angles and the included side of one triangle are equal to the corresponding parts of another triangle.
Two congruent triangles.. To prove it, use the SSS Postulate.
sssThere are five methods for proving the congruence of triangles. In SSS, you prove that all three sides of two triangles are congruent to each other. In SAS, if two sides of the triangles and the angle between them are congruent, then the triangles are congruent. In ASA, if two angles of the triangles and the side between them are congruent, then the triangles are congruent. In AAS, if two angles and one of the non-included sides of two triangles are congruent, then the triangles are congruent. In HL, which only applies to right triangles, if the hypotenuse and one leg of the two triangles are congruent, then the triangles are congruent.
LEGS
The first thing you prove about congruent triangles are triangles that have same side lines (SSS) is congruent. (some people DEFINE congruent that way). You just need to show AAS is equivalent or implies SSS and you are done. That's the first theorem I thought of, don't know if it works though, not a geometry major.
To prove two triangles congruent by the Hypotenuse-Leg (HL) theorem, you need to know that both triangles are right triangles. Additionally, you must establish that the lengths of their hypotenuses are equal and that one pair of corresponding legs is also equal in length. With this information, you can confidently apply the HL theorem to conclude that the triangles are congruent.
AAS: If Two angles and a side opposite to one of these sides is congruent to thecorresponding angles and corresponding side, then the triangles are congruent.How Do I know? Taking Geometry right now. :)
When trying to prove two triangles congruent, you can use SSS, SAS, ASA, AAS, HL, and HA patterns. However, the pattern A S S doesn't work. Instead of spelling or saying this word in class, you can refer to it as "the donkey theorem". You can look at the pattern in the two triangles and say "these two triangles are not congruent because of the donkey theorem." You CANNOT prove triangles incongruent with 'the donkey theorem', nor can you prove them congruent. It's mostly sort of a joke, you could say, but it's never useful. The reason is that if the two triangles ARE congruent, then of course there will be an unincluded congruent angle as well as two congruent sides. The theorem doesn't do anything left, right, forward or backward. It's not even really a theorem. :P
You can use a variety of postulates or theorems, among others: SSS (Side-Side-Side) ASA (Angle-Side-Angle - any two corresponding sides* and a corresponding angle) SAS (Side-Angle-Side - the angle MUST be between the two sides, except:) RHS (Right angle-Hypotenuse-Side - this is only ASS which works) * if two corresponding angles are the same, then the third corresponding angle must also be the same (as the angles of a triangle always sum to 180°), and that can be substituted for one angle of ASA to get AAS or SAA.