Which of the following conditions is always true that the two triangles are congruent

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS   (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)

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

2. SAS   (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:

is congruent to:

(See Solving SAS Triangles to find out more)

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA   (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

is congruent to:

(See Solving ASA Triangles to find out more)

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS   (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

is congruent to:

(See Solving AAS Triangles to find out more)

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

5. HL   (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

• the same length of hypotenuse and
• the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:

is congruent to:

(See Pythagoras' Theorem to find out more)

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.

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When triangles are congruent (identical), one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and corresponding angles will be congruent.

When triangles are congruent, six facts are always true.
Corresponding sides are congruent.	Corresponding angles are congruent.

The good news is that when trying to verify that two triangles congruent , it is not necessary to show that all six of these facts to be true. There are certain ordered combinations of these facts that are sufficient to verify triangles to be congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.

If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.	If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.	AAS (or SAA) Angle-Angle-Side If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. This is an extension of ASA. In ASA, since you know two sets of angles are congruent, you automatically know the third sets are also congruent since there are 180º in each triangle.
If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.	The "included angle" in SAS is the angle formed by the two sides of the triangle being used. The "included side" in ASA is the side between the angles being used. It is the side where the rays of the angles overlap. The "non-included" side in AAS can be either of the two sides that are not directly between the two angles being used.

This method will NOT prove triangles congruent! The AAA combination will show that the triangles are the same SHAPE (similar), but will NOT show that the triangles are the same size. Example: Consider these two equilateral triangles that satisfy the AAA combination. They are the same shape, but are not the same size. Thus, they are not congruent. (They are similar.) Yes, it is possible that the sides "could" be the same length and the triangles would be congruent, but this would be the exception, not the rule.

SSA or ASS Side-Side-Angle This method will NOT always prove triangles congruent! The SSA (or ASS) combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem". SSA (or ASS) is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same. The problem with SSA (or ASS), is that the lengths of the sides being used, and their position in relation to the angle, do not always establish congruent triangles. Since the angle is not between the sides, side b, as shown below, is free to "swing" into one of two different positions. Two different triangles can be formed. Since this method allows for the possibility of creating triangles of various shapes (and even no triangles at all in some cases), this method is not an accepted method to prove triangles congruent. For a more detailed explanation of the possible shapes issue, see http://mathbitsnotebook.com/Geometry/CongruentTriangles/CTtriangleMethods.html