Can we conclude that the two triangles are congruent if their corresponding angles are congruent Why or why not?

When proving that triangles are congruent, it is not necessary to prove that all three pairs of corresponding angles and all three pairs of corresponding sides are congruent. There are shortcuts. For example, if two pairs of corresponding angles are congruent, then the third angle pair is also congruent, since all triangles have 180 degrees of interior angles. The following three methods are shortcuts for determining congruence between triangles without having to prove the congruence of all six corresponding parts. They are called SSS, SAS, and ASA.

SSS (Side-Side-Side)

The simplest way to prove that triangles are congruent is to prove that all three sides of the triangle are congruent. When all the sides of two triangles are congruent, the angles of those triangles must also be congruent. This method is called side-side-side, or SSS for short. To use it, you must know the lengths of all three sides of both triangles, or at least know that they are equal.

SAS (Side-Angle-Side)

A second way to prove the congruence of triangles is to show that two sides and their included angle are congruent. This method is called side-angle-side. It is important to remember that the angle must be the included angle--otherwise you can't be sure of congruence. When two sides of a triangle and the angle between them are the same as the corresponding parts of another triangle there is no way that the triangles aren't congruent. When two sides and their included angle are fixed, all three vertices of the triangle are fixed. Therefore, two sides and their included angle is all it takes to define a triangle; by showing the congruence of these corresponding parts, the congruence of each whole triangle follows.

ASA (Angle-Side-Angle)

The third major way to prove congruence between triangles is called ASA, for angle-side-angle. If two angles of a triangle and their included side are congruent, then the pair of triangles is congruent. When the side of a triangle is determined, and the two angles from which the other two sides point, the whole triangle is already determined, there is only one point, the third vertex, where those other sides could possibly meet. For this reason, ASA is also a valid shortcut/technique for proving the congruence of triangles.

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There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.

AA (Angle-Angle)

If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear.

Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.

SAS (Side-Angle-Side)

If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional.

Conclusion

These are the main techniques for proving congruence and similarity. With these tools, we can now do two things.

• Given limited information about two geometric figures, we may be able to prove their congruence or similarity.
• Given that figures are congruent or similar, we can deduce information about their corresponding parts that we didn't previously know.

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Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule.
In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

The following diagrams show the Rules for Triangle Congruency: SSS, SAS, ASA, AAS and RHS. Take note that SSA is not sufficient for Triangle Congruency. Scroll down the page for more examples, solutions and proofs.

Side-Side-Side (SSS) Rule

Side-Side-Side is a rule used to prove whether a given set of triangles are congruent.

The SSS rule states that:
If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ.

Side-Angle-Side (SAS) Rule

Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent.

The SAS rule states that:
If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is an angle formed by two given sides.

Included Angle           Non-included angle

For the two triangles below, if AC = PQ, BC = PR and angle C< = angle P, then by the SAS rule, triangle ABC is congruent to triangle QRP.

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

The ASA rule states that:
If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Rule

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

The AAS rule states that:
If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagrams below, if AC = QP, angle A = angle Q, and angle B = angle R, then triangle ABC is congruent to triangle QRP.

Three Ways To Prove Triangles Congruent

A video lesson on SAS, ASA and SSS.

1. SSS Postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
2. SAS Postulate: If there exists a correspondence between the vertices of two triangles such that the two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
3. ASA Postulate: If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

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Using Two Column Proofs To Prove Triangles Congruent

Triangle Congruence by SSS How to Prove Triangles Congruent using the Side Side Side Postulate?

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

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Triangle Congruence by SAS How to Prove Triangles Congruent using the SAS Postulate?

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

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Prove Triangle Congruence with ASA Postulate How to Prove Triangles Congruent using the Angle Side Angle Postulate?

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

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Prove Triangle Congruence by AAS Postulate How to Prove Triangles Congruent using the Angle Angle Side Postulate?

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

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