Two triangles given in the figure are congruent give the correspondence between the triangles

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(b) Let us now find the correspondence between the vertices, sides and angles of $\Delta XYZ\cong \Delta MNO$ .Correspondence between the vertices can be given as:$X\leftrightarrow M,Y\leftrightarrow N,Z\leftrightarrow O$We can write the correspondence between the sides as:$\begin{align} & XY=MN \\ & YZ=NO \\ & XZ=MO \\ \end{align}$Now, let us state the correspondence between the angles.$\begin{align} & \angle X=\angle M \\ & \angle Y=\angle N \\ & \angle Z=\angle O \\ \end{align}$This can be clearly seen from the following figure.

Note: We must be aware that the correspondence between vertices, sides and angles are applicable only when the two triangles are congruent. The order of letters in the name of two triangles will indicate the correspondence between the vertices of two triangles.

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To prove that two triangles have the same shape, certain parts of one triangle must coincide with certain parts of the other triangle. Specifically, the vertices of each triangle must have a one-to-one correspondence. This phrase means that the measure of each side and angle of each triangle corresponds to a side or angle of the other triangle. As we will see, triangles don't necessarily have to be congruent to have a one-to-one correspondence; but when they are congruent, it is necessary to know the correspondence of the triangles to know exactly which sides and which angles are congruent.

As you know, when a triangle's name is derived from the letters given to either its angles or sides (ex. triangle ABC). Until now, it didn't seem to matter which letters were there--as long as all three vertices were in the name, we knew which triangle we were talking about. Now, when we want to say that a given triangle, like triangle ABC, is congruent to another triangle, like triangle DEF, the order of the vertices in the name makes a big difference.

Congruent triangles ABC and DEF When two triangle are written this way, ABC and DEF, it means that vertex A corresponds with vertex D, vertex B with vertex E, and so on. This means that side CA, for example, corresponds to side FD; it also means that angle BC, that angle included in sides B and C, corresponds to angle EF. These relationships aren't especially important when triangles aren't congruent or similar. But when they are congruent, the one-to-one correspondence of triangles determines which angles and sides are congruent.

When a triangle is said to be congruent to another triangle, it means that the corresponding parts of each triangle are congruent. By proving the congruence of triangles, we can show that polygons are congruent, and eventually make conclusions about the real world.

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Congruent Triangles
More Geometry Lessons

Congruent Triangles

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.

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.
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.
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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