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Related Pages Congruent TrianglesCongruent 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) RuleSide-Side-Side is a rule used to prove whether a given set of triangles are congruent. The SSS rule states that: In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ. Side-Angle-Side (SAS) RuleSide-Angle-Side is a rule used to prove whether a given set of triangles are congruent. The SAS rule states that: An included angle is an angle formed by two given sides.
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: Angle-Angle-Side (AAS) RuleAngle-side-angle is a rule used to prove whether a given set of triangles are congruent. The AAS rule states that: 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 CongruentA video lesson on SAS, ASA and SSS.
Using Two Column Proofs To Prove Triangles CongruentTriangle 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.
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.
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.
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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Congruent triangles are those two triangles which are said to be congruent if and only if one of them can be made to superpose on the other so as to cover it exactly. Let ∆ABC and ∆DEF be two congruent triangles, then we can superpose ∆ABC on ∆DEF so as to cover it exactly. The vertices of ∆ABC fall on the vertices of ∆DEF in the following order A ↔ D, B ↔ E, C ↔ F. Thus, the order in which vertices match automatically determines the correspondence between the sides and angles of the triangle. Corresponding parts are also called matching parts of triangles. So, we have six equalities Corresponding sides are congruent: AB = DE BC = EF CA = FD Corresponding angles are congruent: ∠A = ∠D ∠B = ∠E ∠C = ∠F In the congruent triangles we will observe six correspondences between their verities. The symbol used to denote correspondence is ‘↔’ A ↔ D B ↔ E C ↔ F written as ABC ↔ DEF A ↔ E B ↔ F C ↔ D written as ABC ↔ EFD A ↔ F B ↔ D C ↔ E written as ABC ↔ FDE A ↔ D B ↔ F C ↔ E written as ABC ↔ DFE A ↔ E B ↔ D C ↔ F written as ABC ↔ EDF A ↔ F B ↔ E C ↔ D written as ABC ↔ FED Therefore, ∆ABC ≅ ∆DEF Note: The order of letters in the name of two triangles will indicated the correspondence between the vertices of two triangles. Thus, two triangles are congruent only if there exist a correspondence between their vertices such that the correspondence sides and correspondence angles of two triangles are equal. Congruent Shapes Congruent Line-segments Congruent Angles Congruent Triangles Conditions for the Congruence of Triangles Side Side Side Congruence Side Angle Side Congruence Angle Side Angle Congruence Angle Angle Side Congruence Right Angle Hypotenuse Side congruence Pythagorean Theorem Proof of Pythagorean Theorem Converse of Pythagorean Theorem 7th Grade Math Problems 8th Grade Math Practice From Congruent Triangles to HOME PAGE
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Identify the one to one correspondence of vertices in which the two triangles are congruent
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