The concept of 'congruency of triangles' is a familiar one in geometry. Essentially, two triangles are considered congruent if their corresponding sides and angles are identical in measurement. Despite their orientation or position, if you were to cut out these triangles and overlay them, they would match up perfectly. This article delves into two key criteria for congruence of triangles – RHS (Right angle – Hypotenuse – Side) and SSS (Side – Side – Side) .
Congruence of Triangles: Understanding RHS and SSS Rules
Last Updated on Jun 17, 2024
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The RHS Congruence Rule
Theorem: In the case of two right-angled triangles, if the hypotenuse and one side of one triangle are equal in length to the corresponding hypotenuse and side of the other triangle, then the two triangles are congruent.
The SSS Congruence Rule
Theorem : If in two triangles, the three sides of one triangle are equal to the corresponding three sides of the other triangle, then the triangles are congruent.
Example to Illustrate
Problem: Given a figure where AB = BC and AD = CD. Demonstrate that BD bisects AC at right angles.
Solution: Our task is to establish that ∠BEA = ∠BEC = 90° and AE = EC.
Let's consider ∆ABD and ∆CBD,
AB = BC (Given)
AD = CD (Given)
BD = BD (Common)
Hence, ∆ABD ≅ ∆CBD (By SSS congruency)
∠ABD = ∠CBD (By CPCT)
Now, let's consider ∆ABE and ∆CBE,
AB = BC (Given)
∠ABD = ∠CBD (Proved above)
BE = BE (Common)
Therefore, ∆ABE≅ ∆CBE (By SAS congruency)
∠BEA = ∠BEC (CPCTC)
Also, ∠BEA +∠BEC = 180° (Linear pair)
2∠BEA = 180° (∠BEA = ∠BEC)
∠BEA = 180°/2 = 90° = ∠BEC
AE = EC (CPCTC)
Therefore, BD is a perpendicular bisector of AC.
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Frequently Asked Questions
What is the RHS congruence rule?
In two right-angled triangles, if the length of the hypotenuse and one side of one triangle, is equal to the length of the hypotenuse and corresponding side of the other triangle, then the two triangles are congruent.
What is the SSS congruence rule?
In two triangles, if the three sides of one triangle are equal to the corresponding three sides of the other triangle, then the two triangles are congruent.
What are congruent triangles?
Two triangles are said to be congruent to each other if the measurements of their three sides and their three angles are exactly the same. They may be rotated or flipped, but when placed one on top of the other, they cover each other perfectly.
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