Congruence, Inequality On sides Of Triangles
Right Angle-Hypotenuse-Side(RHS) Congruence Criterion:
The Right Angle-Hypotenuse-Side (RHS) Congruence Criterion is a principle used in geometry to determine the congruence of right-angled triangles. This criterion states that if the hypotenuse and one of the legs of one right-angled triangle are equal to the hypotenuse and the corresponding leg of another right-angled triangle, then the two triangles are congruent. Here are some important facts about the RHS congruence criterion:
Criterion Statement:
If in two right-angled triangles, one pair of corresponding right angles is equal, and the hypotenuse and one leg of one triangle are equal to the hypotenuse and the corresponding leg of the other triangle, then the triangles are congruent.
Formal Notation:
The RHS congruence criterion is often expressed symbolically as follows:
If \(
\Delta ABC\,\,and\,\,\Delta DEF
\) are right angles such that
here \(
\left| \!{\underline {\,
B \,}} \right. = \left| \!{\underline {\,
E \,}} \right. = 90^O ,AC = DF
\) and BC = EF
\(
\therefore \Delta ABC \cong \Delta DEF
\)
Unique Solution:
The RHS criterion typically results in a unique solution. If two right-angled triangles satisfy the RHS condition, there is only one way to match the right angles, hypotenuses, and legs such that the triangles are congruent.
Application in Geometry:
The RHS congruence criterion is particularly useful when dealing with right-angled triangles. It provides a straightforward way to establish congruence based on specific side measurements without requiring angle measurements.
Comparison with Other Criteria:
The RHS criterion is a special case of the Side-Angle-Side (SAS) criterion, where the angle included is a right angle. It is important to consider the specific context of right-angled triangles when applying the RHS criterion.
Avoiding Ambiguity:
Unlike general triangles, right-angled triangles have a unique side, the hypotenuse, opposite the right angle. This helps avoid ambiguity when applying the RHS criterion.
Converse Statement:
The RHS criterion has a converse statement. If two right-angled triangles have congruent hypotenuses and legs, then they have congruent angles, including the right angle.