Examples on RHS Congruence Rule

Example 1: P is a point equidistant from two lines l and m intersecting at point A. Show that the line AP bisects the angle between them.

Solution:

In the question we are given that lines l and m intersect each other at A.

Let PB ⊥ l, PC ⊥ m.

It is also given that PB = PC.

We need to show that ∠ PAB = ∠ PAC

Considering Δ PAB and Δ PAC,

PB = PC (Given)

∠PBA = ∠PCA = 90° (Given)

PA = PA (Common)

So, by RHS congruency rule,

Δ PAB ≅ Δ PAC

Therefore we prove, ∠ PAB = ∠ PAC (∵ CPCT)

Example 2: State and prove whether given triangles in the following image are congruent or not.

Solution:

In the given triangles, △ZXY and △PQR,

• XZ = PQ [side]
• YZ = PR [hypotenuse]
• ∠ZXY= ∠PQR=90° [right angle]

∴△ZXY≅△PQR, by RHS congruence criterion.

Hence proved.

Question 3: In the given triangle, △ABD, if AC bisects side BD and CE=CF, prove that the area of triangles △BCE and △DCF are equal.

Solution:

Two congruent triangles are always equal in area. SO, we need to prove that both the triangles are congruent for solving this question.

△BCE and △DCF are right triangles, in which,

• CB = CD (as AC bisects BD)
• CE = CF (given)
• ∠CEB=∠CFD=90°

∴ △ BCE ≅ △ DCF (by RHS congruence criterion)

Hence, △BCE and △DCF are equal in area. [Hence Prooved]

RHS Congruence Rule is also known as the HL (Hypotenuse-Leg) Congruence Theorem. It states the criteria for any two right-angle triangles to be congruent.

This rule states that if in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of the other triangle, then the triangles are congruent. In this article, we will discuss the criteria of congruence of right-angle triangles in detail including proof and examples.