Right Triangle Congruence Theorem

Right Triangle Congruence Theorem:

A plane figure bounded by three finite line segments to form a closed figure is known as triangle. A right angled triangle is a special case of triangles. In a right angled triangle, one of the interior angles measure 90°.Two right triangles are said to be congruent if they are of same shape and size. In other words, two right triangles are said to be congruent if the measure of the length of their corresponding sides and their corresponding angles is equal.

In the fig. 1 given below, ∆ABC ≅ ∆RPQ since ∠A= ∠R, ∠C= ∠Q and ∠B= ∠P.

Also, AB = RP, BC = PQ and AC = QR.

Right Triangle - Congruence Theorem

Now, we will discuss about the different methods by which we can draw a triangle congruent to ∆ABC which is right angled at B. This leads to a very important criterion known as the RHS congruence or right triangle congruence theorem.

According to the RHS Criterion:

Two right angled triangles are said to be congruent to each other if the hypotenuse and one side of the right triangle are equal to the hypotenuse and the corresponding side of the other right angled triangle.

Let us understand right triangle congruence theorem with help of following example:

Question: Consider two triangles, ΔABC and ΔXYZ such that: ∠B = ∠Y = 90°, AC = XZ and AB = XY. Prove that ΔABC ≅ ΔXYZ.

Right Triangle - Congruence Theorem

Solution:

Proof:

In ΔABC and ΔXYZ it is given that: AC = XZ, BC = YZ and ∠B = ∠Y. Thus by right triangle congruence theorem, since the hypotenuse and the corresponding bases of the given right triangles are equal therefore both these triangles are congruent to each other. Thus, ΔABC ≅ ΔXYZ.

Therefore by using right triangle congruence theorem we can easily deduce of two right triangles are congruent or not.

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Practise This Question

The three sides of a triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent by which condition?