t us consider the two right-angled triangles, ABC and PQR, drawn below.
et us now discuss some more examples based on the above concept.
Example 1:
ΔABC and ΔLMN are two right-angled triangles. In ΔABC, ∠B = 90°, 
, and
. In ΔLMN, ∠M = 90°, 
, and
. Examine whether the two triangles are congruent.
Solution:
On the basis of the given information, the two triangles can be drawn as follows.
From ΔABC and ΔLMN, we obtain
∠B = ∠M = 90° (Right angle)
= 
= 2.5 cm (Given)
However, 
Hence, ΔABC and ΔLMN are not congruent.
Example 2:
ΔABC ≅ΔFED
What is the value of x?
Solution:
It is given that ΔABC ≅ ΔFED.
Now, we know that when two triangles are congruent, their corresponding sides are equal.
∴ 
= 
= 5.9 cm
Thus, the value of x is 5.9 cm.
Example 3:
ABC is an isosceles triangle where
. If 
is perpendicular to
, then show that
.
Solution:
In ΔABD and ΔACD,
∠ADB = ∠ADC = 90° (Right angle)
= (Given)
= (Common side)
∴ ΔABD ≅ ΔACD (By RHS congruence criterion)
∴ 
(Corresponding sides of congruent triangles)
Example 4:
Is the given pair of right-angled triangles congruent?
Solution:
Here, the hypotenuse of the first right-angled triangle is not equal to the hypotenuse of the second right-angled triangle. Therefore, the two triangles are not congruent.
Example 5:
In the following figure, ∠A = ∠D = 90° and
. Show that ∠BCA = ∠CBD.
Solution:
In ΔBCA and ΔCBD,
∠A = ∠D = 90° (Right angle)
= 
(Common side)
= 
(Given)
∴ ΔBCA ≅ ΔCBD (By RHS congruence criterion)
∴ ∠BCA = ∠CBD (Corresponding angles of congruent triangles)
Can we say that the given triangles are congruent?
Let us find out.
And, ∠B = ∠Q = 90°
Here, ∠B is the angle included between the sides and
. Similarly, ∠Q is the angle included between the sides
and
.
Therefore, by SAS congruence criterion, ΔABC ≅ ΔPQR
Two right-angled triangles are said to be congruent under a correspondence if the hypotenuse and one side of one right-angled triangle is equal to the hypotenuse and one side of the other right-angled triangle. |
