Consider the following statements relating to the congruency of two right-angled triangles.
I. Equality of two sides of one triangle with some two sides of the second makes the triangles congruent.
II. Equality of the hypotenuse and a side of one triangle with the hypotenuse and a side of the second respectively makes the triangles congruent.
III. Equality of the hypotenuse and an acute angle of one triangle with the hypotenuse and an angle of the second respectively makes the triangles congruent.
Which of the above statements are true?

I, II and III

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I and II only

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I and III only

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II and III only

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Solution

The correct option is D
II and III only

According to RHS congruence rule, “In two right-angled triangles, if the length of the hypotenuse and one side of the triangle is equal to the length of the hypotenuse and the corresponding side of the other triangle, then the two triangles are congruent”.
I. Equality of two sides of one triangle with some two sides of the second makes the SS congruency, which is not valid.
II. Equality of the hypotenuse and a side of one triangle with the hypotenuse and a side of the second respectively makes the triangles congruent by the RHS congruence rule.
III.Both triangles are right-angled triangles, so one angle is $90 °$ in each.
Also given that the hypotenuse and an acute angle of one triangle are equal to the hypotenuse and an angle of the second respectively.
Thus, It makes the triangles congruent by the AAS congruence rule.
Hence, the correct statements are II and III only.

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Similar questions

Q. Which of the following statements are true and which of them are false?

(i) All squares are congruent.
(ii) If two squares have equal areas, they are congruent.
(iii) If two figures have equal areas, they are congruent.
(iv) If two triangles are equal in area, they are congruent.
(v) If two sides and one angle of a triangle are equal to the corresponding two sides and angle of another triangle, the triangle are congruent.
(vi) If two angles and any side of a triangle are equal to the corresponding angles and the side of another triangle then the triangles are congruent.
(vii) If three angles of a triangle are equal to the corresponding angles of another triangle then the triangles are congruent.
(viii) If the hypotenuse and an acute angle of a right triangle are equal to the hypotenuse and the corresponding acute angle of another right triangle then the triangle are congruent.
(ix) If the hypotenuse of a right triangle is equal to the hypotenuse of another right triangle then the triangles are congruent.
(x) If two triangles are congruent then their corresponding sides and their corresponding angles are congruent.

Which of the following statements are true (T) and which are false (F):

(i) Sides opposite to equal angles of a triangle may be unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilateral triangle is $60^{\circ}$

(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.

(v) The bisectors of two equal angles of a triangle are equal.

(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

(vii) The two altitudes corresponding to two equal sides of the triangle need not be equal

(viii) If any two sides of a right triangle are respectively equal to two sides of another right triangle, then the two triangles are congruent.

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

Question 104

State whether the statement is True or False.

If the hypotenuse and an acute angle of one right-angled triangle are equal to the hypotenuse and an acute angle of another right-angled triangle, then the triangles are congruent.

Q. Consider the following statements relating to the congruency of two right triangles.
(1) Equality of two sides of one triangle with any two sides of the second makes the triangle congruent.
(2) Equality of the hypotenuse and a side of one triangle with the hypotenuse and a side of the second respectively makes the triangle congruent.
(3) Equality of the hypotenuse and an acute angle of one triangle with the hypotenuse and an angle of the second respectively makes the triangle congruent.
Of these statements:

Q. Which of the following statements are true(T) which are false (F).

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

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