Question

Fill the blanks in the following so that each of the following statements is true.

(i) Sides opposite to equal angles of a triangle are ......

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

(iii) In an equilateral triangle all angles are .....

(iv) In a ΔABC if ∠A = ∠C , then AB = ......

(v) If altitudes CE and BF of a triangle ABC are equal, then AB = ....

(vi) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is …… CE.

(vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then  ΔABC ≅ Δ ……

Answer 1

(i) Sides opposite to equal angles of a triangle are equal

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

(iii) In an equilateral triangle all angles are equal

Reason: Since all sides are equal in a equilateral triangle, the angles opposite to equal sides will be equal

(iv) In a ΔABC if ∠A = ∠C , then AB =BC

Reason: Since, the sides opposite to equal angles are equal, the side opposite to ∠A i.e., BC and ∠C i.e., AB are equal

(v) If altitudes CE and BF of a triangle ABC are equal, then AB =AC

Reason: From RHS congruence criterion ΔBEC≅ΔCFB

⇒∠EBC=∠FCB⇒∠ABC = ∠ACB ⇒ AC = AB

[ ∵ Sides opposite to equal angels are equal]

(vi) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is equal to CE

Reason: Since angles opposite to equal sides are equal, so

So, by ASA congruence criterion

[Corresponding parts of congruent triangles are equal]

(vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then. ΔABC ≅ ΔEFD

Reason: From RHS congruence criterion we have  ΔABC ≅ ΔEFD