Question

Assertion: In the quadrilateral ABCD, AB = BC = CD. ∠B = 100°, ∠C = 60°, then ∠A = 70°.

Reason: Join BD, ΔBCD is equilateral and ΔABD isosceles.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Given AB = BC = CD, so three consecutive sides are equal.

∠B = 100° and ∠C = 60° are given.

If the quadrilateral is such that when BD is joined,

• ΔBCD is equilateral means BC = CD = BD and all angles of ΔBCD are 60°.
• ΔABD is isosceles means AB = BD.

Since AB = BC = CD and BD equals BC and CD (in equilateral ΔBCD), BD = AB also. Therefore, ΔABD is isosceles with AB = BD.

Then calculating ∠A in ΔABD will lead to ∠A = 70° (based on sum of angles in triangles and given angle at B).

Therefore:

• The assertion that ∠A = 70° is true.
• The reason that ΔBCD is equilateral (so BD equals BC and CD) and ΔABD is isosceles is also true.
• Moreover, the reason correctly explains why ∠A = 70° (i.e., joining BD helps determine these properties and angles).