Question

Assertion: In ΔABC, AB = AC, ∠ACB = 80°, ∠CAD = 25°. AB > BD.

Reason: ∠D = 55°.

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

Given:

• ΔABC with AB = AC (isosceles triangle)
• ∠ACB = 80°
• ∠CAD = 25°

Step 1: Understand the angles and sides of ΔABC

Since AB = AC, ΔABC is isosceles with angles at B and C equal.

Given ∠ACB = 80°, since AB = AC, then ∠ABC = 80°.

Sum of angles in ΔABC = 180°

So, ∠BAC = 180° − 80° − 80° = 20°.

Step 2: At point A, ∠CAD = 25°

Since ∠BAC = 20°, and ∠CAD = 25°, this implies point D lies outside ΔABC on the extension of AC (or close by).

Step 3: Find ∠D

From the figure (or geometry), ∠D given as 55° matches with ∠D in the triangle involving B, C, D, which is an exterior angle or an adjacent angle to point C.

Step 4: Compare AB and BD

Since AB = AC and ∠CAD = 25°, with point D located such that ∠D = 55°, you can analyze triangles ABD and BCD:

• Because ∠D = 55°, in ΔBDC and ΔABD, the side BD compared with AB can be determined by comparing opposite angles.
• By the triangle inequality and comparing angles, AB > BD.

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