Question

The diagonals of a rectangle ABCD intersect at P and PAL is an equilateral triangle such that B and L are on the same side of AC. If ∠ACD = 30°, find:

1. ∠ALB
2. ∠ABL

Answer 1

Given:

ABCD is a rectangle, with diagonals intersecting at P

PAL is an equilateral triangle

B and L lie on the same side of diagonal AC

∠ACD = 30°

Step-by-Step reasoning:

i. Find ∠ALB

We are given:

PAL is equilateral ⇒ ∠PAL = 60°

Since B and L lie on the same side of AC, triangle ALB wraps around the vertex A and bends toward B.

Then the angle ∠ALB = Angle at vertex L in triangle ALB

Using angle relationships:

The angle between line segments LA and LB will be:

∠ALB = 180° – ∠PAL + ∠ACD

= 180° – 60° + 0°

= 120°

ii. Find ∠ABL

Now in triangle ABL, angle sum is 180°

We already have:

∠ALB = 120°

∠BAL = 30°   ...(Because ∠PAL = 60° and ∠DAC = 30°, due to rectangle and given angle)

 So, ∠ABL

= 180° – (120° + 30°)

= 30°