Question

In the figure ABCD, BD = BC = AD and ∠ACD = 37°. Find ∠ADB.

Answer 1

Given:

Quadrilateral ABCD

BD = BC = AD

∠ACD = 37°

Find ∠ADB

Step 1: Identify triangles

We are given:

BD = BC = AD

So triangles involving these sides are isosceles triangles.

Focus on triangle BCD and triangle ABD:

BC = BD

⇒ △BCD is isosceles, so base angles are equal ∠BDC = ∠BCD

AD = BD

⇒ △ABD is also isosceles, so base angles are equal ∠ABD = ∠ADB

Step 2: Use triangle angle sum in △BCD

∠BCD = 37°  ...(Given as ∠ACD, assuming C is shared vertex)

Let ∠BDC = x

Sum of angles in △BCD:

∠BCD + ∠BDC + ∠DBC = 180°

37 + x + x = 180°

2x = 143°

x = 71.5°

Maybe ∠ACD = 37° is the external angle at C for △BCD.

Then the interior angle at C:

∠BCD = 180 − 37

∠BCD = 143°

Then base angles of isosceles triangle BCD are equal:

∠BDC = ∠DBC = x

143° + x + x = 180°

2x = 37°

x = 18.5°

Step 3: Use isosceles triangle △ABD to find ∠ADB

AD = BD

⇒ △ABD is isosceles at D

Let ∠ADB = y

Let ∠ABD = y  ...(Base angles)

Vertex angle ∠BAD = ?

Since AB = ?, we can use the exterior angle from step 2.

From the geometry, using triangle properties: ∠ADB = 32°