The value of z in the figure is 125⁰.
How the value of z was determined.
Given
AD = AB and CB = CD
Reflex ∠BAD = 290⁰
Acute ∠BAD = 360 - 290( ∠s at a point)
Acute ∠BAD = 70⁰
∆ABD is issoceles triangle (AD = AB)
∠ABD + ∠BDA + ∠BAD = 180⁰ (sum of interior angles of ∆).
But
∠ABD = ∠BDA(base ∠s of issos. ∆)
2∠BDA + ∠BAD = 180
2∠BDA = 180 -70
2∠BDA = 110
∠BDA = 110/2
= 55⁰
∠BDA + z = 180(linear ∠s)
55 + z = 180
z = 180 - 55
z = 125⁰
The value of z in the figure is 125⁰.