Answer:
in triangle ABR
AB=BR ( Given)
so, angle BAR = BRA = 29 ° ------- i ( angles opp. to equal sides are equal)
Angle ABR = sum of angles of triangle - (29°+29°)
= 180°-58°
= 122° ----------- ii
now angle RBD = x
angle RBD + angle ABR = 180° ( linear pair)
x + 122°=180°
x = 58°Now angle PRS + angle RPQ = 180° ( adjacent angles in a transversal )
Angle PRS = 50°+32°+angle BRA
Angle PRS = 50°+32°+29° ----- using i
Angle PRS = 111°
now ,
angle PRS + angle RPQ = 180°
y + 111°=180°
y = 69° Considering RQ as the tranversal
angle QRS = angle PQR = 32° -------- iii
Now in triangle BRD
angle DBR = 58° (value of x)
angle BRD = 50° (given)
angle BDR = sum of angles of triangle - ( 50°+58°)
= 180°-108°
= 72°
now angle BDR + angle CDQ = 180° ( linera pair)
72° + angle CDQ = 180°
angle CDQ = 180°-72°
angle CDQ = 108°
Now in triangle CDQ
angle CDQ = 108°
angle CQD= 32° -------- using iii
angle QCD = z =sum of angles of triangle - (108+32)
= 180°-140°
Z = 40° therefore the correct ✅option is A) x=58° , y= 69°, z = 40°