急! Properties of Circles 12q8

10) Let ∠BAT = x, then ∠DBA = x (Alt. ∠s, DB//PT) and ∠BDA = x (∠ in alt. segment)

Also ∠BAD = 180 - 118 = 62 (Opp. ∠s, cyclic quad.)

In △BAD, ∠BAD + 2x = 180

2x = 118

x = 59

11a) ∠EPQ = 90 (∠ in semi circle)

Let ∠EQP = x, then ∠EPB = x (∠ in alt. segment)

Also ∠QPA = ∠PQD = 50 + x (Alt ∠s, CD//AB)

On line AB:

∠QPA + ∠EOQ + ∠EPB = 180

50 + x + 90 + x = 180

x = 20

b) ∠QPA = 50 + x = 70

12a) Arc CA is 4/(5 + 4) = 4/9 of the whole circumference, thus ∠AOC = 360 x 4/9 = 160

b) ∠ABC = 1/2 x ∠AOC = 80 (∠ atn centre = twice ∠ at circum)

∠CAP = ∠ABC = 80 (∠ in alt. segment)