急! Properties of Circles 12q12

25a) ∠CDE = ∠ABC (Ext. ∠, cyclic quad.)

∠CDE = ∠EFG (Ext. ∠, cyclic quad.)

Hence ∠ABC = ∠EFG

b) ∠APC + ∠ABC = 180 (Opp. ∠s, cyclic quad.)

∠EPG + ∠EFG = 180 (Opp. ∠s, cyclic quad.)

∠APC + ∠ABC = ∠EPG + ∠EFG

∠APC = ∠EPG

∠APC + ∠CPE = ∠EPG + ∠CPE

∠APE = ∠CPG

26a) ∠FCE = ∠BCE = 34 (Equal chord, equal ∠ at circum)

b) ∠ABE = ∠ECF = 34 (∠ in alt. segment)

With ∠ABE = ∠DCE, ABCD is a cyclic quad. (Converse of ∠ in the same segment)

c) ∠ADE = ∠BCE = 34 (∠ in the same segment)

Hence ∠ABD = ∠ADB and so △ABCD is an isos. △

27a) Join OP, then △OMP and △ONP are congurent (RHS). Hence MP = NP.

b) AB = CD (Equal distance from centre, equal chord)

Also OM and ON bisects AB and BD resp. since they are perp. to each other.

Hence AM = MB = NC = ND

MB = NC

MB - MP = NC - NP

BP = CP