Question

In the figure, two chords AB and CD of the same circle are parallel to each other. P is the centre of the circle Show that angle CPA is congruent with angle DPB

Answer 1

don't worry this question is simple.
just show the chords are parallel and the line joining the centre of the circle and the chord is their transversal. hence the angle which we have to find are alternate angles.

hence by alternate angle property,
the angles are congruent.

just write this in mathematical form. you will get full marks by solving by this method

Answer 2

QUESTION :

P is the centre of the circle . Two chords AB and CD are parallel to each other.To prove that angle CPA is congruent to angle DPB

ANSWER :

GIVEN :

In a circle at centre P

AB || CD

TO PROVE :

angle CPA = angle DPB

OR

angle CPA is congruent to angle DPB

CONSTRUCTION :

join AP,CP,DP,BP

and draw PE perpendicular to AB

and PF perpendicular to CD

such that EF is a line

PROOF :

now in ∆AEP and ∆DPF

angle AEP = angle DFP.......(each 90°)

also angle APE = angleDPF .....(vertically opposite angles)

also PA = PD.......(radius of same circle)

so by AA congruence criteria

∆AEP is congruent to ∆DPF

so by CPCT

angle PAE = angle PDF.....1

AND

now in ∆CFP and ∆BEP

angle CFP = angle BEP.......(each 90°)

also angle CPF = angleBPE .....(vertically opposite angles)

also PC = PB......(radius of same circle)

so by AA congruence criteria

∆CPF is congruent to ∆BPE

so by CPCT

angle PCF = angle PBE......2

NOW

consider ∆ CDF

by exterior angle theorem

angle CPA = angle PCD + angle PDC.....3

Similarly in ∆APB

by exterior angle theorem

angle BPD = angle PAB + angle PBA......4

but from 1 and 2

angle PAE = angle PDF = angle PDC

and angle PCF = angle PBE = angle PBA

so equation 4 becomes

angle BPD = angle PAB + angle PBA

angle BPD = angle PDC + angle PCD....5

but from 3

angle CPA = angle PCD + angle PDC

SO equation 5 becomes

angle BPD = angle PDC + angle PCD

angle BPD = angle CPA

HENCE PROVED

NOTE :

while solving such questions

try to construct the needed things