# Circles and Angles

• Oct 7th 2013, 06:14 AM
wardr1
Circles and Angles
I'm hoping that someone can give me some help with the following problem:

O is the centre of the circle (show on the attachment to this post) and AB is parallel to CD. Find the angles labelled x and y.

With a bit of cheating I was able to find that x = 32o and y = 58o but my questions is 'why' or what is the 'proof' of this?

Many Thanks.
• Oct 7th 2013, 06:59 AM
ebaines
Re: Circles and Angles
If you construct a line from Point B horizontally to intersect DC at right angles at Point E, from symmetry you can see that the length of that line is R sin(64). And from that the angle BDC (which also equals angle x) is arctan(Rsin(64)/(R+Rcos(64)). From that you have angle x.

Length CE is then R-Rcos(64), so angle y is arctan(Rsin(64)/(R-Rcos(64)).

Attachment 29395
• Oct 7th 2013, 10:03 AM
mathssolutions
Re: Circles and Angles
Attachment 29396

ADO = ODA = 58 (isosceles triangle)
DBC = Right angle (Angle on a semi circle)
ABD = 180 -(90 + 58) = 32 (Cyclic quadrilaterals opposite angles are supplementary) [Angle x = 32]
ABD + DBC +BCD = 180 (Supplementary angles between parallel lines)
ABD (32) + DBC (90)= 122 therefore BCD = 180 = 122 = 58 [Angle y = 58]
• Oct 7th 2013, 04:13 PM
votan
Re: Circles and Angles
Quote:

Originally Posted by wardr1
I'm hoping that someone can give me some help with the following problem:

O is the centre of the circle (show on the attachment to this post) and AB is parallel to CD. Find the angles labelled x and y.

With a bit of cheating I was able to find that x = 32o and y = 58o but my questions is 'why' or what is the 'proof' of this?

Many Thanks.

Attachment 29398