Question

Angles problem?

Answer 1

See below.

Explanation:

First note.

The tangent to a point on the circumference of a circle is always perpendicular to the radius at the same point.

From diagram:

`/_OBQ=90^@`

Hence:

`/_CBO=90^@-2X`

For triangle `COB`

`bb(CO)=bb(BO)` radii of circle.

So:

`COB` is isosceles.

Hence:

`/_OCB=/_CBO=90^@-2X`

For triangle `DOC`

`bb(CO)=bb(DO)` radii of circle.

`DOC` is isosceles.

Hence:

`/_DCO=/_CDO=X`

For `/_BCD`

We know:

`/_OCB=90^@-2X`

and

`/_DCO=X`

Hence:

`/_BCD=/_OCB+/_DCO=90^@-2X+X=90^@-X`

`/_COB=180^@-(/_CBO+/_OCB)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(90^@-2X +90^@-2X)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(180^@-4X)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ =4X`

`/_DOC=180^@-(/_CDO+/_DCO)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(X+X)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(2X)`

`\ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-2X`

`/_DOB+/_DOC+/_COB=360^@`

`:.`

`/_DOB=360^@-/_DOC-/_COB`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \=360^@-(180^@-2X)-4X`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \=360^@-180^@+2X-4X`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \=180^@-2X`

`/_ODA = /_OBA=90^@` Perpendicular.

`/_ODA+/_OBA+/_DOB+y=360^@` Sum of angles in quadrilateral

`:.`

`90^@+90^@+(180^@-2X)+y=360^@`

`y=360^@-180^@-180^@+2X`

`y=2X`

Collecting everything:

`/_DCO=X`

`/_CBO=90^@-2X`

`/_OCB=90^@-2X`

`/_BCD=90^@-X`

`/_DOB=180^@-2X`

`y=2X`

I think there is probably a simpler method for this and maybe someone else will give a more concise way, but hope it helps.

Answer 2

see explanation.

Explanation:

let `r` be the radius of the circle,
As `OD=OC=r, => angleDCO=angleCDO=x`,
given `QBA` is tangent to the circle and `angleCBQ=2x`,
`=> angleCBO=90-2x`,
as `OB=OC=r, => angleOCB=angleOBC=90-2x`,
`angleBCD=angleOCB+angleDCO=90-2x+x=90-x`,
As the angle at the center of the circle is twice the angle at the circumference
`angleDOB=2*angleBCD=2*(90-x)=180-2x`

As `ADOB` is a quadrilateral, and the sum of the angles of a quadrilateral always adds up to `180^@`,
`=> y+angleADO+angleDOB+angleABO=360^@`,
`=> y=360-90-(180-2x)-90=360-360+2x=2x`