Question

angle AOB = 125°,
angle COD is equal to​

Answer 1

Answer:

125 because of vertically opposite angle

Answer 2

Given:-

∠AOB = 125°

AB, BC, CD and AD are tangents to the circle with centre O.

ABCD is a quadrilateral.

To find:-

∠COD

Solution:-

Using Angle Sum Property for △OAB,

∠AOB + ∠OBA + ∠OAB = 180°

125° + ∠OBA + ∠OAB = 180° (∵ ∠AOB = 125°)

∠OBA + ∠OAB = 180° - 125°

∠OBA + ∠OAB = 55°

AB = AD (∵ A pair of tangents from an external point to the circle are equal)

Similarly, AB = BC.

∠OAB = ∠OAD (∵ The pair of tangents to a circle from an external point are equal in length. When the line from external point to the centre is drawn, it bisects the angle between the pair of tangents that are equal in length.)➡ (1)

Similarly, ∠OBA = ∠OBC ➡ (2)

∠OAB + ∠OAD = ∠DAB (∵ From (1))

∠OAB + ∠OAB = ∠DAB (∵ ∠OAB = ∠OAD)

2∠OAB = ∠DAB

∠OAB = ∠DAB / 2

∠OBA + ∠OBC = ∠ABC (∵ From (2))

∠OBA + ∠OBA = ∠ABC (∵ ∠OBA = ∠OBC)

2∠OBA  = ∠ABC

∠OBA = ∠ABC / 2

Since ∠OBA + ∠OAB = 55°, ∠BAD/2 + ∠ABC/2 = 55°

                     (∵ ∠OAB = ∠DAB / 2, ∠OBA = ∠ABC / 2)

(∠BAD + ∠ABC) / 2 = 55°

∠BAD + ∠ABC = 55° × 2

∠BAD + ∠ABC = 110°

Using the Angle Sum Property for the quadrilateral ABCD

∠ABC + ∠BCD + ∠ADC + ∠DAB = 360°

(∠ABC + ∠DAB) + ∠BCD + ∠ADC = 360°

110° + ∠BCD + ∠ADC = 360°

∠BCD + ∠ADC = 360° - 110°

∠BCD + ∠ADC = 250°

DC = AD (∵ A pair of tangents from an external point to the circle are equal)

Similarly, DC = BC.

∠ODA = ∠ODC (∵ The pair of tangents to a circle from an external point are equal in length. When the line from external point to the centre is drawn, it bisects the angle between the pair of tangents that are equal in length.)  ➡ (3)

Similarly, ∠OCD = ∠OCB ➡ (4)

∠ODA + ∠ODC = ∠ADC (∵ From (3))

∠ODC + ∠ODC = ∠ADC (∵ ∠ODA = ∠ODC)

2∠ODC = ∠ADC

∠OCD + ∠OCB = ∠BCD (∵ From (4))

∠OCD + ∠OCD = ∠BCD (∵ ∠OCD = ∠OCB)

2∠OCD = ∠BCD

As ∠BCD + ∠ADC = 250°, 2∠ODC + 2∠OCD = 250°

         (∵ 2∠ODC = ∠ADC,  2∠OCD = ∠BCD)

2 (∠ODC + ∠OCD) =  250°

∠ODC + ∠OCD = 250° / 2

∠ODC + ∠OCD = 125°

Using Angle Sum Property for △ODC,

∠COD + ∠ODC + ∠OCD = 180°

∠COD + (∠ODC + ∠OCD) = 180°

∠COD + 125° = 180° (∵ ∠ODC + ∠OCD = 125°)

∠COD = 180° - 125°

∠COD = 55°

∴ The measure of the angle COD is 55°.