In a parallelogram ABCD, the side DC is produced to E and ∠BCE = 105°. Calculate ∠A, ∠B, ∠C, and ∠D.

Consider a Parallelogram ABCD,

It is told that the side BC is produced to E and ∠BCE = 105⁰.

From the figure, we can say that

∠BCE + ∠BCD = 180⁰ (Since ∠DCE is the straight angle)

By substituting the values we get,

⇒ 105⁰ + ∠BCD = 180⁰

⇒ ∠BCD = 180⁰ - 105⁰

⇒ ∠BCD = 75⁰

We know in a parallelogram the opposite angles are equal and the sum of the adjacent angles is 180⁰.

i.e, ∠C = ∠A ...... (1)

∠B = ∠D ...... (2)

∠C + ∠D = 180⁰ ...... (3)

From Eq(1) we can write

∠A = 75⁰.

From Eq(3), we can write

⇒ 75⁰ + ∠D = 180⁰

⇒ ∠D = 180⁰ - 75⁰

⇒ ∠D = 105⁰.

From Eq(2), we can write

∠B = 105⁰.

The angles ∠A, ∠B, ∠C, ∠D are75⁰, 105⁰, 75⁰, 105⁰.