Question

In Fig. 14.114, ABCD is a rectangle in which diagonal AC is produced to E. If ∠ECD=146°, find ∠AOB.

Answer 1

Given : ABCD is a rectangle in which diagonal AC is produced to E and ∠ECD = 146°,  

∠ECD + ∠DCO = 180° (Linear pair of angles)

146° + ∠DCO = 180°

∠DCO = 180° - 146°  

∠DCO = 34° ………(1)

In ∆ COD,  

Since, diagonals of a rectangle are equal and bisect each other. Therefore,

OD = OC  

∠DCO = ∠ODC

[Angles opposite to equal sides are equal]

∠DCO +  ∠ODC + ∠DOC = 180°

[Sum of all the angles of a triangle is 180°]

34° + 34° + ∠DOC = 180°

68 ° + ∠DOC = 180°

∠DOC = 180° - 68°

∠DOC = 112°

∠AOB = ∠DOC = 112°

[Vertically opposite angles]

Hence, ∠AOB is 112°.

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Answer 2

Answer:-

$\:$

Given :

ABCD is a rectangle in which diagonal AC is produced to E and ∠ECD = 146°,  

∠ECD + ∠DCO = 180° (Linear pair of angles)

146° + ∠DCO = 180°

∠DCO = 180° - 146°  

∠DCO = 34° ………(1)

In ∆ COD,  

Since, diagonals of a rectangle are equal and bisect each other. Therefore,

OD = OC  

∠DCO = ∠ODC

[Angles opposite to equal sides are equal]

∠DCO +  ∠ODC + ∠DOC = 180°

[Sum of all the angles of a triangle is 180°]

34° + 34° + ∠DOC = 180°

68 ° + ∠DOC = 180°

∠DOC = 180° - 68°

∠DOC = 112°

∠AOB = ∠DOC = 112°

[Vertically opposite angles]

Hence, ∠AOB is 112°.

$\:$

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