Question
In the given figure, ABCD is a rectangle in which diagonal AC is produced to E. If $\angle\text{ECD}=146^\circ,$ find $\angle\text{AOB}.$
✓
Answer
ABCD is a rectangle With diagonal AC produced to point E. 
We have$\angle1+\angle\text{DCE}=180^\circ$ (Linear pair)
$\angle1+146^\circ=180^\circ$
$\angle1=34^\circ$
We know that the diagonals of a parallelogram bisect each other. Thus OC = OD Also, angles opposite to equal sides are equal. Therefore,$\angle\text{ODC}=34^\circ$
By angle sum property of a traingle$\angle\text{ODC}+\angle1+\text{COD}=180^\circ$
$34^\circ+34^\circ+\text{COD}=180^\circ$
$68^\circ+\angle\text{COD}=180^\circ$
$\angle\text{COD}=112^\circ$
Also, $\angle\text{COD}$ and $\angle\text{AOB}$ are vertically opposite angles. Therefore, $\angle\text{AOB}=112^\circ$ Hence, the required measure for $\angle\text{AOB}$ is 112º.
We have$\angle1+\angle\text{DCE}=180^\circ$ (Linear pair)$\angle1+146^\circ=180^\circ$
$\angle1=34^\circ$
We know that the diagonals of a parallelogram bisect each other. Thus OC = OD Also, angles opposite to equal sides are equal. Therefore,$\angle\text{ODC}=34^\circ$
By angle sum property of a traingle$\angle\text{ODC}+\angle1+\text{COD}=180^\circ$
$34^\circ+34^\circ+\text{COD}=180^\circ$
$68^\circ+\angle\text{COD}=180^\circ$
$\angle\text{COD}=112^\circ$
Also, $\angle\text{COD}$ and $\angle\text{AOB}$ are vertically opposite angles. Therefore, $\angle\text{AOB}=112^\circ$ Hence, the required measure for $\angle\text{AOB}$ is 112º.
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