Question
If ABCD is a rectangle with $\angle\text{BAC}=32^\circ,$ find the measure of $\angle\text{DBC}.$
✓
Answer
Figure is given as : 
Suppose the diagonals AC and BD intersect at O. Since, diagonals of a rectangle are equal and they bisect each other. Therefore, in $\triangle\text{OAB},$ we have OA = OB Angles opposite to equal sides are equal. Therefore,$\angle\text{OAB}=\angle\text{OBA}$
$\angle\text{BAC}=\angle\text{DBA}$
But, $\angle\text{BAC}=32^\circ$$\angle\text{DBA}=32^\circ$
Now,$\angle\text{ABC}=90^\circ$
$\angle\text{DBA}+\angle\text{DBC}=90^\circ$
$32^\circ+\angle\text{DBC}=90^\circ$
$\angle\text{DBC}=58^\circ$
Hence, the measure of $\angle\text{DBC}$ is 58º.
Suppose the diagonals AC and BD intersect at O. Since, diagonals of a rectangle are equal and they bisect each other. Therefore, in $\triangle\text{OAB},$ we have OA = OB Angles opposite to equal sides are equal. Therefore,$\angle\text{OAB}=\angle\text{OBA}$$\angle\text{BAC}=\angle\text{DBA}$
But, $\angle\text{BAC}=32^\circ$$\angle\text{DBA}=32^\circ$
Now,$\angle\text{ABC}=90^\circ$
$\angle\text{DBA}+\angle\text{DBC}=90^\circ$
$32^\circ+\angle\text{DBC}=90^\circ$
$\angle\text{DBC}=58^\circ$
Hence, the measure of $\angle\text{DBC}$ is 58º.
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