Find all the other angles inside the following rectangles. - Vidyadip

Question

Find all the other angles inside the following rectangles.

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Answer

$\begin{array}{l}\angle 1+\angle 9=90^{\circ} \ldots \ldots . . \text { (All corner angles of a rectangle are $90^{\circ}$} \\\angle 1+30^{\circ}=90^{\circ} \\\angle 1=90^{\circ}-30^{\circ} \\\angle 1=60^{\circ} \\\angle 1=\angle 5=60^{\circ} \ldots \ldots . . \text { (Alternate interior angles) } \\\angle 9=\angle 4=30^{\circ} \ldots \ldots . . \text { (Alternate interior angles) }\end{array}$
In $\triangle AOB , OA = OB$, then the angles opposite them are equal
$\therefore \angle 9=\angle 7=30^{\circ}$
$\angle 7=\angle 3=30^{\circ}$. $\qquad$ (Alternate interior angles)
In $\triangle A O D, O A=O D$, then the angles opposite them are equal
$\therefore \angle 2=\angle 1=60^{\circ}$
$\angle 2=\angle 6=60^{\circ}$. $\qquad$ (Alternate interior angles)
$\begin{array}{l}\text { In } \triangle A O B, \\\angle 9+\angle 7+\angle A O B=180^{\circ} \ldots \ldots \ldots . \text { (Sum of angles of a triangle) } \\30^{\circ}+30^{\circ}+\angle A O B=180^{\circ} \\60^{\circ}+\angle A O B=180^{\circ} \\\angle A O B=180^{\circ}-60^{\circ} \\\angle A O B=120^{\circ} \\\\\angle A O B=\angle C O D=120^{\circ} \ldots \ldots \ldots . . \text { (Vertically opposite angles) } \\\\\angle A O B+\angle A O D=180^{\circ} \ldots \ldots \ldots . \text { (Linear pair) } \\120^{\circ}+\angle A O D=180^{\circ} \\\angle A O D=180^{\circ}-120^{\circ} \\\angle A O D=60^{\circ} \\\angle A O D=\angle B O C=60^{\circ} \ldots \ldots \ldots . . \text { (Vertically opposite angles) }\end{array}$
Thus, $\angle 1=\angle 5=\angle 2=\angle 6=\angle AOD =\angle BOC =60^{\circ}$.
$\begin{array}{l}\angle A O B=\angle C O D=120^{\circ} \\\angle 9=\angle 4=\angle 7=\angle 3=30^{\circ}\end{array}$

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