Question 14 Marks
Read the following text carefully and answer the questions that follow:
Harish makes a poster in the shape of a parallelogram on the topic $\text{SAVE ELECTRICITY}$ for an inter$-$school competition as shown in the follow figure.
$i.$ If $\angle A =(4 x +3)^{\circ}$ and $\angle D =(5 x -3)^{\circ}$, then find the measure of $\angle B$.
$ii.$ If $\angle B =(2 y )^{\circ}$ and $\angle D =(3 y -6)^{\circ}$, then find the value of $y.$
$iii.$ If $\angle A=(2 x-3)^{\circ}$ and $\angle C=(4 y+2)^{\circ}$, then find how $x$ and $y$ relate.
OR
If $A B=(2 y-3)$ and $C D=5 \ cm$ then what is the value of $y$ ?
Harish makes a poster in the shape of a parallelogram on the topic $\text{SAVE ELECTRICITY}$ for an inter$-$school competition as shown in the follow figure.
$i.$ If $\angle A =(4 x +3)^{\circ}$ and $\angle D =(5 x -3)^{\circ}$, then find the measure of $\angle B$.
$ii.$ If $\angle B =(2 y )^{\circ}$ and $\angle D =(3 y -6)^{\circ}$, then find the value of $y.$
$iii.$ If $\angle A=(2 x-3)^{\circ}$ and $\angle C=(4 y+2)^{\circ}$, then find how $x$ and $y$ relate.
OR
If $A B=(2 y-3)$ and $C D=5 \ cm$ then what is the value of $y$ ?
Answer
View full question & answer→$i.$ Since, $\text{ABCD}$ is a parallelogram.
$\angle A +\angle D =180^{\circ}($ adjacent angles of a quadrilateral are equal $)$
$(4 x+3)^{\circ}+(5 x+3)^{\circ}=180^{\circ}$
$9 x=180^{\circ}$
$x=20$
$\angle D=(5 x-3)^{\circ}=97^{\circ}$
$\angle D =\angle B ($opposite angles of a parallelogram are equal$)$
Thus, $\angle B =97^{\circ}$
$ii. \angle B =\angle D ($opposite angles of a parallelogram are equal$)$
$\Rightarrow 2 y=3 y-6$
$\Rightarrow 2 y-3 y=-6$
$\Rightarrow-y=-6$
$\Rightarrow y=6$
$iii. \angle A =\angle C ($opposite angles of a parallelogram are equal$)$
$\Rightarrow 2 x-3=4 y+2$
$\Rightarrow 2 x=4 y+5$
$\Rightarrow x=2 y+\frac{5}{2}$
OR
$\text{AB=CD}$
$\Rightarrow 2 y-3=5$
$\Rightarrow 2 y=8$
$\Rightarrow y=4$
$\angle A +\angle D =180^{\circ}($ adjacent angles of a quadrilateral are equal $)$
$(4 x+3)^{\circ}+(5 x+3)^{\circ}=180^{\circ}$
$9 x=180^{\circ}$
$x=20$
$\angle D=(5 x-3)^{\circ}=97^{\circ}$
$\angle D =\angle B ($opposite angles of a parallelogram are equal$)$
Thus, $\angle B =97^{\circ}$
$ii. \angle B =\angle D ($opposite angles of a parallelogram are equal$)$
$\Rightarrow 2 y=3 y-6$
$\Rightarrow 2 y-3 y=-6$
$\Rightarrow-y=-6$
$\Rightarrow y=6$
$iii. \angle A =\angle C ($opposite angles of a parallelogram are equal$)$
$\Rightarrow 2 x-3=4 y+2$
$\Rightarrow 2 x=4 y+5$
$\Rightarrow x=2 y+\frac{5}{2}$
OR
$\text{AB=CD}$
$\Rightarrow 2 y-3=5$
$\Rightarrow 2 y=8$
$\Rightarrow y=4$
