Examples (Revised) - Chapter 3 - Understanding Quadrilaterals - Ncert Solutions class 8 - Maths

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NCERT Solutions for Class 8 Maths Chapter 3 - Understanding Quadrilaterals

Example 1:

Find measure $x$ in Fig 3.3.
Solution:
$
\begin{aligned}
x+90^{\circ}+50^{\circ}+110^{\circ} & =360^{\circ} \\
x+250^{\circ} & =360^{\circ} \\
x & =110^{\circ}
\end{aligned}
$

Example 2:

Find the number of sides of a regular polygon whose each exterior angle has a measure of $45^{\circ}$.
Solution:

Total measure of all exterior angles $=360^{\circ}$
Measure of each exterior angle $=45^{\circ}$
Therefore, the number of exterior angles $=\frac{360}{45}=8$
The polygon has 8 sides.

Example 3:

Find the perimeter of the parallelogram PQRS (Fig 3.16).
Solution:

In a parallelogram, the opposite sides have same length.
Therefore, $\mathrm{PQ}=\mathrm{SR}=12 \mathrm{~cm}$ and $\mathrm{QR}=\mathrm{PS}=7 \mathrm{~cm}$
So, Perimeter $=\mathrm{PQ}+\mathrm{QR}+\mathrm{RS}+\mathrm{SP}$

$
=12 \mathrm{~cm}+7 \mathrm{~cm}+12 \mathrm{~cm}+7 \mathrm{~cm}=38 \mathrm{~cm}
$

Example 4:

In Fig 3.20, BEST is a parallelogram. Find the values $x, y$ and $z$.
Solution:

$\mathrm{S}$ is opposite to $\mathrm{B}$.
So,
$
\begin{aligned}
& x=100^{\circ} \text { (opposite angles property) } \\
& y=100^{\circ} \text { (measure of angle corresponding to } \angle x \text { ) } \\
& z=80^{\circ} \quad \text { (since } \angle y, \angle z \text { is a linear pair) }
\end{aligned}
$

We now turn our attention to adjacent angles of a parallelogram.
In parallelogram ABCD, (Fig 3.21).
$\angle \mathrm{A}$ and $\angle \mathrm{D}$ are supplementary since

$\overline{\mathrm{DC}} \| \overline{\mathrm{AB}}$ and with transversal $\overline{\mathrm{DA}}$, these two angles are interior opposite.
$\angle \mathrm{A}$ and $\angle \mathrm{B}$ are also supplementary. Can you say 'why'?

$\overline{\mathrm{AD}} \| \overline{\mathrm{BC}}$ and $\overline{\mathrm{BA}}$ is a transversal, making $\angle \mathrm{A}$ and $\angle \mathrm{B}$ interior opposite.
Identify two more pairs of supplementary angles from the figure.
Property: The adjacent angles in a parallelogram are supplementary.

Example 5:

In a parallelogram RING, (Fig 3.22) if $m \angle R=70^{\circ}$, find all the other angles.
Solution:

Given $m \angle \mathrm{R}=70^{\circ}$
Then
$
m \angle \mathrm{N}=70^{\circ}
$
because $\angle \mathrm{R}$ and $\angle \mathrm{N}$ are opposite angles of a parallelogram.
Since $\angle \mathrm{R}$ and $\angle \mathrm{I}$ are supplementary,
$
m \angle \mathrm{I}=180^{\circ}-70^{\circ}=110^{\circ}
$

Also,
$m \angle \mathrm{G}=110^{\circ}$ since $\angle \mathrm{G}$ is opposite to $\angle \mathrm{I}$

Thus,
$
m \angle \mathrm{R}=m \angle \mathrm{N}=70^{\circ} \text { and } m \angle \mathrm{I}=m \angle \mathrm{G}=110^{\circ}
$

Example 6:

In Fig 3.25 HELP is a parallelogram. (Lengths are in cms). Given that $\mathrm{OE}=4$ and $\mathrm{HL}$ is 5 more than $\mathrm{PE}$ ? Find $\mathrm{OH}$.
Solution :

If $\mathrm{OE}=4$ then $\mathrm{OP}$ also is 4 (Why?)
So $\quad \mathrm{PE}=8$,
Therefore $\quad \mathrm{HL}=8+5=13$

Hence
$
\mathrm{OH}=\frac{1}{2} \times 13=6.5(\mathrm{cms})
$

Example 7:
RICE is a rhombus (Fig 3.30). Find $x, y, z$. Justify your findings.
Solution:
$
\begin{aligned}
& x=\mathrm{OE} \quad y=\mathrm{OR} \quad z=\text { side of the rhombus } \\
& =\mathrm{OI} \text { (diagonals bisect) }=\mathrm{OC} \text { (diagonals bisect) }=13 \text { (all sides are equal } \\
& =5 \quad=12 \\
&
\end{aligned}
$

Example 8:

RENT is a rectangle (Fig 3.35). Its diagonals meet at O. Find $x$, if $\mathrm{OR}=2 x+4$ and $\mathrm{OT}=3 x+1$.

Solution:

$\overline{\mathrm{OT}}$ is half of the diagonal $\overline{\mathrm{TE}}$,
$\overline{\mathrm{OR}}$ is half of the diagonal $\overline{\mathrm{RN}}$.
Diagonals are equal here. (Why?)
So, their halves are also equal.

Therefore
$
3 x+1=2 x+4
$
or

x=3