NCERT Solutions Class 8 For Maths Chapter 3 Exercise 3.3 Understanding Quadrilaterals - FREE PDF Download

Exercise 3.3

1. Give a parallelogram \[\text{ABCD}\]. Complete each statement along with the definition of property used

\[\left( \text{i} \right)\text{AD=}\]

Ans: As the given figure is a parallelogram and we know that that in parallelogram, the opposite sides are equal to each other, so \[\text{AD=BC}\]

\[\left( \text{ii} \right)\angle \text{DAB}\]

Ans: As the given figure is a parallelogram and we know that that in parallelogram, the opposite angles are equal to each other, so \[\angle \text{DCB=}\angle \text{DAB}\]

\[\left( \text{iii} \right)\text{OC=}\]

Ans: As the given figure is a parallelogram and we know that that in parallelogram, diagonals bisect each other, so \[\text{OC=OA}\]

\[\left( \text{iv} \right)\text{m}\angle \text{DAB+m}\angle \text{CDA=}\]

Ans: As the given figure is a parallelogram and we know that that in parallelogram, the adjacent angles are supplementary to each other, so

\[\text{m}\angle \text{DAB+m}\angle \text{CDA=18}{{\text{0}}^{\text{o}}}\]

2. Complete the following parallelograms

\[\left( \text{i} \right)\]

Ans: We know that in parallelograms adjacent angles are supplementary, we got $\text{x+10}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x=18}{{\text{0}}^{\text{o}}}\text{-10}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x=8}{{\text{0}}^{\text{o}}}  $

\[\left( \text{ii} \right)\]

Ans: We know that in parallelograms adjacent angles are supplementary, we got

 $\text{y+5}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{y=18}{{\text{0}}^{\text{o}}}\text{-5}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x=13}{{\text{0}}^{\text{o}}}  $

Now, we got

\[\text{y=x=13}{{\text{0}}^{\text{o}}}\], as the opposite angles are equal

And 

\[\text{z=x=13}{{\text{0}}^{\text{o}}}\], as they are corresponding angles

\[\left( \text{iii} \right)\]

Ans: From observing the figure, we got

\[\text{x=9}{{\text{0}}^{\text{o}}}\], as vertically opposite angle

Now, applying angle sum property of triangle, we got

$\text{x+y+3}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{9}{{\text{0}}^{\text{o}}}\text{+y+3}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

$\Rightarrow \text{y+12}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

$\Rightarrow \text{y=6}{{\text{0}}^{\text{o}}}  $

\[\text{z=y=6}{{\text{0}}^{\text{o}}}\], as they are alternate interior angles

\[\left( \text{iv} \right)\]

Ans: From observing the figure, we get

\[\text{z=8}{{\text{0}}^{\text{o}}}\], as they are corresponding angles

\[\text{y=8}{{\text{0}}^{\text{o}}}\], as the opposites angles are equal

As we know adjacent angles are supplementary, we get

 $\text{x+y=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x=18}{{\text{0}}^{\text{o}}}\text{-8}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x=10}{{\text{0}}^{\text{o}}}  $

\[\left( \text{v} \right)\]

From the figure, we got that

\[\text{y=11}{{\text{2}}^{\text{o}}}\], as opposite angles are equal

Now applying angle sum property of triangle, we got,

$\text{x+y+4}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x+11}{{\text{2}}^{\text{o}}}\text{+4}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

$\Rightarrow \text{x=18}{{\text{0}}^{\text{o}}}\text{-15}{{\text{2}}^{\text{o}}} $ 

$\Rightarrow \text{x=2}{{\text{8}}^{\text{o}}}  $

And

\[\text{z=x=2}{{\text{8}}^{\text{o}}}\], as they are alternate interior angles.

3. Can a quadrilateral \[\text{ABCD}\] be a parallelogram if,

\[\left( \text{i} \right)\angle \text{D+}\angle \text{B=18}{{\text{0}}^{\text{o}}}\]

Ans: According to the condition given, the quadrilateral may or may not be a parallelogram.

For a quadrilateral to be a parallelogram, there are some more conditions are present that is,

The sum of the adjacent angles should be \[\text{18}{{\text{0}}^{\text{o}}}\] and opposite angles should be of same length.

If these conditions are also fulfilled, then we can say that quadrilateral is also a parallelogram.

\[\left( \text{ii} \right)\text{AB=DC=8}\] cm, \[\text{AD=4}\] cm and \[\text{BC=4}\text{.4}\] cm

Ans: As we know that in parallelogram, the opposite sides are equal

So, according to the given condition, the opposite sides aren’t equal

Therefore, we can’t say that given quadrilateral is a parallelogram.

\[\left( \text{iii} \right)\angle \text{A=7}{{\text{0}}^{\text{o}}}\] and \[\angle \text{C=6}{{\text{5}}^{\text{o}}}\]

Ans: As we know that in parallelogram, the opposite angles are equal

So, according to the given condition, the opposite angles aren’t equal

Therefore, we can’t say that given quadrilateral is a parallelogram.

4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

Ans: 

As we can see, we have drawn a quadrilateral \[\text{ABCD}\] with \[\angle \text{B}\] and \[\angle \text{D}\] as opposite angles which are also equal to each other.

Now, we can also see that the quadrilateral drawn isn’t a parallelogram as in parallelogram, opposite sides are also equal to each other and we can also see that the \[\angle \text{A}\] and \[\angle \text{C}\] aren’t equal to each other.

5. The measures of two adjacent angles of a parallelogram are in the ratio \[\text{3:2}\]. Find the measure of each of the angles of the parallelogram.

Ans: Let us consider the two angles be \[\angle \text{A}\] and \[\angle \text{B}\]

According to the question, angles are in a ratio \[\text{3:2}\]

So, \[\angle \text{A=3x}\] and \[\angle \text{B=2x}\]

As we know that the sum of the adjacent angles in a parallelogram is \[\text{18}{{\text{0}}^{\text{o}}}\], we got

 $\angle \text{A+}\angle \text{B=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{3x+2x=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{5x=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{x=3}{{\text{6}}^{\text{o}}} $

So, we got,

\[\angle \text{A=}\angle \text{C}\Rightarrow \text{3x=10}{{\text{8}}^{\text{o}}}\] and \[\angle \text{B=}\angle \text{D}\Rightarrow \text{2x=7}{{\text{2}}^{\text{o}}}\], as they are opposite angles.

Therefore, the measure of each angles are,

 $\angle \text{A=10}{{\text{8}}^{\text{o}}} $ 

 $\angle \text{B=7}{{\text{2}}^{\text{o}}} $ 

 $\angle \text{C=10}{{\text{8}}^{\text{o}}} $ 

 $\angle \text{D=7}{{\text{2}}^{\text{o}}}  $

6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Ans: As we know that the sum of adjacent angles of parallelogram is \[\text{18}{{\text{0}}^{\text{o}}}\], we got,

\[\angle \text{A+}\angle \text{B=18}{{\text{0}}^{\text{o}}}\] 

According to the question, adjacent angles are of equal measures, we got

 $\angle \text{A+}\angle \text{A=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{2}\angle \text{A=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \angle \text{A=9}{{\text{0}}^{\text{o}}}  $

We know that the adjacent angles are of equal measures

Therefore, we got \[\angle \text{A=}\angle \text{B=9}{{\text{0}}^{\text{o}}}\], \[\angle \text{A=}\angle \text{C=9}{{\text{0}}^{\text{o}}}\](opposite angles) and \[\angle \text{D=}\angle \text{B=9}{{\text{0}}^{\text{o}}}\](opposite angles)

7. The adjacent figure \[\text{HOPE}\] is a parallelogram. Find the angle measure \[\text{x,y}\] and \[\text{z}\]. State the properties you use to find them.

Ans: Now, observing the figure,

From alternate interior angle, we get \[\angle \text{y=4}{{\text{0}}^{\text{o}}}\],

From corresponding angles, we get 

 $\text{7}{{\text{0}}^{\text{o}}}\text{=z+4}{{\text{0}}^{\text{o}}} $ 

 $\text{z=3}{{\text{0}}^{\text{o}}}  $

And from adjacent pair of angles we get, 

 $\text{x+}\left( \text{z+4}{{\text{0}}^{\text{o}}} \right)\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\text{x+7}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\text{x=18}{{\text{0}}^{\text{o}}}\text{-7}{{\text{0}}^{\text{o}}} $ 

 $\text{x=11}{{\text{0}}^{\text{o}}}  $

8. The following figures \[\text{GUNS}\] and \[\text{RUNS}\] are parallelograms find \[\text{x}\] and \[\text{y}\] (lengths are in cm)

(i).

Ans: As we know, the opposite sides in a parallelogram are equal to each other, from this we got,

\[\text{GU=SN}\] and \[\text{GS=UN}\], from the figure we got the length of the sides, so we get

 $\text{3y-1=26} $ 

 $\text{3y=27} $ 

 $\text{y=9}  $

 and 

 $\text{3x=18} $ 

 $\text{x=6}  $

Therefore, \[\text{x=6}\] cm and \[\text{y=9}\] cm.

(ii).

Ans: We know that, in parallelograms, diagonals bisect each other, so we got

 $\text{y+7=20} $ 

 $\text{y=13}$ 

 and 

$\text{x+y=16} $ 

$\text{x+13=16} $ 

$\text{x=3}  $

Therefore, \[\text{x=3}\] cm and \[\text{y=13}\] cm.

9.

In the given figure, both \[\text{RISK}\] and \[\text{CLUE}\] are parallelograms. Find the value of \[\text{x}\].

Ans: As the given figures is a parallelogram and we know that adjacent angles in parallelograms are supplementary and also opposite angles are equal.

Now, in parallelogram \[\text{RISK}\],

 $\angle \text{RKS+}\angle \text{ISK=18}{{\text{0}}^{\text{o}}} $ 

 $\text{12}{{\text{0}}^{\text{o}}}\text{+}\angle \text{ISK=18}{{\text{0}}^{\text{o}}} $ 

 $\angle \text{ISK=6}{{\text{0}}^{\text{o}}}  $

We got \[\angle \text{ISK=6}{{\text{0}}^{\text{o}}}\]

Now in parallelogram \[\text{CLUE}\],

\[\angle \text{ULC+}\angle \text{CEU=7}{{\text{0}}^{\text{o}}}\]

Applying angle sum property, we get

$\text{x+6}{{\text{0}}^{\text{o}}}\text{+7}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\text{x+13}{{\text{0}}^{\text{o}}}\text{=18}{{\text{0}}^{\text{o}}} $ 

 $\text{x=5}{{\text{0}}^{\text{o}}}  $

Therefore, the value of \[\text{x}\] is \[\text{5}{{\text{0}}^{\text{o}}}\].

10. Explain how this figure is a trapezium. Which of its two sides are parallel?

Ans: We know that for lines to be parallel, the transversal line should be parallel to given lines in a way in which the sum of the angles on the same side of transversal is \[\text{18}{{\text{0}}^{\text{o}}}\]

From the given figure we can see that,

\[\angle \text{NML+}\angle \text{MLK=18}{{\text{0}}^{\text{o}}}\]

Therefore, the lines \[\text{NM}\] and \[\text{LK}\] are parallel lines,

Now, we know that for a quadrilateral to be a trapezium, it should have a pair of parallel lines,

Here we have lines \[\text{NM}\] and \[\text{LK}\] are parallel

Therefore, the quadrilateral \[\text{KLMN}\] is a trapezium.

11. Find \[\text{m}\angle \text{C}\] in the following figure, if \[\overline{\text{AB}}\parallel \overline{\text{DC}}\]

Ans: It is given that \[\overline{\text{AB}}\parallel \overline{\text{DC}}\]

We know that the angles on the same side of transversal are supplementary, so we got

 $\angle \text{B+}\angle \text{C=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{12}{{\text{0}}^{\text{o}}}\text{+}\angle \text{C=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \angle \text{C=6}{{\text{0}}^{\text{o}}}  $

Therefore, \[\text{m}\angle \text{C=6}{{\text{0}}^{\text{o}}}\]

12. Find the measure of \[\angle \text{P}\] and \[\angle \text{S}\], if \[\overline{\text{SP}}\parallel \overline{\text{RQ}}\] in the following figure. (If you find \[\text{m}\angle \text{R}\], is there more than one method to find \[\text{m}\angle \text{P}\].)

Ans: Now, we know that sum of angles on the same side of transversal is \[\text{18}{{\text{0}}^{\text{o}}}\],

From the figure we can say that

 $\angle \text{P+}\angle \text{Q=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{13}{{\text{0}}^{\text{o}}}\text{+}\angle \text{P=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \angle \text{P=5}{{\text{0}}^{\text{o}}}  $

 and 

 $\angle \text{R+}\angle \text{S=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \text{9}{{\text{0}}^{\text{o}}}\text{+}\angle \text{S=18}{{\text{0}}^{\text{o}}} $ 

 $\Rightarrow \angle \text{S=9}{{\text{0}}^{\text{o}}}  $

Therefore, the measure of angle \[\text{m}\angle \text{P}\] and \[m\angle \text{S}\] is \[\text{5}{{\text{0}}^{\text{o}}}\] and \[\text{9}{{\text{0}}^{\text{o}}}\] respectively.

There’s one more way to find angle \[\text{m}\angle \text{P}\], as \[\text{m}\angle \text{R}\] and \[\text{m}\angle \text{Q}\] are given,

We can first find the angle \[m\angle \text{S}\] and we also know that the sum of the interior angles is \[{{360}^{\circ }}\], we can apply angle sum property to find the angle \[\text{m}\angle \text{P}\].

Conclusion

In order to help students understand the characteristics and classifications of various quadrilaterals, Vedantu's "NCERT Solutions for class 8 maths ch 3 ex 3.3 - Understanding Quadrilaterals" offers concise explanations and solutions. It emphasises the special qualities of essential geometric shapes including trapeziums, rhombuses, squares, parallelograms, and kites. Knowing the properties of each type of quadrilateral and the angle sum property are important concepts. Concentrate on working through issues involving the identification and computation of angles and sides.

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Chapter-Specific NCERT Solutions for Class 8 Maths

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