# NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3

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Textbook | NCERT |

Board | CBSE |

Category | NCERT Solutions |

Class | Class 8 |

Subject | Maths |

Chapter | Chapter 3 |

Exercise | Class 8 Chapter 3 Understanding Quadrilaterals Exercise 3.3 |

Number of Questions Solved | 6 |

## NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.3

**NCERT TEXTBOOK EXERCISES**

**Question 1. Given a parallelogram ABCD. Complete each statement along with the definition with the definition or property used.**

**(i)** AD = â€¦â€¦â€¦**(ii)** âˆ DCB = â€¦â€¦â€¦â€¦â€¦.**(iii)** OC = â€¦â€¦â€¦â€¦â€¦â€¦.**(iv)** mâˆ DAB + mâˆ CDA = â€¦â€¦â€¦â€¦..

**Solution.**

**(i)** AD = BC

Opposite sides of a parallelogram are equal

**(ii)** âˆ DCB = âˆ DAB

Opposite angles of a parallelogram are equal

**(iii)** OC = OA

âˆµ Diagonals of a parallelogram bisect each other

**(iv)** mâˆ DAB + mâˆ CDA = 180Â°

âˆµ Adjacent angles of a parallelogram are supplementary.

**Question 2. Consider the following paralleloÂ¬grams. Find the values of the unknowns x, y, z.**

**Solution.**

**(i)** y = 100Â°

Opposite angles of a parallelogram are equal

x + 100Â° = 180Â°

Adjacent angles in a parallelogram are supplementary

â‡’ x = 180Â° â€“ 100Â°

â‡’ x = 80Â°

â‡’ z â€“ x = 80Â°

Opposite angles of a parallelogram are of equal measure

**(ii)** x + 50Â° = 180Â°

Adjacent angles in a parallelogram are supplementary

â‡’ x = 180Â° â€“ 50Â° = 130Â°

â‡’ y = x = 130Â°

The opposite angles of a parallelogram are of equal measure

180Â° â€“ z = 50Â°

Opposite angles of a parallelogram are of equal measure

â‡’ z = 180Â° â€“ 50Â° = 130Â°

**(iii)** x = 90Â°

Vertically opposite angles are equal

x + y + 30Â° = 180Â°

By angle sum property of a triangle

â‡’ 90Â° + y + 30Â° = 180Â°

â‡’ 120Â° + y = 180Â°

â‡’ y = 180Â° â€“ 120Â° = 60Â° z + 30Â° + 90Â° â€“ 180Â°

By angle sum property of a triangle

z = 60Â°

**(iv)** y = 80Â°

Opposite angles of a parallelogram are of equal measure

x + 80Â° = 180Â°

Adjacent angles in a parallelogram are supplementary

â‡’ x = 180Â° â€“ 80Â°

â‡’ x = 100Â°

â‡’ 180Â°-2+ 80Â°= 180Â°

Linear pair property and adjacent angles in a parallelogram are supplementary.

z = 80Â°

**(v)** y = 112Â°

Opposite angles of a parallelogram are equal

x + y + 40Â° = 180Â°

By angle sum property of a triangle

â‡’ x + 112Â° + 40Â° = 180Â°

â‡’ x + 152Â° = 180Â°

â‡’ x = 180Â°- 152Â°

â‡’ x = 28Â°

z = x = 28Â°.

Alternate interior angles

**Question 3. Can a quadrilateral ABCD be a parallelogram if**

**(i)** âˆ D + âˆ B = 180Â° ?**(ii)** AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm**(iii)** âˆ A = 70Â° and âˆ C = 65Â°?

**Solution.**

**(i)** Can be, but need not be**(ii)** No: in a parallelogram, opposite sides are equal; but here, AD â‰ BC.**(iii)** No: in a parallelogram, opposite angles are of equal measure; but here âˆ A â‰ âˆ C.

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

**Solution.**

A kite, for example

**Question 5. The measures of two adjacent angles of a parallelogram are in the ratio 3:2. Find the measure of each of the angles of the parallelogram.**

**Solution.**

Let the two adjacent angles be 3xÂ° and 2xÂ°.

Then,

3xÂ° + 2xÂ° = 180Â°

âˆ´ Sum of the two adjacent angles of a parallelogram is 180Â°

â‡’ 5xÂ° = 180Â°

â‡’ { x }^{ \circ }=\frac { { 180 }^{ \circ } }{ 5 }

â‡’ xÂ° = 36Â°

â‡’ 3xÂ° = 3 x 36Â° = 108Â°

and

2xÂ° = 2 x 36Â° = 72Â°.

Since, the opposite angles of a parallelogram are of equal measure, therefore the measures of the angles of the parallelogram are 72Â°, 108Â°, 72Â°, and 108Â°.

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

**Solution.**

Let the two adjacent angles of a parallelogram be xÂ° each.

Then,

xÂ° + xÂ° = 180Â°

âˆ´ Sum of the two adjacent angles of a parallelogram is 180Â°.

â‡’ 2xÂ° = 180Â°

â‡’ ${ x }^{ \circ }=\frac { { 180 }^{ \circ } }{ 2 }$

â‡’ xÂ° = 90Â°.

Since the opposite angles of a parallelogram are of equal measure, therefore the measure of each of the angles of the parallelogram is 90Â°, i.e., each angle of the parallelogram is a right angle.

**Question 7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.**

**Solution.**

x = 180Â° â€“ 70Â° = 110Â°

Linear pair property and the opposite angles of a parallelogram are of equal measure.

âˆµ HOPE is a || gm

âˆ´ HE || OP

and HP is a transversal

âˆ´ y = 40Â°

alternate interior angles

40Â° + z + x = 180Â°

The adjacent angles in a parallelogram are supplementary

â‡’ 40Â° + z + 110Â° = 180Â°

â‡’ z + 150Â° = 180Â°

â‡’ z = 180Â° â€“ 150Â°

â‡’ z = 30Â°.

**Question 8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)**

**(I)**

**(ii)**

**Solution.**

**(i)****For Figure GUNS**

Since the opposite sides of a parallelogram are of equal length, therefore,

â‡’ 3x = 18

â‡’ x=183=6

and, 3y â€“ 1 = 26

â‡’ 3y = 26 + 1

â‡’ 3y = 27

y=273=9

Hence, x = 6; y = 9.

**(ii)****For Figure RUNS**

Since the diagonals of a parallelogram bisect each other, therefore,

â‡’ x + y = 16 â€¦(1)

and, y + 7 = 20 â€¦(2)

From (2),

â‡’ y â€“ 20 â€“ 7 = 13

Putting y = 13 in (1), we get

â‡’ x + 13 = 16 â‡’ x = 16 â€“ 13 = 3.

Hence, x = 3; y = 13.

**Question 9. In the below figure both RISK and CLUE are parallelograms. Find the value of x.**

**Solution.**

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

**Solution.**

âˆµ âˆ KLM + âˆ NML = 80Â° + 100Â° = 180Â°

âˆ´ KL || NM

âˆµ The sum of consecutive interior angles is 180Â°

âˆ´ Figure KLMN is a trapezium.

Its two sides $\overline { KL }$ and $\overline { NM }$ are parallel.

**Question 11. Find mâˆ C in the figure, if $\overline { AB }$ || $\overline { DC }$.**

**Solution.**

âˆµ $\overline { AB }$ || $\overline { DC }$

âˆ´ mâˆ C + mâˆ B = 180Â°

âˆµ The sum of consecutive interior angles is 180Â°

mâˆ C+ 120Â° = 180Â°

â‡’ mâˆ C = 180Â° â€“ 120Â° = 60Â°.

**Question 12. Find the measure of âˆ P and âˆ S, if $\overline { SP }$ || $\overline { RQ }$ in the figure. (If you find mZ R, is there more than one method to find mâˆ P ?)**

**Solution.**

âˆµ $\overline { SP }$ || $\overline { RQ }$

âˆ´ mâˆ P+mâˆ Q = 180Â°

âˆµ The sum of consecutive interior angles is 180Â°

â‡’ mâˆ P + 130Â° = 180Â°

â‡’ mâˆ P = 180Â° â€“ 130Â°

â‡’ mâˆ P = 50Â°

Again, mâˆ R + mâˆ S = 180Â°

âˆµ The sum of consecutive interior angles is 180Â°

â‡’ 90Â° + m Z S = 180Â°

â‡’ mâˆ S = 180Â° â€“ 90Â° = 90Â°

Yes; there is one more method of finding mâˆ P if mâˆ R is given and that is by using the angle sum property of a quadrilateral.

We have,

mâˆ P + mâˆ Q + mâˆ R + mâˆ S = 360Â°

â‡’ mâˆ P + 130Â° + 90Â° + 90Â° = 360Â°

â‡’ mâˆ P + 310Â° = 360Â°

â‡’ mâˆ P = 360Â° â€“ 310Â° = 50Â°.

## NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.3 PDF

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