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

**(i) AD =**

**(ii) âˆ DCB =**

**(iii) OC =**

**(iv) âˆ DAB + âˆ CDA =**

**Solution: **

(i)AD = BC. Because, diagonals bisect each other in a parallelogram.

(ii)âˆ DCB = âˆ BAD. Because, alternate interior angles are equal.

(iii)OC = OA. Because, diagonals bisect each other in a parallelogram.

(iv)âˆ DAB+ âˆ CDA = 180Â°. Because sum of adjacent angles in a parallelogram is 180Â°.

**Question 2. The following figures are parallelograms. Find the degree values of the unknowns x, y, z.**

**Solution: **

(i)From figure we conclude that,âˆ ABC = âˆ y = 100

^{o}(Opposite angles are equal in a parallelogram)âˆ x + âˆ y = 180

^{o}(sum of adjacent angles is = 180Â° in a parallelogram)âˆ x + 100Â° = 180Â°

âˆ x = 180Â° â€“ 100Â° = 80Â°

Hence, âˆ x = 80Â° âˆ y = 100Â° âˆ z = 80Â° (opposite angles are equal in a parallelogram)

(ii)From figure we conclude that,âˆ RSP + âˆ y = 180Â° (sum of adjacent angles is = 180Â° in a parallelogram)

âˆ y + 50Â° = 180Â°

âˆ y = 180Â° â€“ 50Â° = 130Â°

Hence, âˆ x = âˆ y = 130Â° (opposite angles are equal in a parallelogram)

From figure, we conclude that,

âˆ RSP = âˆ RQP = 50Â° (opposite angles are equal in a parallelogram)

âˆ RQP + âˆ z = 180Â° (linear pair)

50Â° + âˆ z = 180Â°

âˆ z = 180Â° â€“ 50Â° = 130Â°

Hence, âˆ x = 130Â°, âˆ y = 130Â° and âˆ z = 130Â°.

(iii)As we know that,In Î”PMN âˆ NPM + âˆ NMP + âˆ MNP = 180Â° (Sum of all the angles of a triangle is 180Â°)

30Â° + 90Â° + âˆ z = 180Â°

âˆ z = 180Â°-120Â° = 60Â°

From figure, we conclude that,

âˆ y = âˆ z = 60Â° (opposite angles are equal in a parallelogram)

âˆ z = 180Â°-120Â° (sum of the adjacent angles is equal to 180Â° in a parallelogram)

âˆ z = 60Â°

âˆ z + âˆ LMN = 180Â° (sum of the adjacent angles is equal to 180Â° in a parallelogram)

60Â° + 90Â°+ âˆ x = 180Â°

âˆ x = 180Â°-150Â° = 30Â°

Hence, âˆ x = 30Â° âˆ y = 60Â° âˆ z = 60Â°

(iv)From figure we conclude that,âˆ x = 90Â° [vertically opposite angles are equal]

In Î”DOC, âˆ x + âˆ y + 30Â° = 180Â° (Sum of all the angles of a triangle is 180Â°)

90Â° + 30Â° + âˆ y = 180Â°

âˆ y = 180Â°-120Â°

âˆ y = 60Â°

âˆ y = âˆ z = 60Â° (alternate interior angles are equal)

Hence, âˆ x = 90Â° âˆ y = 60Â° âˆ z = 60Â°

(v)From figure we conclude that,âˆ x + âˆ POR = 180Â° (sum of the adjacent angles is equal to 180Â° in a parallelogram)

âˆ x + 80Â° = 180Â°

âˆ x = 180Â°-80Â° = 100Â°

âˆ y = 80Â° (opposite angles are equal in a parallelogram)

âˆ SRQ =âˆ x = 100Â°

âˆ SRQ + âˆ z = 180Â° (Linear pair)

100Â° + âˆ z = 180Â°

âˆ z = 180Â°-100Â° = 80Â°

Hence, âˆ x = 100Â°, âˆ y = 80Â° and âˆ z = 80Â°.

(vi)From figure we conclude that,âˆ y = 112Â° (In a parallelogram opposite angles are equal)

âˆ y + âˆ VUT = 180Â° (In a parallelogram sum of the adjacent angles is equal to 180Â°)

âˆ z + 40Â° + 112Â° = 180Â°

âˆ z = 180Â°-152Â° = 28Â°

âˆ z =âˆ x = 28Â° (alternate interior angles are equal)

Hence, âˆ x = 28Â°, âˆ y = 112Â°, âˆ z = 28Â°.

**Question 3. Can the following figures be parallelograms? Justify your answer.**

**Solution: **

(i)No, as we know that opposite angles are equal in a parallelogram.

(ii)Yes, as we know that opposite sides are equal and parallel in a parallelogram.

(iii)No, as we know that the diagonals bisect each other in a parallelogram.

**Question 4. In the adjacent figure HOPE is a parallelogram. Find the angle measures x, y**,** and z. State the geometrical truths you use to find them.**

**Solution: **

As we know that,

âˆ POH + 70Â° = 180Â° (Linear pair)

âˆ POH = 180Â°-70Â° = 110Â°

âˆ POH = âˆ x = 110Â° (opposite angles are equal in a parallelogram)

âˆ x + âˆ z + 40Â° = 180Â° (sum of the adjacent angles is equal to 180Â° in a parallelogram)

110Â° + âˆ z + 40Â° = 180Â°

âˆ z = 180Â° â€“ 150Â° = 30Â°

âˆ z +âˆ y = 70Â°

âˆ y + 30Â° = 70Â°

âˆ y = 70Â°- 30Â° = 40Â°

**Question 5. In the following figures**,** GUNS and RUNS are parallelograms. Find x and y.**

**Solution: **

From figure, we conclude that,

(i)3y â€“ 1 = 26 (opposite sides are of equal length in a parallelogram)3y = 26 + 1

y = 27/3 = 9

3x = 18 (opposite sides are of equal length in a parallelogram)

x = 18/3= 6

Hence, x = 6 and y = 9

(ii)y â€“ 7 = 20 (diagonals bisect each other in a parallelogram)y = 20 + 7 = 27

x â€“ y = 16 (diagonals bisect each other in a parallelogram)

x -27 = 16

x = 16 + 27 = 43

Hence, x = 43 and y = 27

**Question 6. In the following figure RISK and CLUE are parallelograms. Find the measure of x.**

**Solution: **

From figure, we conclude that,

In parallelogram RISK

âˆ RKS + âˆ KSI = 180Â° (sum of the adjacent angles is equal to 180Â° in a parallelogram)

120Â° + âˆ KSI = 180Â°

âˆ KSI = 180Â° â€“ 120Â° = 60Â°

In parallelogram CLUE,

âˆ CEU = âˆ CLU = 70Â° (opposite angles are equal in a parallelogram)

In Î”EOS,

70Â° + âˆ x + 60Â° = 180Â° (Sum of angles of a triangles is 180Â°)

âˆ x = 180Â° â€“ 130Â° = 50Â°

Hence, x = 50Â°

**Question 7. Two opposite angles of a parallelogram are (3x â€“ 2)Â° and (50 â€“ x)Â°. Find the measure of each angle of the parallelogram.**

**Solution: **

As we know that the opposite angles of a parallelogram are equal.

So, (3x â€“ 2)Â° = (50 â€“ x)Â°

3xo â€“ 2Â° = 50Â° â€“ xÂ°

3xÂ° + xo = 50Â° + 2Â°

4xÂ° = 52Â°

xo = 52Â°/4 = 13Â°

The opposite angles are,

(3x â€“ 2)Â° = 3Ã—13 â€“ 2 = 37Â°

(50 â€“ x)Â° = 50 â€“ 13 = 37Â°

As we know that Sum of adjacent angles = 180Â°

Other two angles are 180Â° â€“ 37Â° = 143Â°

Hence, Measure of each angle is 37Â°, 143Â°, 37Â°, 143Â°.

**Question 8. If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.**

**Solution:**

Let us assume that one of the adjacent angle as xÂ°,

then other adjacent angle is = 2xÂ°/3

As we know that sum of adjacent angles = 180Â°

Therefore,

xÂ° + 2xÂ°/3 = 180Â°

(3xÂ° + 2xÂ°)/3 = 180Â°

5xÂ°/3 = 180Â°

xÂ° = 180Â°Ã—3/5 = 108Â°

Other angle is = 180Â° â€“ 108Â° = 72Â°

Hence, Angles of a parallelogram are 72Â°, 72Â°, 108Â°, 108Â°.

**Question 9. The measure of one angle of a parallelogram is 70Â°. What are the measures of the remaining angles?**

**Solution:**

Let us assume that the one of the adjacent angle as xÂ°

Other adjacent angle = 70Â°

As we know that sum of adjacent angles = 180Â°

Therefore,

xÂ° + 70Â° = 180Â°

xÂ° = 180Â° â€“ 70Â° = 110Â°

Hence, Measures of the remaining angles are 70Â°, 70Â°, 110Â° and 110Â°

**Question 10. Two adjacent angles of a parallelogram are as 1: 2. Find the measures of all the angles of the parallelogram.**

**Solution: **

Let us assume that one of the adjacent angle as xÂ°,

then other adjacent angle = 2xÂ°

As we know that sum of adjacent angles = 180Â°

Therefore,

xÂ° + 2xÂ° = 180Â°

3xÂ° = 180Â°

xÂ° = 180Â°/3 = 60Â°

So other angle is 2x = 2Ã—60 = 120Â°

Hence, Measures of the remaining angles are 60Â°, 60Â°, 120Â° and 120Â°

**Question 11. In a parallelogram ABCD, âˆ D= 135Â°, determine the measure of âˆ A and âˆ B.**

**Solution: **

Given that,

one of the adjacent angle âˆ D = 135Â°

Let us assume that other adjacent angle âˆ A be = xÂ°

As we know that sum of adjacent angles = 180Â°

xÂ° + 135Â° = 180Â°

xÂ° = 180Â° â€“ 135Â° = 45Â°

âˆ A = xÂ° = 45Â°

As we know that the opposite angles are equal in a parallelogram.

Therefore, âˆ A = âˆ C = 45Â°

and âˆ D = âˆ B = 135Â°.

**Question 12. ABCD is a parallelogram in which âˆ A = 70Â°. Compute âˆ B, âˆ C**,** and âˆ D.**

**Solution:**

Given that,

one of the adjacent angle âˆ A = 70Â°

and other adjacent angle âˆ B is = xÂ°

As we know that sum of adjacent angles = 180Â°

xÂ° + 70Â° = 180Â°

xÂ° = 180Â° â€“ 70Â° = 110Â°

âˆ B = xÂ° = 110Â°

As we know that the opposite angles are equal in a parallelogram.

Therefore, âˆ A = âˆ C = 70Â°

and âˆ D = âˆ B = 110Â°

**Question 13. The sum of two opposite angles of a parallelogram is 130Â°. Find all the angles of the parallelogram.**

**Solution: **

From figure, we conclude that, ABCD is a parallelogram

âˆ A + âˆ C = 130Â°

Here âˆ A and âˆ C are opposite angles

Therefore âˆ C = 130/2 = 65Â°

As we know that sum of adjacent angles is 180

âˆ B + âˆ D = 180

65 + âˆ D = 180

âˆ D = 180 â€“ 65 = 115

âˆ D = âˆ B = 115 (Opposite angles)

Hence, âˆ A = 65Â°, âˆ B = 115Â°, âˆ C = 65Â° and âˆ D = 115Â°.

**Question 14. All the angles of a quadrilateral are equal to each other. Find the measure of each. Is the quadrilateral a parallelogram? What special type of parallelogram is it?**

**Solution: **

Let us assume that each angle of a parallelogram as xo

As we know that sum of angles = 360Â°

xÂ° + xÂ° + xÂ° + xÂ° = 360Â°

4 xÂ° = 360Â°

xÂ° = 360Â°/4 = 90Â°

Hence, each angle is 90Â°

Yes, this quadrilateral is a parallelogram.

Since each angle of a parallelogram is equal to 90Â°, so it is a rectangle.

**Question 15. Two adjacent sides of a parallelogram are 4 cm and 3 cm respectively. Find its perimeter.**

**Solution: **

As we know that opposite sides of a parallelogram are parallel and equal.

therefore, Perimeter = Sum of all sides (there are 4 sides)

Perimeter = 4 + 3 + 4 + 3 = 14 cm

Hence, Perimeter is 14 cm.