468 + Interview Questions in QUADRILATERALS IN GENERAL AWARENESS Page 4 InterviewSolution

151.	If an angle of a parallelogram is four fifths of its adjacent angle, find the angles of the parallelogram.
Answer» Consider ABCD as a parallelogram If ∠ A = x^o We know that ∠ B is adjacent to A which can be written as 4/5 x^o Opposite angles are equal in a parallelogram So we get ∠ A = ∠ C = x^o and ∠ B = ∠ D = 4/5 x^o We know that the sum of all the angles of a parallelogram is 360^o It can be written as ∠ A + ∠ B + ∠ C + ∠ D = 360^o By substituting the values in the above equation x + (4/5) x + x + (4/5) x = 360^o By addition we get 2x + (8/5) x = 360^o By taking the LCM as 5 (18/5) x = 360^o By cross multiplication x = (360 × 5)/18 On further calculation x = 100^o By substituting the value of x So we get ∠ A = ∠ C = x = 100^o ∠ B = ∠ D = 4/5 x^o = (4/5) (100^o) = 80^o Therefore, ∠ A = ∠ C = x = 100^o and ∠ B = ∠ D = 80^o.

151.

If an angle of a parallelogram is four fifths of its adjacent angle, find the angles of the parallelogram.

Answer»

Consider ABCD as a parallelogram

If ∠ A = x^o

We know that ∠ B is adjacent to A which can be written as 4/5 x^o

Opposite angles are equal in a parallelogram

So we get

∠ A = ∠ C = x^o and ∠ B = ∠ D = 4/5 x^o

We know that the sum of all the angles of a parallelogram is 360^o

It can be written as

∠ A + ∠ B + ∠ C + ∠ D = 360^o

By substituting the values in the above equation

x + (4/5) x + x + (4/5) x = 360^o

By addition we get

2x + (8/5) x = 360^o

By taking the LCM as 5

(18/5) x = 360^o

By cross multiplication

x = (360 × 5)/18

On further calculation

x = 100^o

By substituting the value of x

So we get

∠ A = ∠ C = x = 100^o

∠ B = ∠ D = 4/5 x^o = (4/5) (100^o) = 80^o

Therefore, ∠ A = ∠ C = x = 100^o and ∠ B = ∠ D = 80^o.

Discussion

152.	Find the measure of each angle of a parallelogram, if one of its angles is 30° less than twice the smallest angle.
Answer» Consider ABCD as a parallelogram Let us take ∠ A as the smallest angle So we get ∠ B = 2 ∠ A – 30^o We know that the opposite angles are equal in a parallelogram ∠ A = ∠ C and ∠ B = ∠ D = 2 ∠ A – 30^o We know that the sum of all the angles of a parallelogram is 360^o It can be written as ∠ A + ∠ B + ∠ C + ∠ D = 360^o By substituting the values in the above equation ∠ A + (2 ∠ A – 30^o) + ∠ A + (2 ∠ A – 30^o) = 360^o On further calculation ∠ A + 2 ∠ A – 30^o + ∠ A + 2 ∠ A – 30^o = 360^o So we get 6 ∠ A – 60^o = 360^o By addition 6 ∠ A = 360^o + 60^o 6 ∠ A = 420^o By division ∠ A = 70^o By substituting the value of ∠ A ∠ A = ∠ C = 70^o ∠ B = ∠ D = 2 ∠ A – 30^o = 2 (70^o) – 30^o ∠ B = ∠ D = 110^o Therefore, ∠ A = ∠ C = 70^o and ∠ B = ∠ D = 110^o.

152.

Find the measure of each angle of a parallelogram, if one of its angles is 30° less than twice the smallest angle.

Answer»

Consider ABCD as a parallelogram

Let us take ∠ A as the smallest angle

So we get

∠ B = 2 ∠ A – 30^o

We know that the opposite angles are equal in a parallelogram

∠ A = ∠ C and ∠ B = ∠ D = 2 ∠ A – 30^o

We know that the sum of all the angles of a parallelogram is 360^o

It can be written as

∠ A + ∠ B + ∠ C + ∠ D = 360^o

By substituting the values in the above equation

∠ A + (2 ∠ A – 30^o) + ∠ A + (2 ∠ A – 30^o) = 360^o

On further calculation

∠ A + 2 ∠ A – 30^o + ∠ A + 2 ∠ A – 30^o = 360^o

So we get

6 ∠ A – 60^o = 360^o

By addition

6 ∠ A = 360^o + 60^o

6 ∠ A = 420^o

By division

∠ A = 70^o

By substituting the value of ∠ A

∠ A = ∠ C = 70^o

∠ B = ∠ D = 2 ∠ A – 30^o = 2 (70^o) – 30^o

∠ B = ∠ D = 110^o

Therefore, ∠ A = ∠ C = 70^o and ∠ B = ∠ D = 110^o.

Discussion

153.	Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
Answer» Given: One angle of a parallelogram is 24° less than twice the smallest angle. Let x be the smallest angle, then x + 2x – 24° = 180° 3x – 24° = 180° 3x = 108° + 24° 3x = 204° x = 204°/3 = 68° So, x = 68° Another angle = 2x – 24° = 2(68°) – 24° = 112° Hence, four angles are 68°, 112°, 68°, 112°.

153.

Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.

Answer»

Given: One angle of a parallelogram is 24° less than twice the smallest angle.

Let x be the smallest angle, then

x + 2x – 24° = 180°

3x – 24° = 180°

3x = 108° + 24°

3x = 204°

x = 204°/3 = 68°

So, x = 68°

Another angle = 2x – 24° = 2(68°) – 24° = 112°

Hence, four angles are 68°, 112°, 68°, 112°.

Discussion

154.	If an angle of a parallelogram is two-third ofits adjacent angle, find the angles of the parallelogram.A. `108^(@)`B. `54^(@)`C. `72^(@)`D. `81^(@)`
Answer» Correct Answer - C

Discussion

155.	Find the measure of all the angles of a parallelogram, if one angle is 24° less than the twice of the smallest angle.
Answer» Then its consecutive angle = 180 – x° By problem (180 – x)° = (2x- 24)° (∵ opp. angles are equal) 180 + 24 = 2x + x 3x = 204 x = 204/3 = 68° ∴ The angles are 68°; (2 x 68 – 24)°; 68°; (2 x 68 – 24)° = 68°, 112°, 68°, 112°

155.

Find the measure of all the angles of a parallelogram, if one angle is 24° less than the twice of the smallest angle.

Answer»

Then its consecutive angle = 180 – x°

By problem (180 – x)° = (2x- 24)°

(∵ opp. angles are equal)

180 + 24 = 2x + x

3x = 204

x = 204/3 = 68°

∴ The angles are 68°; (2 x 68 – 24)°; 68°; (2 x 68 – 24)°

= 68°, 112°, 68°, 112°

Discussion

156.	Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
Answer» Given, Let the smallest angle = x^∘ Than the other angle = (2x – 24)° So, x^∘+x^∘+2x – 24 + 2x – 24 = 360^∘ 6x = 360 + 48 = 408 x = \(\frac{408^∘}{6}\) = 68^∘ Other angles = 2x – 24 = 2×68 – 24 = 136-24 = 112^∘ Angles are = 68^∘, 68^∘, 112^∘, 112^∘

156.

Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.

Answer»

Given,

Let the smallest angle = x^∘

Than the other angle = (2x – 24)°

So,

x^∘+x^∘+2x – 24 + 2x – 24 = 360^∘

6x = 360 + 48 = 408

x = \(\frac{408^∘}{6}\) = 68^∘

Other angles = 2x – 24 = 2×68 – 24 = 136-24 = 112^∘

Angles are = 68^∘, 68^∘, 112^∘, 112^∘

Discussion

157.	Which of the following statements are true for a rectangle?(i) It has two pairs of equal sides.(ii) It has all its sides of equal length.(iii) Its diagonals are equal.(iv) Its diagonals bisect each other.(v) Its diagonals are perpendicular.(vi) Its diagonals are perpendicular and bisect each other.(vii) Its diagonals are equal and bisect each other.(viii) Its diagonals are equal and perpendicular, and bisect each other.(ix) All rectangles are squares.(x) All rhombuses are parallelograms.(xi) All squares are rhombuses and also rectangles.(xii) All squares are not parallelograms.
Answer» (i) True, In a rectangle two pairs of sides are equal. (ii) False, In a rectangle two pairs of sides are equal. (iii) True, In a rectangle diagonals are of equal length. (iv) True, In a rectangle diagonals bisect each other. (v) False, Diagonals of a rectangle need not be perpendicular. (vi) False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each other. (vii) True, Diagonals are of equal length and bisect each other. (viii) False, Diagonals are of equal length and bisect each other. Diagonals of a rectangle need not be perpendicular (ix) False, In a square all sides are of equal length. (x) True, All rhombuses are parallelograms, since opposite sides are equal and parallel. (xi) True, All squares are rhombuses, since all sides are equal in a square and rhombus. All squares are rectangles, since opposite sides are equal and parallel. (xii) False, All squares are parallelograms, since opposite sides are parallel and equal.

157.

Which of the following statements are true for a rectangle?(i) It has two pairs of equal sides.(ii) It has all its sides of equal length.(iii) Its diagonals are equal.(iv) Its diagonals bisect each other.(v) Its diagonals are perpendicular.(vi) Its diagonals are perpendicular and bisect each other.(vii) Its diagonals are equal and bisect each other.(viii) Its diagonals are equal and perpendicular, and bisect each other.(ix) All rectangles are squares.(x) All rhombuses are parallelograms.(xi) All squares are rhombuses and also rectangles.(xii) All squares are not parallelograms.

Answer»

(i) True, In a rectangle two pairs of sides are equal.

(ii) False, In a rectangle two pairs of sides are equal.

(iii) True, In a rectangle diagonals are of equal length.

(iv) True, In a rectangle diagonals bisect each other.

(v) False, Diagonals of a rectangle need not be perpendicular.

(vi) False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each other.

(vii) True, Diagonals are of equal length and bisect each other.

(viii) False, Diagonals are of equal length and bisect each other. Diagonals of a rectangle need not be perpendicular

(ix) False, In a square all sides are of equal length.

(x) True, All rhombuses are parallelograms, since opposite sides are equal and parallel.

(xi) True, All squares are rhombuses, since all sides are equal in a square and rhombus. All squares are rectangles, since opposite sides are equal and parallel.

(xii) False, All squares are parallelograms, since opposite sides are parallel and equal.

Discussion

158.	Which of the following statements are true for a rhombus?(i) It has two pairs of parallel sides.(ii) It has two pairs of equal sides.(iii) It has only two pairs of equal sides.(iv) Two of its angles are at right angles.(v) Its diagonals bisect each other at right angles.(vi) Its diagonals are equal and perpendicular.(vii) It has all its sides of equal lengths.(viii) It is a parallelogram.(ix) It is a quadrilateral.(x) It can be a square.(xi) It is a square.
Answer» (i) True, Rhombus is a parallelogram. (ii) True, Rhombus has all four sides equal. (iii) False, Rhombus has all four sides equal. (iv) False, In rhombus no angle is right angle. (v) True, in rhombus diagonals bisect each other at right angles. (vi) False, in rhombus diagonals are of unequal length. (vii) True, Rhombus has all four sides equal. (viii) True, Rhombus is a parallelogram since opposite sides equal and parallel. (ix) True, Rhombus is a quadrilateral since it has four sides. (x) True, Rhombus becomes square when any one angle is 90°. (xi) False, Rhombus is never a square. Since in a square each angle is 90°.

158.

Which of the following statements are true for a rhombus?(i) It has two pairs of parallel sides.(ii) It has two pairs of equal sides.(iii) It has only two pairs of equal sides.(iv) Two of its angles are at right angles.(v) Its diagonals bisect each other at right angles.(vi) Its diagonals are equal and perpendicular.(vii) It has all its sides of equal lengths.(viii) It is a parallelogram.(ix) It is a quadrilateral.(x) It can be a square.(xi) It is a square.

Answer»

(i) True, Rhombus is a parallelogram.

(ii) True, Rhombus has all four sides equal.

(iii) False, Rhombus has all four sides equal.

(iv) False, In rhombus no angle is right angle.

(v) True, in rhombus diagonals bisect each other at right angles.

(vi) False, in rhombus diagonals are of unequal length.

(vii) True, Rhombus has all four sides equal.

(viii) True, Rhombus is a parallelogram since opposite sides equal and parallel.

(ix) True, Rhombus is a quadrilateral since it has four sides.

(x) True, Rhombus becomes square when any one angle is 90°.

(xi) False, Rhombus is never a square. Since in a square each angle is 90°.

Discussion

159.	Which of the following statements are true for a square?(i) It is a rectangle.(ii) It has all its sides of equal length.(iii) Its diagonals bisect each other at right angle.(v) Its diagonals are equal to its sides.
Answer» (i) True, square is a rectangle, since opposite sides are equal and parallel and each angle is right angle. (ii) True, In a square all sides are of equal length. (iii) True, in a square diagonals bisect each other at right angle. (v) False, in a square diagonals are of equal length. Length of diagonals is not equal to the length of sides

159.

Which of the following statements are true for a square?(i) It is a rectangle.(ii) It has all its sides of equal length.(iii) Its diagonals bisect each other at right angle.(v) Its diagonals are equal to its sides.

Answer»

(i) True, square is a rectangle, since opposite sides are equal and parallel and each angle is right angle.

(ii) True, In a square all sides are of equal length.

(iii) True, in a square diagonals bisect each other at right angle.

(v) False, in a square diagonals are of equal length. Length of diagonals is not equal to the length of sides

Discussion

160.	In a parallelogram the sum of the angle bisectors of two adjacent angle is: (A) 300 (B) 450 (C) 600 (D) 900
Answer» The correct option is (D).

Discussion

161.	Fill in the blanks in each of the following, so as to make the statement true :(i) A rectangle is a parallelogram in which ________.(ii) A square is a rhombus in which __________.(iii) A square is a rectangle in which ___________.
Answer» (i) A rectangle is a parallelogram in which opposite sides are parallel and equal. (ii) A square is a rhombus in which all the sides are of equal length. (iii) A square is a rectangle in which opposite sides are equal and parallel and each angle is a right angle.

161.

Fill in the blanks in each of the following, so as to make the statement true :(i) A rectangle is a parallelogram in which .(ii) A square is a rhombus in which .(iii) A square is a rectangle in which _____.

Answer»

(i) A rectangle is a parallelogram in which opposite sides are parallel and equal.

(ii) A square is a rhombus in which all the sides are of equal length.

(iii) A square is a rectangle in which opposite sides are equal and parallel and each angle is a right angle.

Discussion

162.	From the figure find the value of ∠SQP and ∠QSP of parallelogram PQRS.(A) 600 , 500 (B) 600 , 450 (C) 700 , 350 (D) 350 , 700
Answer» The correct option is (A).

Discussion

163.	In the adjoining parallelogram ABCD, the angles x and y are:(A) 600 , 300 (B) 300 , 600 (C) 450 , 450 (D) 900 , 900
Answer» The correct answer is (A).

Discussion

164.	Fill in the blanks to make the following statements correct: (i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is ____ (ii) The triangle formed by joining the mid-points of the sides of a right triangle is ______ (iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is _____
Answer» (i) Isosceles (ii) Right triangle (iii) Parallelogram

164.

Fill in the blanks to make the following statements correct: (i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is (ii) The triangle formed by joining the mid-points of the sides of a right triangle is (iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is ___

Answer»

(i) Isosceles

(ii) Right triangle

(iii) Parallelogram

Discussion

165.	ABC is an isosceles triangle in which `A B =A C`. AD bisects exterior angle PAC and `C D \|\|A B`. Show that(i) `/_D A C =/_B C A ` and (ii) ABCD is a parallelogram.
Answer» ABC is a isosceles triangle `/_ABC=/_ACB` 1)`/_DAC=/_BCA` `/_PAC=/_ACB+/_ABC` `/_PAC=theta+theta=2theta` 2)ABCD is a parallelogram. AD\|\|AC CD\|\|AB.

Discussion

166.	In a rhombus PQRS, if PQ = 3x – 7 and QR = x + 3 find PS.
Answer» PQ = QR [Sides of a rhombus] 3x – 7 = x + 3 3x – x = 3 + 7 2x = 10 x = \(\frac{10}{2}\) x = 5 PQ = 3x – 7 = 3 × 5 – 7 =15 – 7 PQ = 8cm ∴ SP = 8

166.

In a rhombus PQRS, if PQ = 3x – 7 and QR = x + 3 find PS.

Answer»

PQ = QR [Sides of a rhombus]

3x – 7 = x + 3

3x – x = 3 + 7

2x = 10

x = \(\frac{10}{2}\)

x = 5

PQ = 3x – 7 = 3 × 5 – 7 =15 – 7

PQ = 8cm

∴ SP = 8

Discussion

167.	In the figure, both RISK and CLUE are parallelograms. Find the value of x.
Answer» Solution: In parallelogram RISK Angle ISK = 180° - 120° = 60° Similarly, in parallelogram CLUE Angle CEU = 180° - 70° = 110° Now, in the triangle x = 180° - (110° - 60°) = 10°

Discussion

168.	Rhombus is a parallelogram. Justify.
Answer» In a rhombus opposite sides are equal and parallel to each other therefore it is a parallelogram.

Discussion

169.	The side of square ABCD is 5cm and another square PQRS has perimeter equal to 40cm. Find the ratio of perimeter ABCD to perimeter of PQRS. Find the ratio of the area of ABCD to the area of PQRS.
Answer» Side of ABCD = 5 cm Perimeter of ABCD = 5 × 4 = 20 cm Perimeter of PQRS = 40 cm 4 × side = 40 side = \(\frac{40}{4}\) side = 10cm Perimeter of ABCD : Perimeter PQRS = 20 : 40 = 1 : 2 Area of ABCD:Area of PQRS = 52 : 102 = 25 : 100 = 1 : 4

169.

The side of square ABCD is 5cm and another square PQRS has perimeter equal to 40cm. Find the ratio of perimeter ABCD to perimeter of PQRS. Find the ratio of the area of ABCD to the area of PQRS.

Answer»

Side of ABCD = 5 cm

Perimeter of ABCD = 5 × 4 = 20 cm

Perimeter of PQRS = 40 cm 4 × side = 40

side = \(\frac{40}{4}\)

side = 10cm

Perimeter of ABCD : Perimeter PQRS

= 20 : 40 = 1 : 2

Area of ABCD:Area of PQRS

= 52 : 102 = 25 : 100 = 1 : 4

Discussion

170.	In a parallelogram ABCD, if ∠D = 115°, then write the measure of ∠A.
Answer» Given, In parallelogram ABCD ∠D = 115° Given ∵∠A & ∠D are adjacent angles of parallelogram ∴∠A +∠D = 180° ∠A = 180°-115° = 65°

170.

In a parallelogram ABCD, if ∠D = 115°, then write the measure of ∠A.

Answer»

Given,

In parallelogram ABCD

∠D = 115° Given

∵∠A & ∠D are adjacent angles of parallelogram

∴∠A +∠D = 180°

∠A = 180°-115° = 65°

Discussion

171.	The measure of angles of a hexagon are x°, (x-5)°, (x-5)°, (2x-5)°, (2x-5)°, (2x+20)°. Find value of x.
Answer» The sum of interior angles of a polygon = (n – 2) × 180° where n = number of sides of polygon.Now, we know, a hexagon has 6 sides. So, The sum of interior angles of a hexagon = (6 – 2) × 180° = 4 × 180° = 720° therefore, we have x°+ (x - 5)°+ (x - 5)°+ (2x - 5)°+ (2x - 5)°+ (2x + 20)° = 720° x°+ x°- 5°+ x° - 5°+ 2x° - 5°+ 2x° - 5°+ 2x° + 20° = 720° 9x° = 720° x = \(\frac{720°}{9}\) x = 80°

171.

The measure of angles of a hexagon are x°, (x-5)°, (x-5)°, (2x-5)°, (2x-5)°, (2x+20)°. Find value of x.

Answer»

The sum of interior angles of a polygon = (n – 2) × 180°

where n = number of sides of polygon.Now, we know, a hexagon has 6 sides. So,

The sum of interior angles of a hexagon = (6 – 2) × 180° = 4 × 180° = 720°

therefore, we have

x°+ (x - 5)°+ (x - 5)°+ (2x - 5)°+ (2x - 5)°+ (2x + 20)° = 720°

x°+ x°- 5°+ x° - 5°+ 2x° - 5°+ 2x° - 5°+ 2x° + 20° = 720°

9x° = 720°

x = \(\frac{720°}{9}\)

x = 80°

Discussion

172.	Each angle of a square is ………………… A) 90° B) 180° C) 45° D) 60°
Answer» Correct option is: A) 90°

Discussion

173.	The angles of a quadrilateral are x°, (x + 10°), (x + 20°), (x + 30°) then ∠x = ……………. A) 65° B) 75° C) 85° D) 70°
Answer» Correct option is: B) 75°

Discussion

174.	ABCD is a parallelogram. ∠A + ∠D = A) 90° B) 180° C) 360° D) 270°
Answer» Correct option is: B) 180°

Discussion

175.	A square field has side 20m. Find the length of the wire required to fence it four times.
Answer» Perimeter = 4a = 4 × 20 = 80m Length of the wire required to fence 4 times. = 4 × 80 = 320 m.

175.

A square field has side 20m. Find the length of the wire required to fence it four times.

Answer»

Perimeter = 4a = 4 × 20 = 80m

Length of the wire required to fence 4 times.

= 4 × 80 = 320 m.

Discussion

176.	ABCD is a parallelogram in which AB=9.5 cm and its perimeter is 30 c. Find the length of each side of the parallelogram.
Answer» Correct Answer - `AB=9.5 cm = DC, BC=5.5cm =DA`

Discussion

177.	The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is(a) 72° (b) 144° (c) 36° (d) 18°
Answer» (c) 36^o We know that, sum of all interior angle of quadrilaterals is equal to 360^o. Let us assume the angles be x, 2x, 3x, and 4x Then, x + 2x + 3x + 4x = 360^o 10x = 360^o x = 360/10 x = 36 Therefore the angles are x = 36^o 2x = 2 × 36 = 72^o 3x = 3 × 36 = 108^o 4x = 4 × 36 = 144^o

177.

The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is(a) 72° (b) 144° (c) 36° (d) 18°

Answer»

(c) 36^o

We know that, sum of all interior angle of quadrilaterals is equal to 360^o.

Let us assume the angles be x, 2x, 3x, and 4x

Then,

x + 2x + 3x + 4x = 360^o

10x = 360^o

x = 360/10

x = 36

Therefore the angles are x = 36^o

2x = 2 × 36 = 72^o

3x = 3 × 36 = 108^o

4x = 4 × 36 = 144^o

Discussion

178.	In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 : 5. Find the measure of each angle of the quadrilateral.
Answer» Sum of angles of a quadrilateral is 360° Let angle is x° Therefore each angle is x°, 2x°, 4x° and 5x° x° + 2x° + 4x° + 5x° = 360° 12x° = 360° x° = \(\frac{360°}{12}\) x° = 30° Therefore angles are: x = 30° 2x = 2 × 30° = 60° 4x = 4 × 30 = 120° 9x = 5 × 30 = 150°

178.

In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 : 5. Find the measure of each angle of the quadrilateral.

Answer»

Sum of angles of a quadrilateral is 360°

Let angle is x°

Therefore each angle is x°, 2x°, 4x° and 5x°

x° + 2x° + 4x° + 5x° = 360°

12x° = 360°

x° = \(\frac{360°}{12}\)

x° = 30°

Therefore angles are: x = 30°

2x = 2 × 30° = 60°

4x = 4 × 30 = 120°

9x = 5 × 30 = 150°

Discussion

179.

List out the difference between square and rhombus

Answer»

Square	Rhombus
i. All the angles are equal	i. Opposite angles are equal
ii. All the angle are right angles	ii. No angle is a right angle
iii. Diagonals are equal	iii. Diagonals are not equal.

Discussion

180.	From the given figure, ∠x = ……?A) 60° B) 45° C) 30° D) 55°
Answer» Correct option is: A) 60°

Discussion

181.	In a quadrilateral ABCD, the angles A, B, C and D are in the ratio of 1:2:4:5. Find the measure of each angles of the quadrilateral.
Answer» Let the angles of the quadrilaterals are A = x, B = 2x, C = 4x and D = 5x We know, sum of all angles of a quadrilateral = 360° A + B + C + D = 360° x + 2x + 4x + 5x = 360° 12x = 360° x = 360°/12 = 30° Therefore, A = x = 30° B = 2x = 60° C = 4x = 120° D = 5x = 15°

181.

In a quadrilateral ABCD, the angles A, B, C and D are in the ratio of 1:2:4:5. Find the measure of each angles of the quadrilateral.

Answer»

Let the angles of the quadrilaterals are A = x, B = 2x, C = 4x and D = 5x

We know, sum of all angles of a quadrilateral = 360°

A + B + C + D = 360°

x + 2x + 4x + 5x = 360°

12x = 360°

x = 360°/12 = 30°

Therefore,

A = x = 30°

B = 2x = 60°

C = 4x = 120°

D = 5x = 15°

Discussion

182.	If the adjacent angles of a parallelogram are x + 29°, 2x + 31° then ∠x = …………… A) 30° B) 40° C) 60° D) 120°
Answer» Correct option is: B) 40°

Discussion

183.	Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle.
Answer» Three angles of a quadrilateral are 110°, 50° and 40° Let the fourth angle be ‘x’ We know, sum of all angles of a quadrilateral = 360° 110° + 50° + 40° + x° = 360° ⇒ x = 360° – 200° ⇒x = 160° Therefore, the required fourth angle is 160°.

183.

Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle.

Answer»

Three angles of a quadrilateral are 110°, 50° and 40°

Let the fourth angle be ‘x’

We know, sum of all angles of a quadrilateral = 360°

110° + 50° + 40° + x° = 360°

⇒ x = 360° – 200°

⇒x = 160°

Therefore, the required fourth angle is 160°.

Discussion

184.	Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angles.
Answer» Given, Three angles of quadrilateral = 110^∘, 50^∘, 40^∘ Let fourth angle be = X^∘ As we know sum of all angles of a quadrilateral = 360^∘ So, 110^∘+ 50^∘+ 40^∘+ X^∘ = 360^∘ X^∘= 360^∘ - 200^∘= 160^∘

184.

Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angles.

Answer»

Given,

Three angles of quadrilateral = 110^∘, 50^∘, 40^∘
Let fourth angle be = X^∘

As we know sum of all angles of a quadrilateral = 360^∘

So,

110^∘+ 50^∘+ 40^∘+ X^∘ = 360^∘

X^∘= 360^∘ - 200^∘= 160^∘

Discussion

185.	The angles of a quadrilateral are in the ratio 2: 4: 5: 7. Find the angles.
Answer» Consider the angles of a quadrilateral as 2x, 4x, 5x and 7x In a quadrilateral we know that the sum of all the angles is 360^o So we can write it as 2x + 4x + 5x + 7x = 360^o By addition 18x = 360^o By division x^o = 20^o Now by substituting the value of x^o 2x = 2(20^o) = 40^o 4x = 4(20^o) = 80^o 5x = 5(20^o) = 100^o 7x = 7(20^o) = 140^o Therefore, the angles are 40^o, 80^o, 100^o and 140^o.

185.

The angles of a quadrilateral are in the ratio 2: 4: 5: 7. Find the angles.

Answer»

Consider the angles of a quadrilateral as 2x, 4x, 5x and 7x

In a quadrilateral we know that the sum of all the angles is 360^o

So we can write it as

2x + 4x + 5x + 7x = 360^o

By addition

18x = 360^o

By division

x^o = 20^o

Now by substituting the value of x^o

2x = 2(20^o) = 40^o

4x = 4(20^o) = 80^o

5x = 5(20^o) = 100^o

7x = 7(20^o) = 140^o

Therefore, the angles are 40^o, 80^o, 100^o and 140^o.

Discussion

186.	In a parallelogram ABCD , if `/_A=(2x+25)^@` and `/_B=(3x-5)^@`. Find the value of x and measure of each angle of the parallelogram.
Answer» Correct Answer - `x=32; angle A =89^(@), angle B=91^(@), angle C= 89^(@) and angle D=91^(@)`

Discussion

187.	The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the measure of each of these angles.
Answer» Let x be the common multiple. As per question, \(\angle\)A = 3x \(\angle\)B = 5x \(\angle\)C = 7x \(\angle\)D = 9x As we know that, Sum of all four angles of quadrilateral is 360^o . \(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360° 3x + 5x + 7x + 9x = 360° 24x = 360° X = 360/24 = 15° \(\angle\)A = 3 × 15° = 45° \(\angle\)B = 5 × 15° = 75° \(\angle\)C = 7 × 15° = 105° \(\angle\)D = 9 × 15° = 135° So, Angles of quadrilateral are 45°, 75°, 105° and 135°.

187.

The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the measure of each of these angles.

Answer»

Let x be the common multiple.

As per question,

\(\angle\)A = 3x

\(\angle\)B = 5x

\(\angle\)C = 7x

\(\angle\)D = 9x

As we know that, Sum of all four angles of quadrilateral is 360^o .

\(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360°

3x + 5x + 7x + 9x = 360°

24x = 360° X = 360/24

= 15°

\(\angle\)A = 3 × 15° = 45°

\(\angle\)B = 5 × 15° = 75°

\(\angle\)C = 7 × 15° = 105°

\(\angle\)D = 9 × 15° = 135°

So, Angles of quadrilateral are 45°, 75°, 105° and 135°.

Discussion

188.	Three angles of a quadrilateral are 75°, 90° and 75°. Find the measure of the fourth angle.
Answer» Consider the fourth angle as x^o In a quadrilateral we know that the sum of all the angles is 360^o So we can write it as x^o + 75^o + 90^o + 75^o = 360^o On further calculation we get x^o = 360^o – 75^o – 90^o – 75^o By subtraction x^o = 360^o – 240^o x^o = 120^o Therefore, the measure of the fourth angle is 120^o.

188.

Three angles of a quadrilateral are 75°, 90° and 75°. Find the measure of the fourth angle.

Answer»

Consider the fourth angle as x^o

In a quadrilateral we know that the sum of all the angles is 360^o

So we can write it as

x^o + 75^o + 90^o + 75^o = 360^o

On further calculation we get

x^o = 360^o – 75^o – 90^o – 75^o

By subtraction

x^o = 360^o – 240^o

x^o = 120^o

Therefore, the measure of the fourth angle is 120^o.

Discussion

189.	The angles of a quadrilateral are in the ratio `2:4:5:7.` Find the angles.
Answer» Correct Answer - `40^(@),80^(@),100^(@),140^(@)`

Discussion

190.	In a quadrilateral ABCD, AO & DO are angle bisectors of ∠A and ∠D and given that ∠C = 1050 , ∠B = 700 then the ∠AOD is: (A) 67.50 (B) 77.50 (C) 87.50 (D) 99.750
Answer» The correct option is (C).

Discussion

191.	Take two triangular pieces of paper such that . one side of one triangle is equal to one side of the other. Let us suppose that in ∆ABC and ∆PQR, sides AC and PQ are the equal sides. Join the triangles so that their equal sides lie B side by side. What figure do we get?
Answer» If we place the triangles together such that the equal sides overlap, the two triangles form a quadrilateral.

Discussion

192.	Two angles of a quadrilateral measure 85° and 75° respectively. The other two angles are equal. Find the measure of each of these equal angles.
Answer» Let x be the common angle of quadrilateral. As per question, \(\angle\)A = 85° \(\angle\)B = 75° \(\angle\)C = \(\angle\)D = x As we know that, Sum of all four angles of quadrilateral is 360^o \(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360° 85° + 75°+ x + x = 360° 2x = 360° - (85° + 75°) 2x = 200° X = 200 / 2 = 100° \(\angle\)C = \(\angle\)D = 100° So, Two angles of quadrilateral whose measuring’s are equal is 100°.

192.

Two angles of a quadrilateral measure 85° and 75° respectively. The other two angles are equal. Find the measure of each of these equal angles.

Answer»

Let x be the common angle of quadrilateral.

As per question,

\(\angle\)A = 85°

\(\angle\)B = 75°

\(\angle\)C = \(\angle\)D = x

As we know that, Sum of all four angles of quadrilateral is 360^o

\(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360°

85° + 75°+ x + x = 360°

2x = 360° - (85° + 75°)

2x = 200°

X = 200 / 2

= 100°

\(\angle\)C = \(\angle\)D = 100°

So, Two angles of quadrilateral whose measuring’s are equal is 100°.

Discussion

193.	A quadrilateral has three acute angles, each measuring 75°. Find the measure of the fourth angle.
Answer» Three angles are acute angle and each measuring is 75° means \(\angle\)A = \(\angle\)B = \(\angle\)C = 75° (Acute angle is angle whose measuring is greater than 0 and less than 90.) As we know that, Sum of all four angles of quadrilateral is 360^o \(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360° 75° + 75° + 75° + D = 360° \(\angle\)D = 360° - (75° + 75° + 75°) = 360° - 225° = 135° So, fourth angle of quadrilateral is 135°.

193.

A quadrilateral has three acute angles, each measuring 75°. Find the measure of the fourth angle.

Answer»

Three angles are acute angle and each measuring is 75° means

\(\angle\)A = \(\angle\)B = \(\angle\)C = 75°

(Acute angle is angle whose measuring is greater than 0 and less than 90.)

As we know that, Sum of all four angles of quadrilateral is 360^o

\(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360°

75° + 75° + 75° + D = 360°

\(\angle\)D = 360° - (75° + 75° + 75°)

= 360° - 225°

= 135°

So, fourth angle of quadrilateral is 135°.

Discussion

194.	Three angles of a quadrilateral are equal and the measure of the fourth angle is 120°. Find the measure of each of the equal angles.
Answer» Let x be the common angle of quadrilateral. As per question, \(\angle\)A = \(\angle\)B = \(\angle\)C + \(\angle\)D = 360° x + x + x + 120° = 360° 3x = 360° - 120 3x = 240° X = 240 / 3 = 80° \(\angle\)A = \(\angle\)B = \(\angle\)C = 80° So, Three Angles of quadrilateral whose measuring’s are equal is 80°.

194.

Three angles of a quadrilateral are equal and the measure of the fourth angle is 120°. Find the measure of each of the equal angles.

Answer»

Let x be the common angle of quadrilateral.

As per question,

\(\angle\)A = \(\angle\)B = \(\angle\)C + \(\angle\)D = 360°

x + x + x + 120° = 360°

3x = 360° - 120

3x = 240°

X = 240 / 3

= 80°

\(\angle\)A = \(\angle\)B = \(\angle\)C = 80°

So, Three Angles of quadrilateral whose measuring’s are equal is 80°.

Discussion

195.	Three angles of a quadrilateral are `75^(@), 90^(@) and 75^(@)`. Find the measure of the fourth angle.
Answer» Correct Answer - `120^(@)`

Discussion

196.	Three angles of a quadrilateral measure `110^(@), 82^(@) and 68^(@)`. Find the measure of the fourth angle.
Answer» Let the measure of the fourth angle be `x^(@)`. Then , the sum of the angles of a quadrilateral being `360^(@)`, we have : `110+82+68 +x=360 rArr 260 +x =360 rArr x = 100. ` Hence, the measure of the fourth angle is `100^(@).`

Discussion

197.	Among the following the correct statement is……………… i) A Rhombus is a Square ii) Product of two odd integers eveniii) All even numbers are composite iv) 3 is a primeA) i and iii B) ii and iv C) only ii D) only iv
Answer» Correct option is D) only iv

Discussion

198.	Each of a parallelogram, is 90° then it is ……………. A) rhombus B) square C) rectangle D) trapezium
Answer» Correct option is: C) rectangle

Discussion

199.	Three angles of a Quadrilateral are 65°, 120°, 70° then the 4 the angle is ……………….. A) 95° B) 115° C) 105° D) 100°
Answer» Correct option is: C) 105°

Discussion

200.	The angles of a quadrilateral are in the ratio of 1 : 2 : 3 : 4 then the largest angle is ………………. A) 120° B) 108° C) 72° D) 144°
Answer» Correct option is: D) 144°

Discussion

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