13. Chapter 13. Quadrilaterals NIOS Class 10 Mathematics Textbook Solutions

Chapter 13. Quadrilaterals NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation

Chapter 13. QUADRILATERALS

CHECK YOUR PROGRESS 13.1

1. Name each of the following quadrilaterals.

Photo  Fig. 13.10

Answer: (i) Rectangle   (ii) Trapezium   (iii) Rectangle   (iv) Parallelogram   (v) Rhombus   (vi) Square

2. State which of the following statements are correct ?

(i) Sum of interior angles of a quadrilateral is 360°.

(ii) All rectangles are squares,

(iii) A rectangle is a parallelogram.

(iv) A square is a rhombus.

(v) A rhombus is a parallelogram.

(vi) A square is a parallelogram.

(vii) A parallelogram is a rhombus.

(viii) A trapezium is a parallelogram.

(ix) A trapezium is a rectangle.

(x) A parallelogram is a trapezium.

Answer: (i) Sum of interior angles of a quadrilateral is 360°. →True
[ This is a fundamental property of all quadrilaterals.]

(ii) All rectangles are squares. → False .
[ A rectangle has opposite sides equal, but a square has all four sides equal. Only some rectangles are squares. ]

(iii) A rectangle is a parallelogram. → True .
[ A rectangle has both pairs of opposite sides parallel, so it is a parallelogram.]

(iv) A square is a rhombus. → True
[ A square has all sides equal, so it satisfies the definition of a rhombus.]

(v) A rhombus is a parallelogram. → True
[ A rhombus has both pairs of opposite sides parallel, so it is a parallelogram.]

(vi) A square is a parallelogram. → True
[ A square has both pairs of opposite sides parallel, so it is a parallelogram.]

(vii) A parallelogram is a rhombus. → False
[ A parallelogram does not necessarily have all sides equal (only opposite sides are equal). Only some parallelograms are rhombuses.]

(viii) A trapezium is a parallelogram. →False
[ A trapezium has only one pair of opposite sides parallel, whereas a parallelogram has two pairs.]

(ix) A trapezium is a rectangle. → False
[ A rectangle has two pairs of parallel sides and right angles, while a trapezium has only one pair of parallel sides.]

(x) A parallelogram is a trapezium. → False
[ A trapezium is defined as a quadrilateral with at least one pair of opposite sides parallel. Since a parallelogram has two pairs.]

3. In a quadrilateral, all its angles are equal. Find the measure of each angle.

Solution: Let, x be the measure of each angle.

We know that, the sum of all interior angles of a quadrilateral is 360°.

∴  x+x+x+x=360°

⇒4x=360°

⇒4x=360°

⇒x=360°4

⇒x=90°

Therefore, each angle of the quadrilateral measures 90°.

4. The angles of a quadrilateral are in the ratio 5:7:7: 11. Find the measure of each angle.

Solution: Let, 5x , 7x , 7x and 11x are the angles of a quadrilateral .

We know that, the sum of all interior angles of a quadrilateral is 360°.

⇒5x+7x+7x+11x=360°

⇒30x=360°

⇒x=360°30

⇒x=12°

5x=5×12°=60° ,  7x=7×12°=84° , 7x=7×12°=84° and 11x=11×12°=132°

Therefore, the angles of a quadrilateral are 60° , 84° , 84° and 132° .

5. If a pair of opposite angles of a quadrilateral are supplementary, what can you say about the other pair of angles?

Solution:  If one pair of opposite angles of a quadrilateral are supplementary (sum = 180°), then the other pair of opposite angles must also be supplementary. This is because the total sum of all four interior angles is 360°, so the remaining two angles also add up to 360° − 180° = 180°.

CHECK YOUR PROGRESS 13.2

1. In a parallelogram ABCD, ∠A = 62° . Find the measures of the other angles.

Solution: Since, ABCD is a parallelogram . Here, ∠A = 62°

So, ∠A = ∠C = 62° and ∠B = ∠D

AB||DC , we have

∠A+∠D=180°

⇒∠D=180°-∠A

⇒∠D=180°-62°

⇒∠D=118°

∠B=∠D=118°

Hence, ∠B=118°  , ∠C=62° , ∠D=118° .

2. The sum of the two opposite angles of a parallelogram is 150° . Find all the angles of the parallelogram.

Solution: Let, ABCD is a parallelogram .

So, ∠A=∠C  and ∠B=∠D

A/Q,  ∠A+∠C=150°

⇒∠A+∠A=150°

⇒2∠A=150°

⇒∠A=150°2

⇒∠A=75°

∠A=∠C=75°

 ∴     AB ∥DC

  ∠A+∠D=180°

⇒75°+∠D=180°

⇒∠D=180°-75°

⇒∠D=105°

  ∠B=∠D=105°

Hence, ∠A=75°  , ∠B=105° , ∠C=75°  and ∠D=105° 

3. In a parallelogram ABCD, ∠A = (2x + 10)° and ∠C = (3x – 20)° . Find the value of x.

Solution: Given, ∠A = (2x + 10)° and ∠C = (3x – 20)°

Since, ABCD is a parallelogram .

So, ∠A=∠C

⇒2x+10°=3x-20°

⇒2x+10=3x-20°

⇒10°+20°=3x-2x

⇒x=30°

Therefore, the value of x is 30° .

4. ABCD is a parallelogram in which ∠DAB = 70° and ∠CBD = 55° . Find ∠CDB and ∠ADB.

Solution: Here, ∠DAB = 70° and ∠CBD = 55°

Since, ABCD is a parallelogram .

∠DAB =∠BCD=70°  

In ∆BCD , we have

 ∠BCD+∠CBD+∠CDB=180° 

⇒70°+55°+∠CDB=180°

⇒125°+∠CDB= 180°

⇒∠CDB=180°-125°

⇒∠CDB=55°

 AD∥BC  and BD is transversal .

 ∠CBD=∠ADB=55°

Therefore,  ∠CDB=55° and  ∠ADB=55°

5. ABCD is a rhombus in which ∠ABC = 58°. Find the measure of ∠ACD.

Solution: Here, ∠ABC = 58°

Since, ABCD is a rhombus . So, AB = BC = CD = AD  and ∠ABC=∠ADC=58°

In ∆ADC , we have

 ∠CAD+∠ADC+∠ACD=180°

⇒∠ACD+58°+∠ACD=180°  AD=CD  

⇒2∠ACD=180°-58°=122°

⇒∠ACD=122° 2

⇒∠ACD=61°

6. In Fig. 13.22, the diagonals of a rectangle PQRS intersect each other at O. If ∠ROQ= 40° , find the measure of ∠OPS.

Photo

Solution: Here, ∠ROQ= 40° and OP=OQ=OR=OS

∴   ∠ROQ=∠POS=40° [Vertically opposite angles]

In ∆POS , we have

∠OPS+∠POS+∠OSP=180°

⇒∠OPS+40°+∠OPS=180° OP=OS

 ⇒2∠OPS=180°-40°=140°

⇒∠OPS=140°2

⇒∠OPS=70°

7. AC is one diagonal of a square ABCD. Find the measure of ∠CAB.

Solution: Since, AC is one diagonal of a square ABCD .

So, ∠CAB=∠CAD

∠BAD=90°

⇒∠CAB+∠CAD=90°

⇒∠CAB+∠CAB=90°

⇒2∠CAB=90°

⇒∠CAB=90°2

⇒∠CAB=45° 

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