3. Understanding Quadrilaterals

Exercise 3.1

1. Given here are some figures.

Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon

Answer:  (a)  Figure (1) , Figure (2) , Figure (3) , Figure (5) , Figure (6) and Figure (7) are simple curve .

(b) Figure (1) , Figure (2) , Figure (3) , Figure (5) , Figure (6) and Figure (7) are simple closed curve .

(c) Figure (1) and Figure (2)are polygon .
(d) Figure (2) is convex polygon .

(e) Figure (1) is concave polygon

2. How many diagonals does each of the following have?
(a) A convex quadrilateral (b) A regular hexagon (c) A triangle

Answer: (a) A convex quadrilateral have two diagonals .

(b) A regular hexagon has nine diagonals .

(c) A triangle have no diagonal .

[The number of diagonal   , where  is the number of sides of polygon.]

3. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

Answer: The sum of the measures of the angles of a convex quadrilateral is 360° . Yes .

4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

What can you say about the angle sum of a convex polygon with number of sides?
(a) 7 (b) 8 (c) 10 (d) n

Solution:  We know that , the angle sum of a convex polygon

(a)  900°      (b) 1080°    (c) 1440°     (d) 

5. What is a regular polygon? State the name of a regular polygon of : (i) 3 sides (ii) 4 sides (iii) 6 sides

Answer:  A regular polygon is equiangular and equilateral . For example : a square has sides of equal length and angles of equal measures .

(i) Equilateral triangle (for 3 sides)   (ii) Square (for 4 sides)    (iii) Regular hexagon (for 6 sides)
6. Find the angle measure x in the following figures.

Solution:  (a) Given figure, 

We know that , the sum of the measures of the four angles of a quadrilateral is 360° .

 

Therefore, the value of  is 60° .

(b) Given figure, 

 

We know that , the sum of the measures of the four angles of a quadrilateral is 360° .

Therefore, the value of  is 140° .

(c) Given figure, 

We know tha,

So,

 Again ,

Therefore, the value of  is 140° .

(d) Given figure,

 

We have , 

 

Therefore, the value of  is 108° .

7.

Solution: (a) Given figure,

  We know that , Interior angle+Exterior angle=180° 

So,

 

Again ,

The sum of the three interior angle of the polygon is 180° .

 

And 

 

 

 

 Solution: (b) Given, figure,

We know that ,  

 

 

 

 

 

 

 

 

 

We know that , the sum of the measures of the four angles of a quadrilateral is 360° .

 

 

 

 

 And  

 

 

Exercise 3.2

1. Find x in the following figures.

Solution:  (a) Given figure ,

Solution: We know that , the sum of the measure of the external angles of any polygon is 360° .

 Therefore, the value of  is 110° .

(b) Given figure,

(b) We know that , the sum of the measure of the external angles of any polygon is 360° .

 

Therefore, the value of  is 50° .

2. Find the measure of each exterior angle of a regular polygon of
    (i) 9 sides              (ii) 15 sides

Solution:  We know that, the measure of each external angle , where  is the number of sides .

(i) the measure of each exterior angle of a regular

(ii) the measure of each exterior angle of a regular

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution:  We know that , the number of sides of a regular polygon

4. How many sides does a regular polygon have if each of its interior angles is 165°?

Solution: We know that, interior angle + exterior angle = 180°

             Exterior angle = 180° – 165° = 15°

Therefore, the number of sides of a regular polygon  

5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?

Solution : (a) We know that , the number of sides of a regular polygon

No . because 22° is not divisor of 360° .

(b) No , Because   Exterior angle = 180° – 22° = 158° is not a divisor of 360° . 

6. (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?

Solution: (a) The minimum interior angle possible for a regular polygon is an equilateral triangle . Because , the equilateral triangle being a regular polygon of 3 sides has the least measure of an interior angle = 60° .

(b) The maximum exterior angle possible for a regular polygon is 120° .

Exercise 3.3

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

(i) AD = ...... (ii) ∠ DCB = ......
(iii) OC = ...... (iv) m ∠DAB + m ∠CDA = ...... 

Solution: (i) Since , the opposite sides of a parallelogram are of equal length .

So ,  AD = BC

(ii) Since, the opposite angles of a parallelogram are of equal measure .

So,   

(iii) Since, the diagonal of a parallelogram bisect each other .

So, OC = OA

(iv) Since , the adjacent angles in a parallelogram are supplementary .

So,  

2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

Solution: (i) Since , the adjacent angles in a parallelogram are supplementary .

 

 

Again , the opposite angles of a parallelogram are of equal measure .

 

and   

Thus , the value of   ,  and

(ii) Since , the adjacent angles in a parallelogram are supplementary .

 

 

Again , the opposite angles of a parallelogram are of equal measure .

    

and  [ Corresponding angles are equal]

Thus, the value of  ,  and  .

(iii) In figure,

 [Vertically opposite angle]

Since , the sum of all the angle of triangle is 180° .

    [Alternative interior angle]

Thus, the value of ,  and  .

(iv) We have ,  [Corresponding angle are equal]

 The opposite angles of a parallelogram are of equal measure .

 

The adjacent angles in a parallelogram are supplementary .

 

Therefore, the value of  ,  and  .

(v) We have,  [Opposite angles are equal]

The adjacent angles in a parallelogram are supplementary .

 

 [Alternative interior angles are equal]

Therefore, the value of  ,   and  .

3. Can a quadrilateral ABCD be a parallelogram if
(i) ∠D + ∠B = 180°? (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) ∠A = 70° and ∠C = 65°?

Solution: (i)   

 ABCD quadrilateral may be a parallelogram , if the condition of the parallelogram are fulfilled .

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm

Since , the opposite sides of a parallelogram are equal .

So, , but .

(iii)  and  

Since , the opposite angles of a parallelogram are equal .

But  .

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

Solution:

  

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  and  be the angles of the parallelogram.
Since, the adjacent angles in a parallelogram  are supplementary .

  

Thererfore, the angle are :  and

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

Solution: let  be the angles of a parallelogram .

Since, the adjacent angles in a parallelogram  are supplementary .

 

 

Therefore, the measure of the angles of the parallelogram are : 90° and 90° .

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:  Since, the corresponding angle of a parallelogram are equal .

    [alternative interior angle]

Again, the adjacent angles in a parallelogram  are supplementary .

 

Therefore, the angle are :   ,  and

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

Solution:  (i) Since, the opposite sides of a parallelogram are of equal length .

And  

Therefore, the value of and .

Solution: (ii) Since, the diagonals of a parallelogram bisect each other .

and  

 

Therefore, the value of  and .

9.

In the above figure both RISK and CLUE are parallelograms. Find the value of x.

In the above figure both RISK and CLUE are parallelograms. Find the value of x.
Solution: We have,  [Opposite angle of a parallelogram are equal]

Since , the adjacent angles in a parallelogram  are supplementary .

 

 

 

 

Therefore, the value of  

10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)
     

Solution: Since, the adjacent angles in a parallelogram  are supplementary .

 

So , KLMN is a trapezium .

11. Find in Fig 3.33 if AB DC .   Photo

Solution: Given ,  

So,

 

 

12. Find the measure of and   if SPRQ in Fig 3.34.  Photo
(If you find  , is there more than one method to find ?

Solution: Given ,  

So,

 

And 

 

 and 

Exercise 3.4

1. State whether True or False.
(a) All rectangles are squares (e) All kites are rhombuses.
(b) All rhombuses are parallelograms (f) All rhombuses are kites.
(c) All squares are rhombuses and also rectangles (g) All parallelograms are trapeziums.
(d) All squares are not parallelograms. (h) All squares are trapeziums.

Answer:  (a) False    (b) True  (c) True  (d) False   (e) False   (f) True  (g) True   (h) True

2. Identify all the quadrilaterals that have.
(a) four sides of equal length (b) four right angles

Answer: (a) Square and Rhombus

(b) Square and Rectangle .

3. Explain how a square is.
(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle

Answer: (i) A square is four equal sides . So , A square is a quadrilateral .

(ii)  A square has its opposite sides parallel and equal  , So, Square is a parallelogram .

(iii)  A square is a parallelogram with all the four sides equal and opposite angles are equal . So , A square is a rhombus .

(iv)  A square is a parallelogram with each angle a right angle . So , A square is a rectangle .

4. Name the quadrilaterals whose diagonals.
(i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal

Answer:  (i)  Parallelogram , rhombus , square and rectangle .

(ii)  Square  and rhombus

(iii)  Square and rectangle

5. Explain why a rectangle is a convex quadrilateral.

Answer: Because , both of its diagonal lie in its interior .
6. ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you). Photo

Solution:  Since ,  ,   

So, ABCD is a parallelogram and the mid-point of diagonal AC is O .