Gujarat Board GSEB Textbook Solutions Class 8 Maths Chapter 3 Understanding Quadrilaterals InText Questions and Answers.

## Gujarat Board Textbook Solutions Class 8 Maths 3 Understanding Quadrilaterals InText Questions

Try These (Page 43)

Question 1 .

Take a regular hexagon as shown in the figure:

1. What is the sum of the measures of its exterior angles x, y, z, p, q, r?

2. Is x = y = z = p = q = r? Why?

3. What is the measure of each?

(i) exterior angle

(ii) interior angle

4. Repeat this activity for the cases of

(i) A regular octagon

(ii) a regular 20-gon

Solution:

1. âˆ x + âˆ y + âˆ z + âˆ p + âˆ q + âˆ r = 360Â°

[âˆµ Sum of exterior angles of a polygon = 360Â°]

2. Since, all the sides of the polygon are equal.

âˆ´ It is a regular hexagon.

So, its interior angles are equal.

x = (180Â° – a) y = (180Â° – a)

z = (180Â° – a) p = (180Â° – a)

q = (180Â° – a) r = (180Â° – a)

âˆ´ x – y = z = p = q = r

3.

(i) âˆµ x + y + z + p + q + r = 360Â°

[âˆµ sum of exterior angles = 360Â°] and all these angles are equal.

âˆ´ Measure of each exterior angle

= \(\frac{360^{\circ}}{6}\) = 60Â°

(ii) âˆµ Exterior angle = 60Â°

âˆ´ 180Â° – 60Â° = Interior angle

or 120Â°= Interior angle

or Measure of interior angle = 120Â°

4.

(i) In a regular octagon, number of sides (n) = 8

âˆ´Each exterior angle = \(\frac{360^{\circ}}{8}\) = 45Â°

âˆ´Each interior angle = 180Â° – 45Â° = 135Â°

(ii) For a regular 20-gon, the number of sides (n) = 20

âˆ´Each exterior angle = \(\frac{360^{\circ}}{20}\) = 18Â°

Thus, each interior angle = 180Â° – 18Â° = 162Â°

Question 2.

Find the number of sides of a regular polygon whose each exterior angle has a measure of 40Â°?

Solution:

Since, the given polygon is a regular polygon.

âˆ´ Its each exterior angle is equal.

âˆµ Sum of all the exterior angles = 360Â°

âˆ´ Number of exterior angles = \(\frac{360^{\circ}}{40^{\circ}}\) = 9

â‡’ Number of sides = 9

Thus, it is a nonagon.

Try These (Page 47)

Question 1.

Take two identical set squares with angles 30Â° – 60Â° – 90Â° and place them adjacently to form a parallelogram as shown in Figure? Does this help to you to verify the above property?

Solution:

The given figure helps us to verify that Opposite sides of a parallelogram are of equal length.

Try These (Page 48)

Question 1.

Take two identical 30Â° – 60Â° – 90Â° set – squares and form a parallelogram as before. Does the figure obtained help you to confirm the above property?

Solution:

Above figure also help us to confirm that: opposite angles of a parallelogram are equal.

Try These (Page 50)

Question 1.

After showing mâˆ R = mâˆ N = 70Â°, can you find mâˆ I and mâˆ G by any other method?

Solution:

Yes, without using the property of a parallelogram, we can also find mâˆ I and mâˆ G as given below:

âˆµ mâˆ R = mâˆ N = 70Â° and RG || IN. the transversal RI intersecting them.

âˆ´ mâˆ R + mâˆ I = 180Â°

[Sum of interior opposite angles is 180Â°]

or 70Â° + mâˆ I = 180Â°

mâˆ I = 180Â° – 70Â° = 110Â°

Similarly, mâˆ G = 110Â°

Question 2.

In the figure. ABCD is a parallelogram. Given that OD = 5 cm and AC is 2 cm less than BD. Find OA?

Solution:

âˆµ Diagonals of a parallelogram bisect each other.

âˆ´ OD = OB =5 cm

or OB = 5cm

or BD = 5 cm Ã— 2 = 10 cm

âˆµ AC = BD – 2cm

âˆ´ AC = (10 – 2) cm = 8 cm

0r \(\frac{1}{2}\) AC = \(\frac{1}{2}\) Ã— 8 cm = 4cm

or OA = 4 cm.

Try These (Page 56)

Question 1.

A mason has made a concrete slab. He needs it to be rectangular In what different ways can he make sure thai it is rectangular?

Solution:

He can make sure thai it is rectangular using the following different ways:

- By making opposite sides of equal length.
- By keeping each angle at the corners as 900.
- By keeping the diagonals of equal length.
- By making opposite sides parallel and ensuring one angle as 900 in measure.
- By making all angles equal and ensuring the measure of one angle as 900.

Question 2.

A square was defined as a rectangle with all sides equal. Can we dfine it as rhombus with equal angles? Explore this idea?

Solution:

Yes, because a rhombus becomes a square if its all angles are equal.

Question 3.

Can a trapezium have all angles equal? Can it have all sides equal? Explain?

Solution:

Yes, if it is a rectangle. It can have all sides equal, when it becomes a square.