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. ∠x + ∠y + ∠z + ∠p + ∠q + ∠r = 360°

[∵ Sum of exterior angles of a polygon = 360°]

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

∴ It is a regular hexagon.

So, its interior angles are equal.

x = (180° – a) y = (180° – a)

z = (180° – a) p = (180° – a)

q = (180° – a) r = (180° – a)

∴ x – y = z = p = q = r

3.

(i) ∵ x + y + z + p + q + r = 360°

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

∴ Measure of each exterior angle

= \(\frac{360^{\circ}}{6}\) = 60°

(ii) ∵ Exterior angle = 60°

∴ 180° – 60° = Interior angle

or 120°= Interior angle

or Measure of interior angle = 120°

4.

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

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

∴Each interior angle = 180° – 45° = 135°

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

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

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

Question 2.

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

Solution:

Since, the given polygon is a regular polygon.

∴ Its each exterior angle is equal.

∵ Sum of all the exterior angles = 360°

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

⇒ Number of sides = 9

Thus, it is a nonagon.

Try These (Page 47)

Question 1.

Take two identical set squares with angles 30° – 60° – 90° 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 30° – 60° – 90° 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 m∠R = m∠N = 70°, can you find m∠I and m∠G by any other method?

Solution:

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

∵ m∠R = m∠N = 70° and RG || IN. the transversal RI intersecting them.

∴ m∠R + m∠I = 180°

[Sum of interior opposite angles is 180°]

or 70° + m∠I = 180°

m∠I = 180° – 70° = 110°

Similarly, m∠G = 110°

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:

∵ Diagonals of a parallelogram bisect each other.

∴ OD = OB =5 cm

or OB = 5cm

or BD = 5 cm × 2 = 10 cm

∵ AC = BD – 2cm

∴ AC = (10 – 2) cm = 8 cm

0r \(\frac{1}{2}\) AC = \(\frac{1}{2}\) × 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.