The document NCERT Solutions(Part- 2)- Understanding Quadrilaterals Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.

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**Question 1: **Take a regular hexagon as shown in the figures:

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 = 360°/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 = 360°/8 = 45°

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

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

∴ Each exterior angle = 36°/2 = 18°

Thus, each interior angle = 180° – 18° = 162°**Example 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 angles is equal.

∵ Sum of all the exterior angles = 360°

∴ Number of exterior angles = 360°/40° = 9

⇒ Numbers of sides = 9

Thus, its is a nonagon.

**EXERCISE 3.2****Question 1. **Find x in the following figures.**Solution:** (a) Sum of all the exterior angles of a polygon = 360°

∴ 125° + 125° + x = 360°

or 250° + x = 360°

or x = 360° – 250° =110°

(b) ∵ x + 90° + 60° + 90° + 70° = 360°

or x + 310° = 360°

or x = 360° – 310° = 50°**Question 2.** Find the measure of each exterior angle of a regular polygon of

(i) 9 sides

(ii) 15 sides **Solution: **(i) Number of sides (n) = 9

∴ Number of exterior angles = 9

Since, sum of all the exterior angles = 360°

∵ The given polygon is a regular polygon.

∴ All the exterior angles are equal.

∴ Measure of an exterior angle = 360°/9° = 40°

(ii) Number sides of regular polygon = 15

∴ Number of equal exterior angles = 15 The sum of all the exterior angles = 360°

∴ The measure of each exterior angle = 360°/15 = 24°**Question 3.** How many sides does a regular polygon have if the measure of an exterior angle is 24°?**Solution:** For a regular polygon, measure of each angle is equal.

∴ Sum of all the exterior angles = 360° Measure of an exterior angle = 24°

∴ Number of angles = 60°/24° = 15

Thus, there are 15 sides of the polygon.**Question 4.** How many sides does a regular polygon have if each of its interior angles is 165°?**Solution:** The given polygon is regular polygon.

Each interior angle = 165°

∴ Each exterior angle = 180° – 165° = 15°

∴ Number of sides = 360°/15° = 24°

Thus, there are 24 sides of the polygon.**Question 5.** (a) Is it possible to have a regular polygon with measure of each exterior angle is 22°?

(b) Can it be an interior angle of a regular polygon? Why?**Solution:** (a) Each exterior angle = 22°

∴ Number of sides = 360°/22° = 180/11

If it is a regular polygon, then its number of sides must be a whole number.

Here, 180/11 is not a whole number.

∴ 22° cannot be an exterior angle of a regular polygon.

(b) If 22° is an interior angle, then 180° – 22°, i.e. 158° is exterior angle.

∴ Number of sides = 360°/158° = 180°/79°

Which is not a whole number.

Thus, 22° cannot be an interior angle of a regular polygon.**Question 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 number of sides of a polygon = 3

The regular polygon of 3-sides is an equilateral.

∵ Each interior angle of an equilateral triangle = 60°

Hence, the minimum possible interior angle of a polynomial = 60°

(b) ∵ The sum of an exterior angle and its corresponding interior angle is 180°.

And minimum interior angle of a regular polygon = 60°

∴ The maximum exterior angle of a regular polygon = 180° – 60° = 120°**Question 7. **In the figure, PQRS is a parallelogram. Find the values of x, y and z.**Solution: **Using the property, ‘Opposite angles of a parallelogram are equal’, we have:

z = 110°

Since, sum of all the angles of parallelogram = 360°

∴ 2z + 2x = 360°

or 2(110°) + 2x = 360°

or 220° + 2x = 360° or 2x = 360° – 220° = 140°

or

Now, y + z = 180° [Adjacent angles of a || gm are Supplementary]

or y + 110° = 180°

∴ y = 180° – 110° = 70°

Thus, **Question 8. **In a parallelogram JKLM, if m∠K = 80°, find all other angles.**Solution: **Using the property, “Opposite angles of a parallelogram are equal”, we have:

m∠K= m∠M

or m∠M = 80°

Again, m∠J + m∠L + m∠K + m∠M = 360° [Sum of all angles of a quadrilateral is 360°]

or m∠J + m∠L + 80° + 80° = 360°

or m∠J + m∠L = 360° – 80° – 80° = 200°

But m∠J = m∠L [Opposite angles of parallelogram]

∴ m∠J= m∠L = 200°/2 = 100°

Thus, m∠J = 100°, m∠L = 100° and m∠M = 80°.**Question 9:** 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 parallelogram, we can also solve to 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 10.** 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 = 5 cm

or BD = 5 cmx 2 = 10 cm

∵ AC = BD – 2 cm

∴ AC = (10 – 2) cm = 8 cm

or

or OA = 4 cm.

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