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**Question 1:** In the adjoining figure, find x + y + z + w

**Solution:** Since, the sum of the measures of interior angles of a quadrilateral is 360°.

Also, 115° + 70° + 60° = 245°

∴ 245° +∠ABC = 360°

⇒∠ABC = 360° – 245° = 115°

Now, x = ext. ∠BCD

= 180° – ∠BCD

= 180° – 115° = 65°

Similarly, y = 180° – 70° = 110°

z = 180° – 60° = 120°

w = 180° – 115° = 65°

∴ x + y + z + w = 65° + 110° + 120° + 65°

= 360°**Question 2:** In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 3 : 4. Find the measure of each angle of the quadrilateral.**Solution:** ∵∠A : ∠B : ∠C : ∠D = 1 : 2 : 3 : 4

∴ Let us suppose that

∠A = 1x°, ∠B = 2x°

∠C = 3x°, –D = 4x°

Since, ∠A +∠B + ∠C +∠D = 360°

∴ x + 2x + 3x + 4x = 360°

⇒ 10x = 360°

⇒ x = 360/10 = 36°

∴ Angles are:

∠A = x° = 36°

∠B = 2x° = 2 x 36° = 72°

∠C = 3x° = 3 x 36° = 108°

∠D = 4x° = 4 x 36° = 144°

Thus, the measure of the angle of the quad. are 36°, 72°, 108° and 144°**Question 3:** The interior angle of a regular is 108°. Find the number of sides of the polygon.**Solution:** Let there are ‘n’ sides of the regular polygon.

∴ Measure of each exterior angle = (360/n)^{o}

Since, the measure of each of the interior angle = 108°

∴ Measure of each exterior angle = 180° – 108° = 72°

⇒ 360/n = 72 ⇒ n = 360/72 = 5

Thus, the required number of sides of the regular polygon is 5.**ALTERNATE METHOD**

Let there be ‘n’ sides of the regular polygon.

Since, the measure of each interior angle is

∴

⇒ (2n-4) x 90 = 108 x n

⇒ 180n - 360 = 108n

⇒ 180n - 108n = 360

⇒ 72n = 360 ⇒ n= (360/72) = 5

Thus, there are 5 sides of the given regular polygon.**Question 4:** Two regular polygons are such that the ratio of the measures their interior angles is 4 : 3 and the ratio between their number of sides is 2 : 1. Find the number of sides of each polygon.**Solution:** Let 2n and n be the number of sides of the regular polygons.

∴ Their interior angles are

Since the ratio of the interior angles is 4 : 3

∴

⇒

⇒

⇒

⇒

⇒ 3(n-1) = 4(n-2)

⇒ 3n -3 = 4n -8

⇒ 3n - 4n = -8 +3

⇒ -n = -5 ⇒ n=5

⇒ 2n = 2 x 5 = 10

Thus, the number of sides of the polygons are 10 and 5 respectively.**Question 5: **The exterior angle of a regular polygon is one-fifth of its interior angle. How many sides has the polygon?**Solution:** Let the number of sides be ‘n’.

∴ Exterior angle of polygon = (360/n)^{o}

^{And interior angle of the polygon = }

since, Exterior angle = 1/5 (Interior angle)

⇒

⇒

⇒

⇒ 2n- 4 = 20 ⇒ 2n = 20 +4 = 24

⇒ n= 24/2 = 12

Thus, the polygon is having 12 sides.**Question 6:** The measures of two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of the angles of the parallelogram.**Solution: **Let ABCD be a parallelogram such that ∠A and ∠B are 4x and 5x respectively.

Since, the adjacent angles are supplementary,

∴ ∠A+ ∠B = 180^{0}

⇒ 4x + 5x = 180^{o}

⇒ 9x+ = 180^{0 . }⇒ x = 180^{o}/9 = 20^{o}

∴ ∠A = 4x = 4 x 20° = 80°

and ∠B = 5x = 5 x 20° = 100°

We know that opposite angles of a parallelogram are equal.

∴ ∠C = ∠A = 80º

And ∠D= ∠B = 100°

Thus, ∠A = 80°, ∠B = 100°, ∠C = 80° and D = 100°**Question 7:** In a quadrilateral ABCD, DO and CO are the bisectors of ∠D and ∠C respectively.

Prove that

**Solution:** In ΔCOD, we have

⇒ ∠COD + ∠1 + ∠2 = 180°

⇒ ∠COD = 180° – [–1 + –2]

⇒

⇒ But ∠A+ ∠B+∠C+∠D = 360^{o}

⇒ ∠C+∠D = 360^{o} - (∠A+∠B)

∴

Thus,

**PRACTICE EXERCISE****Q 1. If an exterior angle of a regular polygon is 45°, then find the number of its sides.**

**Ans: **8**Q 2. If an interior angle of a regular polygon is 162°, then find the number of its sides.Ans: **20

** Ans: **70

**Q 4. Find the measure of an interior angle of a regular polygon having 15 sides.**

**Ans:** 156^{o}**Q 5. Find the measure of each of the (i) exterior angle. (ii) interior angle of an octagon.**

Ans: (i) 45° (ii) 135°

**Q 6. An angle of a parallelogram measures 70°. Find the measure of the remaining three angles.**

**Ans:** 110°, 70°, 110°

**Q 7. In the following figure, ABCD is a parallelogram. Find measures of x, y and z.**

**Ans: **x = 80°, y = 100°, z = 80°**Q 8. In the following figure, find the measures of x, y, z and w.**** ****Ans: **x = 120°, y = 100°, z = 120°, w = 80°**Q 9. One angle of a quadrilateral is 111° and the remaining three angles are equal. Find three angles. **

**Ans: **81° each**Q.10. What is the ratio of the interior angles of a pentagon and a decagon.**

**Note:** (i) A pentagon has 5 sides. (ii) A decagon has 10 sides.

**Ans: **3 : 4

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