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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 360Â°.

Also, 115Â° + 70Â° + 60Â° = 245Â°

âˆ´ 245Â° +âˆ ABC = 360Â°

â‡’âˆ ABC = 360Â° â€“ 245Â° = 115Â°

Now, x = ext. âˆ BCD

= 180Â° â€“ âˆ BCD

= 180Â° â€“ 115Â° = 65Â°

Similarly, y = 180Â° â€“ 70Â° = 110Â°

z = 180Â° â€“ 60Â° = 120Â°

w = 180Â° â€“ 115Â° = 65Â°

âˆ´ x + y + z + w = 65Â° + 110Â° + 120Â° + 65Â°

= 360Â°**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:** âˆµâˆ A : âˆ B : âˆ C : âˆ D = 1 : 2 : 3 : 4

âˆ´ Let us suppose that

âˆ A = 1xÂ°, âˆ B = 2xÂ°

âˆ C = 3xÂ°, â€“D = 4xÂ°

Since, âˆ A +âˆ B + âˆ C +âˆ D = 360Â°

âˆ´ x + 2x + 3x + 4x = 360Â°

â‡’ 10x = 360Â°

â‡’ x = 360/10 = 36Â°

âˆ´ Angles are:

âˆ A = xÂ° = 36Â°

âˆ B = 2xÂ° = 2 x 36Â° = 72Â°

âˆ C = 3xÂ° = 3 x 36Â° = 108Â°

âˆ D = 4xÂ° = 4 x 36Â° = 144Â°

Thus, the measure of the angle of the quad. are 36Â°, 72Â°, 108Â° and 144Â°**Question 3:** The interior angle of a regular is 108Â°. Find the number of sides of the polygon.**Solution:** Let there are â€˜nâ€™ sides of the regular polygon.

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

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

âˆ´ Measure of each exterior angle = 180Â° â€“ 108Â° = 72Â°

â‡’ 360/n = 72 â‡’ n = 360/72 = 5

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

Let there be â€˜nâ€™ sides of the regular polygon.

Since, the measure of each interior angle is

âˆ´

â‡’ (2n-4) x 90 = 108 x n

â‡’ 180n - 360 = 108n

â‡’ 180n - 108n = 360

â‡’ 72n = 360 â‡’ 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.

âˆ´ Their interior angles are

Since the ratio of the interior angles is 4 : 3

âˆ´

â‡’

â‡’

â‡’

â‡’

â‡’ 3(n-1) = 4(n-2)

â‡’ 3n -3 = 4n -8

â‡’ 3n - 4n = -8 +3

â‡’ -n = -5 â‡’ n=5

â‡’ 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 â€˜nâ€™.

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

^{And interior angle of the polygon = }

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

â‡’

â‡’

â‡’

â‡’ 2n- 4 = 20 â‡’ 2n = 20 +4 = 24

â‡’ 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 âˆ A and âˆ B are 4x and 5x respectively.

Since, the adjacent angles are supplementary,

âˆ´ âˆ A+ âˆ B = 180^{0}

â‡’ 4x + 5x = 180^{o}

â‡’ 9x+ = 180^{0 . }â‡’ x = 180^{o}/9 = 20^{o}

âˆ´ âˆ A = 4x = 4 x 20Â° = 80Â°

and âˆ B = 5x = 5 x 20Â° = 100Â°

We know that opposite angles of a parallelogram are equal.

âˆ´ âˆ C = âˆ A = 80Âº

And âˆ D= âˆ B = 100Â°

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

Prove that

**Solution:** In Î”COD, we have

â‡’ âˆ COD + âˆ 1 + âˆ 2 = 180Â°

â‡’ âˆ COD = 180Â° â€“ [â€“1 + â€“2]

â‡’

â‡’ But âˆ A+ âˆ B+âˆ C+âˆ D = 360^{o}

â‡’ âˆ C+âˆ D = 360^{o} - (âˆ A+âˆ B)

âˆ´

Thus,

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

**Ans: **8**Q 2. If an interior angle of a regular polygon is 162Â°, 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) 45Â° (ii) 135Â°

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

**Ans:** 110Â°, 70Â°, 110Â°

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

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

**Ans: **81Â° 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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