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Q1. The measure of interior angle of a regular polygon is five times the measure of its exterior angle. Find the number of sides in the polygon.

Solution

From the given condition, we have:

Q2. If two adjacent angles of a parallelogram are equal, then it is a 

• 1) kite or rhombus
• 2) rectangle or rhombus
• 3) rectangle or square
• 4) square or rhombus

Solution

In a rectangle as well as in square, each angle is 90^o, so their adjacent angles are supplementary as well as equal.

Q3. For the quadrilateral PQRS, which of the following option is incorrect?

• 1)
• 2)
• 3)
• 4)

Solution

Q4. The measure of two angles of a quadrilateral are 115^o and 45^o , and the other two angles are equal. Find the measure of each of the equal angles.

Solution

Let the measure of the each equal angle be x^o ∴ 115 + 45 + x + x = 360 (As the sum of angles of quadrilateral is 360^o) ⇒ 160 + 2x = 360 ⇒ 2x = 360 - 160  ⇒ x = 100 Hence, the measure of each of the equal angles is 100^o .

Q5. For the given parallelogram MNOP, which of the following option is incorrect?  

• 1)
• 2)
• 3)
• 4)

Solution

Q6. For a rhombus, which of the following statement is correct?

• 1) Diagonals do not bisect each other
• 2) Opposite sides and angles are equal
• 3) Diagonals are equal
• 4) Each of the angles is a right angle

Solution

A rhombus has all the properties of a parallelogram. Hence,  opposite sides and opposite angles are equal in a rhombus. Also diagonals bisect each other but they are not equal.

Q7.

• 1) 50^o
• 2) 40^o
• 3) 20^o
• 4) 80^o

Solution

Q8. The sum of the angles in a quadrilateral is always

• 1) 360°
• 2) 180°
• 3) None of the above
• 4) 270°

Solution

The sum of the angles in a quadrilateral is 360^o. 

Q9. A regular polygon with each exterior angle of 18^o has number of sides equal to 

• 1) 20
• 2) 18
• 3) 12
• 4) 10

Solution

We know that sum of all exterior angles of a polygon is 360^o. Since, the polygon is regular, so all the exterior angles are equal. Measure of each exterior angle = 18^o     (Given) Let there be n exterior angles. Therefore, n x 18^o = 360^o n = 20

Q10. In the figure, □ABCD is a rectangle. Which of the following is not true?  

• 1) Diagonals are congruent.
• 2) Opposite sides are equal.
• 3) Measure of each interior angle is 90°.
• 4) Diagonal bisect each other at right angles.

Solution

Diagonals of a rectangle do not bisect at right angles.

Q11. The sum of interior angles of a regular polygon is 1800^o. Find the measure of each interior angle of a regular polygon with half the number of sides of the given polygon.

Solution

Q12. Name the type of quadrilateral associated with each of the following properties: 1. Only one pair of opposite sides is parallel. 2. All sides are equal and all angles are equal. 3. One of the diagonal bisects the other.

Solution

1. Trapezium. Since, in a trapezium one pair of opposite sides is parallel.   2. Square. Since, all sides and all angles are equal in case of square.   3. Kite. Since in a kite one of the diagonal bisects the other.

Q13. Given ABCD is a rectangle with diagonals AC and DB. From the given statements, choose the one which is correct.  

• 1) DO = BC
• 2) AD = AB
• 3) OD = OC
• 4) AO = AD

Solution

In a rectangle, diagonals are equal and they bisect each other. Therefore, AC = BD or, 2OC = 2 OD or, OC = OD

Q14. In the adjoining figure, ABCD is a quadrilateral. (i) How many pairs of adjacent sides are there? Name them. (ii) How many pairs of opposite sides are there? Name them.

Solution

(i) There is 4 pair of adjacent sides in quadrilateral ABCD. (AB, BC), (BC, CD), (CD, DA) and (DA, AB). (ii) There are only two pairs of opposite sides in given figure. (AB, DC) and (AD, BC).

Q15. If the sum of the interior angles of a convex polygon is (4 × 180°). What kind of polygon is it?

• 1) Pentagon
• 2) Hexagon
• 3) Triangle
• 4) Quadrilateral

Solution

We know that the sum of the interior angles of a polygon of n sides is equal to (n - 2) × 180^o Given, that the sum of the interior angles of the polygon is (4 × 180^o). On comparing, we get n - 2 = 4 n = 4 + 2 = 6 Thus, the polygon is a hexagon.

Q16. The measure of two adjacent angles of a parallelogram is in the ratio 2 : 3. Find the measure of each of the angle of the parallelogram.

Solution

Let the adjacent angles of the parallelogram be 2x and 3x. We know that sum of adjacent angles of a parallelogram is 180^o. ⇒ 2x + 3x = 180^o ⇒ 5x = 180^o ⇒ x = 36^o Therefore, adjacent angles are 2 × 36^o = 72^o and 3 × 36^o = 108^o We know that the opposite angles of a parallelogram are equal. Thus, the angles of the parallelogram are 72^o, 108^o, 72^o, 108^o.

Q17. In □LMNO, if m ∠L = 100° and m ∠M = 80°, then m ∠N +m ∠O =?

• 1) 270°
• 2) 360°
• 3) 180°
• 4) 90°

Solution

The sum of angles of a quadrilateral is 360^o. m∠L +m ∠M = 100° + 80° = 180°  ∴m ∠N +m ∠O = 180° 

Q18. In the given figure, ABCD is a rhombus whose diagonals AC and BD intersect at a point O. If side AB =10 cm and diagonal BD =16 cm, find the length of diagonal AC.  

Solution

We know that the diagonals of a rhombus bisect each other at right angles. ∴ BO =BD = cm = 8 cm, AB =10 cm and ∠AOB = 90⁰ . From right triangle OAB, we have AB² = AO² + BO² ⇒AO² = (AB² - BO²) = {(10)² -(8)²} cm² = (100 -64) cm² = 36 cm² ⇒ AO = cm = 6 cm. ∴ AC = 2 × AO = (2×6) cm = 12 cm.

Q19. The length of a rectangle is 8 cm and each of its diagonals measures 10 cm. Find its breadth.

Solution

Let ABCD be the given rectangle in which length AB = 8 cm and diagonal AC = 10 cm. Since each angle of a rectangle is a right angle, we have ∠ABC = 90⁰ . From right angled triangle ABC , we have AB² + BC² = AC² ⇒BC² = AC² - AB² = {(10)² -(8)²} = (100 - 64) = 36 ⇒BC = = 6 cm. Hence , breadth = 6 cm.

Q20. Given figure EFGH is a rectangle with diagonals HF = 4x + 2 and EG = 5x-1. What is the length of OH and OE?

Solution

Since the diagonals are equal in rectangle So, HF = EG 4x + 2 = 5x-1 2+1 = 5x-4x 3 = x So, HF = 4×3 + 2 = 14 cm And EG = 5x3 - 1 = 14 cm Now, it is also known that the diagonals of a rectangle bisect each other. Therefore, OH = × HF = x14 cm = 7 cm OE = × EG = x 14 cm = 7 cm

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