Chapter 5

Class 8 Mathematics · 201 questions · 186 with answers

Solved examples

example-1Multiple choice

The number of diagonals in a polygon of n sides is n (n 1) n (n 2) n (n 3)

  • (a)
  • (b)
  • (c)
  • (d)n (n–3). 2 2 2
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Solution : The correct answer is (c).

example-2Multiple choice

The angles of a quadrilateral ABCD taken in an order are in the ratio 3 : 7 : 6 : 4. Then ABCD is a

  • (a)kite
  • (b)parallelogram
  • (c)rhombus
  • (d)trapezium
Show solution

Solution : The correct answer is (d).

example-3Multiple choice

If the diagonals of a quadrilateral bisect each other at right angles, it will be a

  • (a)rhombus
  • (b)trapezium
  • (c)rectangle
  • (d)kite
Show solution

Solution : The correct answer is (a).

example-4Multiple choice

The sum of the angles of a quadrilateral is

  • (a)180°
  • (b)270°
  • (c)360°
  • (d)300°
Show solution

Solution : The correct answer is (c).

example-5Multiple choice

In a square ABCD, the diagonals meet at point O. The ∆AOB is

  • (a)isosceles right triangle
  • (b)equilateral triangle
  • (c)isosceles triangle but not right triangle
  • (d)scalene right triangle.
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Solution : The correct answer is (a). Quadrilaterals with certain properties are given additional names. A trapezium has exactly 1 pair of parallel sides. A parallelogram has 2 pairs of parallel sides. A rectangle has 4 right angles. A rhombus has 4 congruent sides. A square has 4 congruent sides and 4 right angles.

example-6Multiple choice

ABCD is a quadrilateral in which AB = 5 cm, CD = 8 cm and the sum of angle A and angle D is 180°. What is the name of this quadrilateral?

  • (a)Parallelogram
  • (b)Trapezium
  • (c)Rhombus
  • (d)Can not be determined
Show solution

Solution : The correct answer is (b).

example-7Multiple choice

Rukmini has a farm land which is triangular in shape. What is the sum of all the exterior angles taken in an order of the farm land?

  • (a)90°
  • (b)180°
  • (c)360°
  • (d)Can not be determined.
Show solution

Solution : The correct answer is (c).

example-8Multiple choice

How many sides does an octagon have?

  • (A)7
  • (b)8
  • (c)9
  • (d)10
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Solution : The correct answer is (b) In examples 9 and 13, fill in the blanks to make the statements true.

example-9Fill in the blanks

The diagonals of a rhombus bisect each other at _____ angles.

Show solution

Solution : Right.

example-10Fill in the blanks

For getting diagonals through vertex A of a pentagon ABCDE, A is joined to _________.

Show solution

Solution : C and D.

example-11Fill in the blanks

For constructing a unique quadrilateral at least __________ measurements are required.

Show solution

Solution : Five.

example-12Fill in the blanks

If diagonals of a quadrilateral bisect at right angles it is a __________.

Show solution

Solution : Rhombus (or square).

example-13Fill in the blanks

The diagonals of a __________ intersect at right angles.

Show solution

Solution : Kite. In examples 14 to 23, state whether the statements are true (T) or false (F).

example-14Short answer

Every rectangle is a parallelogram.

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Solution : True.

example-15Short answer

Every rhombus is a kite.

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Solution : True.

example-16Short answer

Every parallelogram is a trapezuim.

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Solution : True.

example-17Short answer

Every kite is a trapezium.

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Solution : False.

example-18Short answer

Every kite is a parallelogram.

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Solution : False.

example-19Short answer

Diagonals of a rectangle are perpendicular to each other.

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Solution : False.

example-20Short answer

For constructing a unique parallelogram lengths of only two sides should be given.

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Solution : False. Diagonals of a — Parallelogram bisect each other bisect each other are perpendicular Rhombus to each other are equal Rectangle bisect each other bisect each other are perpendicular to Square each other are equal

example-21Short answer

is a simple closed curve.

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Solution : False.

example-22Short answer

is a concave polygon.

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Solution : True.

example-23Short answer

A triangle is not a polygon.

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Solution : False.

example-24Short answer

The sides AB and CD of a quadrilateral ABCD are extended to points P and Q respectively. Is ∠ADQ + ∠CBP = ∠A + ∠C? Give reason.

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Solution : Join AC, then Q C ∠CBP = ∠BCA + ∠BAC and ∠ADQ = ∠ACD + ∠DAC (Exterior angles of triangles) A B P Therefore, ∠CBP + ∠ADQ = ∠BCA + ∠BAC + ∠ACD + ∠DAC = (∠BCA + ∠ACD) + (∠BAC + ∠DAC) = ∠C + ∠A Angles in a Quadrilateral A diagonal of a quadrilateral is a segment that joins two vertices of the quadrilateral but is not a side. You can use a diagonal of a quadrilateral to show that the sum of the angle measures in a quadrilateral is 360°. Cut a quadrilateral The sum of the angle Quadrilateral with along a diagonal to measures in each 2 pairs of parallel form two triangles. triangle is 180°. sides.

example-25Short answer

If AM and CN are perpendiculars on the diagonal BD of a parallelogram ABCD, Is ∆AMD ≅ ∆CNB? Give reason.

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Solution : In triangles AMD and CNB, AD = BC (opposite sides of parallelogram) ∠AMB = ∠CNB = 900 ∠ADM = ∠NBC (AD || BC and BD is transversal.) So, ∆AMD ≅ ∆CNB (AAS)

example-26Short answer

Construct a quadrilateral ABCD in which AB = AD = 5cm, BC = CD = 7cm and BD = 6cm. What type of quadrilateral is this?

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Solution : Looking at the rough figure, draw a line segment BD = 6cm. Taking B and D as centres and 5 cm radius, draw arcs to intersect at the point A, then taking B and D as centres and 7 cm radius, draw arcs in the opposite side of A to intersect at the point C. Join AB, AD and BC, DC. Then ABCD is the required quadrilateral. It is a kite.

example-27Short answer

Find x in the following figure.

Show solution

Solution : In the given figure ∠1 + 90° = 180° (linear pair) ∠1 = 90° Now, sum of exterior angles of a polygon is 360°, therefore, x + 60° + 90° + 90° + 40° = 360° x + 280° = 360° x = 80° Classifying Plane Figures Triangle Trapezoid Parallelogram Closed figure with 3 Quadrilateral with Quadrilateral with straight sides that 1 pair of parallel 2 pairs of parallel connect 3 points sides sides Rhombus Rectangle Square Circle Parallelogram Set of all points Parallelogram Parallelogram with 4 sides of in a plane that with 4 sides of with 4 right equal length and are at the same equal length angles 4 right angles distance from a fixed point

example-28Short answer

Two adjacent angles of a parallelogram are in the ratio 4:5. Find their measures.

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Solution : Let the angles be 4x and 5x. Then, 4x + 5x = 180° 9x = 180° x = 20° So, angles are 4 × 20° = 80° and 5 × 20° =100°.

example-29Short answer

The four angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find the angles.

Show solution

Solution : Let angles be 3x, 4x, 5x, 6x. Thus, 3x + 4x + 5x + 6x = 360° since sum of the angles of a quadrilateral is 360°. So, 18x = 360° or, x = 20° Thus, angles are 60°, 80°, 100°, 120°.

example-30Short answer

In a parallelogram PQRS, the bisectors of ∠P and ∠Q meet at O. Find ∠POQ.

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Solution : Since OP and OQ are the bisectors of ∠P and ∠Q respectively (see figure on the right),

example-31Short answer

Three angles of a quadrilateral are 50°, 40° and 123°. Find its fourth angle.

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Solution : Let fourth angle be x. Then 500 + 400 + 1230 + x = 3600. or x = 3600 – 500 – 400 – 1230 = 3600 – 2130 = 1470. A quadrilateral is a closed plane figure with four sides that are line segments. The figures below are special types of quadrilaterals. Special Quadrilaterals Diagram Trapezium A trapezium is a quadrilateral with exactly 1 pair of parallel sides. Parallelogram A Parallelogram is a quadrilateral with 2 pairs of parallel sides. Rhombus A rhombus is a parallelogram with

example-32Short answer

The ratio of exterior angle to interior angle of a regular polygon is 1:4. Find the number of sides of the polygon.

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Solution : Let the exterior angle of the polygon be x Then, the interior angle of polygon = 180° – x According to question, x 1 180 x 4 or, 4x = 180° – x or, 5x = 180° 180 or, x = So, x = 36° 360 Number of sides of polygon = exterior angle 360 = = 10 36

example-33Short answer

Each interior angle of a polygon is 108°. Find the number of sides of the polygon.

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Solution : Since interior angle = 108° so, exterior angle = 1800 – 1080 = 72° 360 3600 Number of sides = 5 exterior angle 720

example-34Short answer

Construct a rhombus PAIR, given that PA = 6 cm and angle ∠A = 110°.

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Solution : Since in a rhombus, all sides are equal so, PA = AI = IR = RP = 6cm Also, rhombus is a parallelogram so, adjacent angle, ∠I = 180° – 110° = 70° Steps of construction 1. Draw AI = 6 cm 2. Draw ray AX such that ∠IAX = 110° and draw IY such that ∠AIY = 70°. 3. With A and I as centres and radius 6cm draw arcs intersecting AX and IY at P and R respectively. 4. Join PR. Thus, PAIR is the required rhombus.

example-35Short answer

One of the diagonals of a rhombus and its sides are equal. Find the angles of the rhombus.

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Solution : Let PQRS be a rhombus such that its diagonal PR is equal to its side, that is, PQ = QR = RS = PS = PR So, ∆PRS and ∆PQR are equilateral. ∠S = ∠Q = 60° [Each angle of an equilateral triangle is 60°.] ∠P = ∠1 + ∠2 = 60° + 60° = 120° = ∠R Hence ∠S = ∠Q = 60° and ∠P = ∠R = 120°

example-36Short answer

In the figure, HOPE is a rectangle. Its diagonals meet at G. If HG = 5x + 1 and EG = 4x + 19, find x.

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Solution : Since diagonals of a rectangle bisect each other, HP = 2HG = 2 (5x + 1) = 10x +2 and OE = 2EG = 2(4x +19) = 8x + 38 Diagonals of a rectangle are equal. So HP = OE or 10x + 2 = 8x + 38 or 2x = 36 or x = 18

example-37Short answer

Application on the problem strategy C RICE is a rhombus. Find x, y, z. Justify your findings. Hence, find the perimeter of the rhombus.

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Solution : Understand and explore the problem E l 5 O x+2 We have to find the values of x, y, z. i.e. OE, OY and side IR of the rhombus y+x and perimeter of the rhombus. What do we know? R RICE is a rhombus and OC = 12, OE = 5, OI = x + 2, OR = x + y Plan a strategy (1) We have to find the parts of the diagonal. Use diagonals of a rhombus bisect each other. (2) We have to find the side of the rhombus. We use diagonals intersect at right angles and apply pythagoras theorem. (3) Since all sides of a rhombus are equal, perimeter of the rhombus = 4 × side. Solve Step 1. OI = OE ⇒ x + 2 = 5 or x = 5 – 2 = 3. OC = OR ⇒ 12 = y + x or y = 12 – x 12 – 3 = 9 Step 2. EOR is a right triangle ER2 = OE2 + OR2 = 52 + 122 = 25 + 144 = 169 ER = 169 = 13cm Step 3. Since all sides of a rhombus are equal. ∴ RE = RI = IC = CE = 13 cm. Perimeter of RICE = 4 × RE = 4 × 13 cm = 52 cm Revise We have been asked to find x, y and z and we have found that. Checking x + 2 = 5 and x = 3 ⇒ 3 + 2 = 5 Hence value of x is correct. x + y = 12 x = 3 and y = 9 and 3 + 9 = 12 ⇒ value of y is correct. Perimeter of rhombus = 2 d12 + d22 (where d1 and d2 are diagonals) = 2 242 + 102 = 2 576 + 100 = 2 676 = 52 cm (i) If RICE is a parallelogram, not a rhombus can you find x, y and z ? (ii) If RICE is a rhombus with EC = 20 cm and OC = 12 cm, can you find x, y, z ?

example-38Short answer

Application on the problem solution strategy Construct a rhombus with side 4.5cm and diagonal 6cm.

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Solution : Understand and explore the problem What do you know? Here, side of rhombus = 4.5 cm. Diagonal of rhombus = 6 cm. What do we need to make rhombus? 4 sides and its one diagonal Plan a strategy (1) Use property of rhombus— all sides are equal. (2) Make a free hand rough sketch and name it ABCD. Solve Step-1. Draw AB = 4.5 cm. Step-2. With A as centre and radius

Questions

Q1Short answer

1 so, ∠OPQ = ∠P and ∠OQP = ∠Q

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(a) x = 3 (b) Y = 2 (c) Z = 2 (d) P = 1 (e) Q = 6 (f) R = 2

Q2Short answer

2 In ∆POQ, ∠OPQ + ∠PQO + ∠POQ = 180° (Angle sum property) 1 1 i.e. ∠P + ∠POQ + ∠Q = 180° 2 2 i.e. ∠POQ = 180° – (∠P + ∠Q) = 180° – × 180° = 90°

Show answer

1 2

Q4Short answer

sides of equal length. Rectangle A rectangle is a parallelogram with 4 right angles. Square A square is a parallelogram with 4 sides of equal length and 4 right angles.

Show answer

(1) 6 (2) 1 (3) -1 (4) (5) 60 2 7 7 4 (6) -5 (7) - (8) 4 (9) 5 (10) 42 5 5

Q6Multiple choice

cm draw an arc above AB. Step-3. With B as centre draw an arc to cut the arc drawn in step 2 at pt C. Step-4. Join AC and BC. Step-5. With A and C as centre and radius 4.5 cm draw arcs to intersect each other at D. Step-6. ABCD is required rhombus. Checking: Verify your figure by adopting some other property of rhombus. Step 1. Join BD to intersect AC as O. Step 2. Measure ∠AOB. Is it 90°? Step 3. Measure OA and OC. Are they equal? Step 4. Measure OB and OD. Are they equal? If your answer to 2, 3, 4 is yes it means what you have constructed is a right angle. 1. Can you draw this rhombus by using some other property? 2. Can you draw a parallelogram with given measurement? 3. How will you construct this rhombus if instead of side 4.5 cm diagonal 4.5 cm is given? In questions 1 to 52, there are four options, out of which one is correct. Write the correct answer. 1. If three angles of a quadrilateral are each equal to 75°, the fourth angle is

  • (a)150°
  • (b)135°
  • (c)45°
  • (d)75° 2. For which of the following, diagonals bisect each other? (a) Square (b) Kite (c) Trapezium (d) Quadrilateral 3. For which of the following figures, all angles are equal? (a) Rectangle (b) Kite (c) Trapezium (d) Rhombus 4. For which of the following figures, diagonals are perpendicular to each other? (a) Parallelogram (b) Kite (c) Trapezium (d) Rectangle 5. For which of the following figures, diagonals are equal? (a) Trapezium (b) Rhombus (c) Parallelogram (d) Rectangle 6. Which of the following figures satisfy the following properties? - All sides are congruent. - All angles are right angles. - Opposite sides are parallel. (a) P (b) Q (c) R (d) S
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Multiplication

Q7Multiple choice

Which of the following figures satisfy the following property? - Has two pairs of congruent adjacent sides.

  • (a)P
  • (b)Q
  • (c)R
  • (d)S
Show answer

Equivalent equation

Q8Multiple choice

Which of the following figures satisfy the following property? - Only one pair of sides are parallel.

  • (a)P
  • (b)Q
  • (c)R
  • (d)S
Show answer

Identity

Q9Multiple choice

Which of the following figures do not satisfy any of the following properties? - All sides are equal. - All angles are right angles. - Opposite sides are parallel.

  • (a)P
  • (b)Q
  • (c)R
  • (d)S
Show answer

Algebra

Q10Multiple choice

Which of the following properties describe a trapezium?

  • (a)A pair of opposite sides is parallel.
  • (b)The diagonals bisect each other.
  • (c)The diagonals are perpendicular to each other.
  • (d)The diagonals are equal.
Show answer

Standard form

Q11Multiple choice

Which of the following is a property of a parallelogram?

  • (a)Opposite sides are parallel.
  • (b)The diagonals bisect each other at right angles.
  • (c)The diagonals are perpendicular to each other.
  • (d)All angles are equal.
Show answer

(a) Opposite sides are parallel.

(d) All angles are equal.

(b) The diagonals bisect each other at right angles.

(c) The diagonals are perpendicular to each other.

Q12Multiple choice

What is the maximum number of obtuse angles that a quadrilateral can have ?

  • (a)1
  • (b)2
  • (c)3
  • (d)4
Show answer

(c) 3

Q13Multiple choice

How many non-overlapping triangles can we make in a n-gon (polygon having n sides), by joining the vertices?

  • (a)n –1
  • (b)n –2
  • (c)n –3
  • (d)n –4
Show answer

(b) n –2

Q14Multiple choice

What is the sum of all the angles of a pentagon?

  • (a)180°
  • (b)360°
  • (c)540°
  • (d)720°
Show answer

(c) 540°

Q15Multiple choice

What is the sum of all angles of a hexagon?

  • (a)180°
  • (b)360°
  • (c)540°
  • (d)720°
Show answer

(d) 720°

Q16Multiple choice

If two adjacent angles of a parallelogram are (5x – 5)° and (10x + 35)°, then the ratio of these angles is

  • (a)1 : 3
  • (b)2 : 3
  • (c)1 : 4
  • (d)1 : 2
Show answer

(a) 1 : 3

Q17Multiple choice

A quadrilateral whose all sides are equal, opposite angles are equal and the diagonals bisect each other at right angles is a __________.

  • (a)rhombus
  • (b)parallelogram
  • (c)square
  • (d)rectangle
Show answer

(a) rhombus

Q18Multiple choice

A quadrialateral whose opposite sides and all the angles are equal is a

  • (a)rectangle
  • (b)parallelogram
  • (c)square
  • (d)rhombus
Show answer

(a) rectangle

Q19Multiple choice

A quadrilateral whose all sides, diagonals and angles are equal is a

  • (a)square
  • (b)trapezium
  • (c)rectangle
  • (d)rhombus
Show answer

(a) square

Q20Multiple choice

How many diagonals does a hexagon have?

  • (a)9
  • (b)8
  • (c)2
  • (d)6
Show answer

(a) 9

Q21Multiple choice

If the adjacent sides of a parallelogram are equal then parallelogram

  • (a)rectangle
  • (b)trapezium
  • (c)rhombus
  • (d)square
Show answer

(c) rhombus

Q22Multiple choice

If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a

  • (a)rhombus
  • (b)rectangle
  • (c)square
  • (d)parallelogram
Show answer

(b) rectangle

Q23Multiple choice

The sum of all exterior angles of a triangle is

  • (a)180°
  • (b)360°
  • (c)540°
  • (d)720°
Show answer

(b) 360°

Q24Multiple choice

Which of the following is an equiangular and equilateral polygon?

  • (a)Square
  • (b)Rectangle
  • (c)Rhombus
  • (d)Right triangle
Show answer

(a) Square

Q25Multiple choice

Which one has all the properties of a kite and a parallelogram?

  • (a)Trapezium
  • (b)Rhombus
  • (c)Rectangle
  • (d)Parallelogram
Show answer

(b) Rhombus

Q26Multiple choice

The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is

  • (a)72°
  • (b)144°
  • (c)36°
  • (d)18°
Show answer

(c) 36°

Q27Multiple choice

In the trapezium ABCD, the measure of ∠D is

  • (a)55°
  • (b)115°
  • (c)135°
  • (d)125°
Show answer

(d) 125°

Q28Multiple choice

A quadrilateral has three acute angles. If each measures 80°, then the measure of the fourth angle is

  • (a)150°
  • (b)120°
  • (c)105°
  • (d)140°
Show answer

(b) 120°

Q29Multiple choice

The number of sides of a regular polygon where each exterior angle has a measure of 45° is

  • (a)8
  • (b)10
  • (c)4
  • (d)6
Show answer

(a) 8

Q30Multiple choice

In a parallelogram PQRS, if ∠P = 60°, then other three angles are

  • (a)45°, 135°, 120°
  • (b)60°, 120°, 120°
  • (c)60°, 135°, 135°
  • (d)45°, 135°, 135°
Show answer

(b) 60°, 120°, 120°

Q31Multiple choice

If two adjacent angles of a parallelogram are in the ratio 2 : 3, then the measure of angles are

  • (a)72°, 108°
  • (b)36°, 54°
  • (c)80°, 120°
  • (d)96°, 144°
Show answer

(a) 72°, 108°

Q32Multiple choice

If PQRS is a parallelogram, then ∠P – ∠R is equal to

  • (a)60°
  • (b)90°
  • (c)80°
  • (d)0°
Show answer

(d) 0°

Q33Multiple choice

The sum of adjacent angles of a parallelogram is

  • (a)180°
  • (b)120°
  • (c)360°
  • (d)90°
Show answer

(a) 180°

Q34Multiple choice

The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30°. The measure of the obtuse angle is

  • (a)100°
  • (b)150°
  • (c)105°
  • (d)120°
Show answer

(b) 150°

Q35Multiple choice

In the given figure, ABCD and BDCE are parallelograms with common base DC. If BC ⊥ BD, then ∠BEC =

  • (a)60°
  • (b)30°
  • (c)150°
  • (d)120°
Show answer

(a) 60°

Q36Multiple choice

Length of one of the diagonals of a rectangle whose sides are 10 cm and 24 cm is

  • (a)25 cm
  • (b)20 cm
  • (c)26 cm
  • (d)3.5 cm
Show answer

(c) 26 cm

Q37Multiple choice

If the adjacent angles of a parallelogram are equal, then the parallelogram is a

  • (a)rectangle
  • (b)trapezium
  • (c)rhombus
  • (d)any of the three
Show answer

(a) rectangle

Q38Multiple choice

Which of the following can be four interior angles of a quadrilateral?

  • (a)140°, 40°, 20°, 160°
  • (b)270°, 150°, 30°, 20°
  • (c)40°, 70°, 90°, 60°
  • (d)110°, 40°, 30°, 180°
Show answer

(a) 140°, 40°, 20°, 160°

Q39Multiple choice

The sum of angles of a concave quadrilateral is

  • (a)more than 360°
  • (b)less than 360°
  • (c)equal to 360°
  • (d)twice of 360°
Show answer

(c) equal to 360°

Q40Multiple choice

Which of the following can never be the measure of exterior angle of a regular polygon?

  • (a)22°
  • (b)36°
  • (c)45°
  • (d)30°
Show answer

(a) 22°

Q41Multiple choice

In the figure, BEST is a rhombus, Then the value of y – x is

  • (a)40°
  • (b)50°
  • (c)20°
  • (d)10°
Show answer

(a) 40°

Q42Multiple choice

The closed curve which is also a polygon is

  • (a)
  • (b)
  • (c)
  • (d)
Show answer

(a)

Q43Multiple choice

Which of the following is not true for an exterior angle of a regular polygon with n sides? 360°

  • (a)Each exterior angle =
  • (b)Exterior angle = 180° – interior angle 360 °
  • (c)n = exterior angle (n – 2) × 180°
  • (d)Each exterior angle =
Show answer

(d) Each exterior angle =

Q44Multiple choice

PQRS is a square. PR and SQ intersect at O. Then ∠POQ is a

  • (a)Right angle
  • (b)Straight angle
  • (c)Reflex angle
  • (d)Complete angle
Show answer

(a) Right angle

Q45Multiple choice

Two adjacent angles of a parallelogram are in the ratio 1:5. Then all the angles of the parallelogram are

  • (a)30°, 150°, 30°, 150°
  • (b)85°, 95°, 85°, 95°
  • (c)45°, 135°, 45°, 135°
  • (d)30°, 180°, 30°, 180°
Show answer

(a) 30°, 150°, 30°, 150°

Q46Multiple choice

A parallelogram PQRS is constructed with sides QR = 6 cm, PQ = 4 cm and ∠PQR = 90°. Then PQRS is a

  • (a)square
  • (b)rectangle
  • (c)rhombus
  • (d)trapezium
Show answer

(b) rectangle

Q47Multiple choice

The angles P, Q, R and S of a quadrilateral are in the ratio 1:3:7:9. Then PQRS is a

  • (a)parallelogram
  • (b)trapezium with PQ || RS
  • (c)trapezium with QR||PS
  • (d)kite
Show answer

(b) trapezium with PQ || RS

Q48Multiple choice

PQRS is a trapezium in which PQ||SR and ∠P=130°, ∠Q=110°. Then ∠R is equal to:

  • (a)70°
  • (b)50°
  • (c)65°
  • (d)55°
Show answer

(a) 70°

Q49Multiple choice

The number of sides of a regular polygon whose each interior angle is of 135° is

  • (a)6
  • (b)7
  • (c)8
  • (d)9
Show answer

(c) 8

Q50Multiple choice

If a diagonal of a quadrilateral bisects both the angles, then it is a

  • (a)kite
  • (b)parallelogram
  • (c)rhombus
  • (d)rectangle
Show answer

(c) rhombus

Q51Multiple choice

To construct a unique parallelogram, the minimum number of measurements required is

  • (a)2
  • (b)3
  • (c)4
  • (d)5
Show answer

(b) 3

Q52Multiple choice

To construct a unique rectangle, the minimum number of measurements required is

  • (a)4
  • (b)3
  • (c)2
  • (d)1 In questions 53 to 91, fill in the blanks to make the statements true.
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(c) 2

Q53Fill in the blanks

In quadrilateral HOPE, the pairs of opposite sides are __________.

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HO and EP, PO and EH

Q54Fill in the blanks

In quadrilateral ROPE, the pairs of adjacent angles are __________.

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RO and OP, OP and PE, PE and ER, ER and RO

Q55Fill in the blanks

In quadrilateral WXYZ, the pairs of opposite angles are __________.

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∠W and ∠Y, ∠X and ∠Z 56.DF and EG

Q56Fill in the blanks

The diagonals of the quadrilateral DEFG are __________ and __________.

Q57Fill in the blanks

The sum of all __________ of a quadrilateral is 360°.

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Angles

Q58Fill in the blanks

The measure of each exterior angle of a regular pentagon is __________.

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720

Q59Fill in the blanks

Sum of the angles of a hexagon is __________.

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7200

Q60Fill in the blanks

The measure of each exterior angle of a regular polygon of 18 sides is __________.

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200

Q61Fill in the blanks

The number of sides of a regular polygon, where each exterior angle has a measure of 36°, is __________.

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100

Q62Fill in the blanks

is a closed curve entirely made up of line segments. The another name for this shape is __________.

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Concave Polygon

Q63Fill in the blanks

A quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure is __________.

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Kite

Q64Fill in the blanks

The measure of each angle of a regular pentagon is __________.

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1080

Q65Fill in the blanks

The name of three-sided regular polygon is __________.

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An equilateral triangle

Q66Fill in the blanks

The number of diagonals in a hexagon is __________.

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9

Q67Fill in the blanks

A polygon is a simple closed curve made up of only __________.

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Line segments

Q68Fill in the blanks

A regular polygon is a polygon whose all sides are equal and all __________ are equal.

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Angles

Q69Fill in the blanks

The sum of interior angles of a polygon of n sides is __________right angles.

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2n–4

Q70Fill in the blanks

The sum of all exterior angles of a polygon is __________.

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3600

Q71Fill in the blanks

__________ is a regular quadrilateral.

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Square

Q72Fill in the blanks

A quadrilateral in which a pair of opposite sides is parallel is __________.

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Trapezium

Q73Fill in the blanks

If all sides of a quadrilateral are equal, it is a __________.

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Rhombus, Square

Q74Fill in the blanks

In a rhombus diagonals intersect at __________ angles.

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Right

Q75Fill in the blanks

__________ measurements can determine a quadrilateral uniquely.

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5

Q76Fill in the blanks

A quadrilateral can be constructed uniquely if its three sides and __________ angles are given.

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2 included

Q77Fill in the blanks

A rhombus is a parallelogram in which __________ sides are equal.

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All

Q78Fill in the blanks

The measure of __________ angle of concave quadrilateral is more than 180°.

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1

Q79Fill in the blanks

A diagonal of a quadrilateral is a line segment that joins two __________ vertices of the quadrilateral.

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Opposite

Q80Fill in the blanks

The number of sides in a regular polygon having measure of an exterior angle as 72° is __________.

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5

Q81Fill in the blanks

If the diagonals of a quadrilateral bisect each other, it is a __________.

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Parallelogram

Q82Fill in the blanks

The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is __________.

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28cm

Q83Fill in the blanks

A nonagon has __________ sides.

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9

Q84Fill in the blanks

Diagonals of a rectangle are __________.

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Equal

Q85Fill in the blanks

A polygon having 10 sides is known as __________.

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Decagon

Q86Fill in the blanks

A rectangle whose adjacent sides are equal becomes a __________.

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Square

Q87Fill in the blanks

If one diagonal of a rectangle is 6 cm long, length of the other diagonal is __________.

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6cm

Q88Fill in the blanks

Adjacent angles of a parallelogram are __________.

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Supplementary

Q89Fill in the blanks

If only one diagonal of a quadrilateral bisects the other, then the quadrilateral is known as __________.

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Kite

Q90Fill in the blanks

In trapezium ABCD with AB||CD, if ∠A = 100°, then ∠D = __________.

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800

Q91Fill in the blanks

The polygon in which sum of all exterior angles is equal to the sum of interior angles is called __________. In questions 92 to 131 state whether the statements are true (T) or (F) false.

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Quadrilateral

Q92Short answer

All angles of a trapezium are equal.

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False

Q93Short answer

All squares are rectangles.

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True

Q94Short answer

All kites are squares.

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False

Q95Short answer

All rectangles are parallelograms.

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True

Q96Short answer

All rhombuses are squares.

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False

Q97Short answer

Sum of all the angles of a quadrilateral is 180°.

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False

Q98Short answer

A quadrilateral has two diagonals.

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True

Q99Short answer

Triangle is a polygon whose sum of exterior angles is double the sum of interior angles.

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True

Q100Short answer

is a polygon.

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False

Q101Short answer

A kite is not a convex quadrilateral.

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True

Q102Short answer

The sum of interior angles and the sum of exterior angles taken in an order are equal in case of quadrilaterals only.

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True

Q103Short answer

If the sum of interior angles is double the sum of exterior angles taken in an order of a polygon, then it is a hexagon.

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True

Q104Short answer

A polygon is regular if all of its sides are equal.

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False

Q105Short answer

Rectangle is a regular quadrilateral.

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False

Q106Short answer

If diagonals of a quadrilateral are equal, it must be a rectangle.

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False

Q107Short answer

If opposite angles of a quadrilateral are equal, it must be a parallelogram.

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False

Q108Short answer

The interior angles of a triangle are in the ratio 1:2:3, then the ratio of its exterior angles is 3:2:1.

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False

Q109Short answer

is a concave pentagon.

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False

Q110Short answer

Diagonals of a rhombus are equal and perpendicular to each other.

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False

Q111Short answer

Diagonals of a rectangle are equal.

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True

Q112Short answer

Diagonals of rectangle bisect each other at right angles.

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False

Q113Short answer

Every kite is a parallelogram.

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False

Q114Short answer

Every trapezium is a parallelogram.

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False

Q115Short answer

Every parallelogram is a rectangle.

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False

Q116Short answer

Every trapezium is a rectangle.

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False

Q117Short answer

Every rectangle is a trapezium.

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True

Q118Short answer

Every square is a rhombus.

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True

Q119Short answer

Every square is a parallelogram.

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True

Q120Short answer

Every square is a trapezium.

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True

Q121Short answer

Every rhombus is a trapezium.

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True

Q122Short answer

A quadrilateral can be drawn if only measures of four sides are given.

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False

Q123Short answer

A quadrilateral can have all four angles as obtuse.

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False

Q124Short answer

A quadrilateral can be drawn if all four sides and one diagonal is known.

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True

Q125Short answer

A quadrilateral can be drawn when all the four angles and one side is given.

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False

Q126Short answer

A quadrilateral can be drawn if all four sides and one angle is known.

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True

Q127Short answer

A quadrilateral can be drawn if three sides and two diagonals are given.

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True

Q128Short answer

If diagonals of a quadrilateral bisect each other, it must be a parallelogram.

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True

Q129Short answer

A quadrilateral can be constructed uniquely if three angles and any two sides are given.

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True

Q130Short answer

A parallelogram can be constructed uniquely if both diagonals and the angle between them is given.

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True

Q131Short answer

A rhombus can be constructed uniquely if both diagonals are given. Solve the following :

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True

Q132Short answer

The diagonals of a rhombus are 8 cm and 15 cm. Find its side.

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8.5cm

Q133Short answer

Two adjacent angles of a parallelogram are in the ratio 1:3. Find its angles.

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450, 1350, 450, 1350

Q134Short answer

Of the four quadrilaterals— square, rectangle, rhombus and trapezium— one is somewhat different from the others because of its design. Find it and give justification.

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Trapezium, Others are parallelogram

Q135Short answer

In a rectangle ABCD, AB = 25 cm and BC = 15. In what ratio does the bisector of ∠C divide AB?

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2:3

Q136Short answer

PQRS is a rectangle. The perpendicular ST from S on PR divides ∠S in the ratio 2:3. Find ∠TPQ.

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360

Q137Short answer

A photo frame is in the shape of a quadrilateral. With one diagonal longer than the other. Is it a rectangle? Why or why not?

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No, in a rectangle diagonals are equal.

Q138Short answer

The adjacent angles of a parallelogram are (2x – 4)° and (3x – 1)°. Find the measures of all angles of the parallelogram.

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700, 1100, 700, 1100

Q139Short answer

The point of intersection of diagonals of a quadrilateral divides one diagonal in the ratio 1:2. Can it be a parallelogram? Why or why not?

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No, diagonals of a parallelogram bisect each other i.e. in the ratio 1:1.

Q140Short answer

The ratio between exterior angle and interior angle of a regular polygon is 1:5. Find the number of sides of the polygon.

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12

Q141Short answer

Two sticks each of length 5 cm are crossing each other such that they bisect each other. What shape is formed by joining their end points? Give reason.

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Parallelogram

Q142Short answer

Two sticks each of length 7 cm are crossing each other such that they bisect each other at right angles. What shape is formed by joining their end points? Give reason.

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Rhombus

Q143Short answer

A playground in the town is in the form of a kite. The perimeter is 106 metres. If one of its sides is 23 metres, what are the lengths of other three sides?

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23 cm, 30 cm, 30 cm

Q144Short answer

In rectangle READ, find ∠EAR, ∠RAD and ∠ROD R D 60° O E A

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300, 600, 1200

Q145Short answer

In rectangle PAIR, find ∠ARI, ∠RMI and ∠PMA.

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550, 700, 700

Q146Short answer

In parallelogram ABCD, find ∠B, ∠C and ∠D.

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1000, 800, 1000

Q147Short answer

In parallelogram PQRS, O is the mid point of SQ. Find ∠S, ∠R, PQ, QR and diagonal PR. 15 cm S R 11 cm m O 6c 60° P Q Y

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1200, 600, 15 cm, 11 cm, 12 cm, 52 cm

Q148Short answer

In rhombus BEAM, find ∠AME and ∠AEM.

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200, 200

Q149Short answer

In parallelogram FIST, find ∠SFT, ∠OST and ∠STO.

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450, 750, 350

Q150Short answer

In the given parallelogram YOUR, ∠RUO = 120° and OY is extended to point S such that ∠SRY = 50°. Find ∠YSR.

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700

Q151Short answer

In kite WEAR, ∠WEA = 70° and ∠ARW = 80°. Find the remaining two angles.

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150 each

Q152Multiple choice

A rectangular MORE is shown below: Answer the following questions by giving appropriate reason.

  • (i)Is RE = OM?
  • (ii)Is ∠MYO = ∠RXE?
  • (iii)Is ∠MOY = ∠REX?
  • (iv)Is ∆MYO ≅ ∆RXE? (v) Is MY = RX?
Show answer

(i) Is RE = OM?

(ii) Is ∠MYO = ∠RXE?

(iii) Is ∠MOY = ∠REX?

(iv) Is ∆MYO ≅ ∆RXE? (v) Is MY = RX?

Q153Short answer

In parallelogram LOST, SN⊥OL and SM⊥LT. Find ∠STM, ∠SON and ∠NSM.

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500, 500, 500

Q154Short answer

In trapezium HARE, EP and RP are bisectors of ∠E and ∠R respectively. Find ∠HAR and ∠EHA. E R 25° 30° H A

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1200

Q155Short answer

In parallelogram MODE, the bisector of ∠M and ∠O meet at Q, find the measure of ∠MQO.

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900

Q156Short answer

A playground is in the form of a rectangle ATEF. Two players are standing at the points F and B where EF = EB. Find the values of x and y.

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1350, 450

Q157Short answer

In the following figure of a ship, ABDH and CEFG are two parallelograms. Find the value of x.

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1000

Q158Short answer

A Rangoli has been drawn on a flor of a house. ABCD and PQRS both are in the shape of a rhombus. Find the radius of semicircle drawn on each side of rhombus ABCD.

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2.5

Q159Short answer

ABCDE is a regular pentagon. The bisector of angle A meets the side CD at M. Find ∠AMC

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900

Q160Short answer

Quadrilateral EFGH is a rectangle in which J is the point of intersection of the diagonals. Find the value of x if JF = 8x + 4 and EG = 24x – 8.

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x = 2

Q161Short answer

Find the values of x and y in the following parallelogram.

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x = 100, y = 200

Q162Short answer

Find the values of x and y in the following kite.

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x = 800, y = 1100

Q163Short answer

Find the value of x in the trapezium ABCD given below.

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x = 800

Q164Short answer

Two angles of a quadrilateral are each of measure 75° and the other two angles are equal. What is the measure of these two angles? Name the possible figures so formed.

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1050 each, Parallelogram

Q165Short answer

In a quadrilateral PQRS, ∠P = 50°, ∠Q = 50°, ∠R = 60°. Find ∠S. Is this quadrilateral convex or concave?

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2000, concave

Q166Short answer

Both the pairs of opposite angles of a quadrilateral are equal and supplementary. Find the measure of each angle.

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900

Q167Short answer

Find the measure of each angle of a regular octagon.

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1350 QTNN

Q168Short answer

Find the measure of an are exterior angle of a regular pentagon and an exterior angle of a regular decagon. What is the ratio between these two angles?

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Ext. angle of regular pentagon = = 720 QTNN Ext. angle of regular decagon = = 360 720 = 2 × 360

Q169Short answer

In the figure, find the value of x.

Show answer

740

Q170Short answer

Three angles of a quadrilateral are equal. Fourth angle is of measure 120°. What is the measure of equal angles?

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800 1 1

Q171Short answer

In a quadrilateral HOPE, PS and ES are bisectors of ∠P and ∠E respectively. Give reason.

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Yes, ∠E + ∠P = 1800 – ∠PSE ⇒ ∠E + ∠P = 3600 – 2∠PSE 2 2 and ∠E + ∠P + ∠O + ∠H = 3600 ⇒ 3600 – 2∠PSE + ∠O + ∠H = 3600

Q172Short answer

ABCD is a parallelogram. Find the value of x, y and z.

Show answer

x = 900, y = 600, z = 300

Q173Short answer

Diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Give a figure to justify your answer.

Show answer

False Trap ABCD  BC in which AD

Q174Short answer

ABCD is a trapezium such that AB||CD, ∠A : ∠D = 2 :1, ∠B : ∠C = 7 : 5. Find the angles of the trapezium.

Show answer

∠A = 1200, ∠B = 1050, ∠C = 750, ∠D = 600 m

Q175Short answer

A line l is parallel to line m and a transversal p interesects them at X, Y respectively. Bisectors of interior angles at X and Y interesct at P and Q. Is PXQY a rectangle? Given reason.

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l ∠DXY = ∠XYA (alt int. ∠S) ∠DXY ∠XYA = (÷2) 2 2 ∠1 = ∠2 (XP and YQ are bisectors) QY ∴ XP (1)  PY Similarly XQ (2) From (1) and (2) PXQY is a parallelogram ∠DXY + ∠XYB = 1800 ∠DXY ∠XYB 1800 + = (÷ by 2) 2 2 2 ∠1 + ∠3 = 900 (4) In ∆XYP ∠1 + ∠3 + ∠P = 1800 900 + ∠P = 1800 (from 4) ∠P = 900 From (3) and (5), PXQY is a rectangle

Q176Short answer

ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.

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∠A = ∠C (opp. Ls of a gm) ∠A ∠C = (÷2) 2 2 ∠1 = ∠2 But ∠2 = ∠3 (all ∠s) ∴ ∠1 = ∠3 But they are a pair of corresponding ∠s  YC ∴ AX (1) XC AY  DC) (2) (AB From (1) and (2) AXCY is a Parallelogram

Q177Short answer

A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.

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Given: (i) ABCD is a gm (ii) ∠1 = ∠2 To Prove: (i) ∠3 = ∠4 (ii) ABCD is rhombus Proof: (i) ∠1 = ∠4 ∠2 = ∠3 (alt ∠s) But ∠1 = ∠2 ∠3 = ∠4 (ii) ∠1 = ∠2 (given alt.) ∠2 = ∠3 ∠1 = ∠3 Hence CD = DA ∴ ABCD is a rhombus

Q178Short answer

The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.

Show answer

1350, 450, 1350, 450

Q179Short answer

ABCD is a rhombus such that the perpendicular bisector of AB passes through D. Find the angles of the rhombus. Hint: Join BD. Then ∆ ABD is equilateral.

Show answer

600, 1200, 600, 1200

Q180Short answer

ABCD is a parallelogram. Points P and Q are taken on the sides AB and AD respectively and the parallelogram PRQA is formed. If ∠C = 45°, find ∠R.

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450

Q181Short answer

In parallelogram ABCD, the angle bisector of ∠A bisects BC. Will angle bisector of B also bisect AD? Give reason.

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Given: ABCD is a gm, bisector of ∠A, bisects BC in F i.e. ∠1 = ∠2, CF = FB BA Const: Draw FE Proof: ABFE is a gm by const. (FE BA) ∠1 = ∠6 (alt. ∠) But ∠1 = ∠2 (given) ∴ ∠2 = ∠6 AB = FB (1) (sides opp to equal ∠s) ∴ ABFE is a rhombus In ∆ABO and ∆BOF AB = BF from (1) BO = BO Common AO = FO Diagonals bisect each other ∆ABO ≅ ∆BOF ∠3 = ∠4 BF = BC (given) BF = AD (BC = AD) AE = AC (BF = AE) ∴ E is mid point of AD

Q182Short answer

A regular pentagon ABCDE and a square ABFG are formed on opposite sides of AB. Find ∠BCF.

Show answer

90

Q183Short answer

Find maximum number of acute angles which a convex, a quadrilateral, a pentagon and a hexagon can have. Observe the pattern and generalise the result for any polygon.

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3, 3, 3. So, maximum number of acute angles is always 3.

Q184Short answer

In the following figure, FD||BC||AE and AC||ED. Find the value of x.

Show answer

(a) 1160

Q185Short answer

In the following figure, AB||DC and AD = BC. Find the value of x.

Show answer

30cm

Q186Short answer

Construct a trapezium ABCD in which AB||DC, ∠A = 105°, AD = 3 cm, AB = 4 cm and CD = 8 cm.

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∠A + ∠D = 1800 1050 + ∠D = 1800 ∠D = 75 Steps of construction 1. Draw AB = 4 cm D 8 cm C 2. Draw AX such that o Y ∠BAX = 105 3 cm 3. Mark a point D on AX such that AD = 3cm 105o A 4 cm B 4. Draw DY such that ∠ADY = 750 5. Mark a point C such that CD = 8cm 6. Join BC. ABCD is the required trapezium. X D 4 cm

Q187Short answer

Construct a parallelogram ABCD in which AB = 4 cm, BC = 5 cm and ∠B = 60°.

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Opp sides of a gm are equal. C AB = DC = 4cm 5 cm 5 cm BC = AD = 5cm Steps of construction A 4 cm B 1. Draw AB = 4 cm 2. Draw ray BX such that ∠ABX = 600 3. Mark a point C such that BC = 5cm 4. With C and A as centre, draw arcs intersecting at a point D respectively ABCD is the required parallelogram.

Q188Short answer

Construct a rhombus whose side is 5 cm and one angle is of 60°.

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∠B = 600 (suppose) ∠A + ∠B = 1800 (sum of co-interior angles) ∠A + 600 = 1800 ∠A = 1200 AB = BC = CD = DA = 5cm Steps of construction 1. Draw AB = 5cm 2. Draw ray AY such that ∠BAY = 1200 3. Mark a point D such that AD = 5cm 4. Draw ray BX such that ∠ABX = 600 5. Mark a point C such that BC = 5cm 6. Joint C and D ∴ ABCD is the required rhombus

Q189Short answer

Construct a rectangle whose one side is 3 cm and a diagonal equal to 5 cm.

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Diagonals of a rectangle are equal. AC = BD = 5 cm Steps of construction 1. Draw AB = 3 cm 2. Draw a ray BX such that ∠ABX = 900 3. Draw an are such that AC = 5cm 4. With B as centre, draw an arc of radius 5cm. With C as centre draw another arc of radius 3cm which intersect first arc at a point, suppose D. 5. Join CD and AD ABCD is the required rectangle. OWNL OWOL U 7.5 cm E 7.5 cm 7.5 cm L 7.5 cm C OWPL 7 cm 5 cm 5 B 6 cm E RA = 5 cm 194. Cyclic quadrilateral ∠B = ∠D = 900 (Angle in a semicircle) ∠A = ∠C = 900 ∠B + ∠D = 1800 ∠A + ∠C = 1800 opposite ∠s are supplementary. OWSL 196. No, In a ∆, sum of two sides always is greater than the third side. AB + BC > AC 197. No, ∠O + ∠R + ∠A = 1200 + 1050 + 1350 = 3600 198. Diagonals bisects at right angle OWWL Fourth angle = 3600 – (600+1100+850) = 3600 – 2550 = 1050 200. K S OPNN 5 cm 6.5 cm R 7 cm I Other side = 5 cm 201. 720 202. ∠I + ∠S = 1800 600 + ∠S = 1800 ∠S = 1200 A 6.4 cm B 3.2 3.2 PNQL 3.2 D 9.6 cm E C BEC is an equilateral triangle ∠A = 1200, ∠B = 600 Across 1. Trapezium 2. Polygon 3. Kite 4. Diagonal 5. Perpendicular 6. Opposite 7. Eight 8. Triangle 9. Regular 10. Nine Down 11. Heptagon 12. Adjacent 13. Parallelogram 14. Equal 15. Pentagon 16. Bisect 17. Rhombus

Q190Short answer

Construct a square of side 4 cm.

Q191Short answer

Construct a rhombus CLUE in which CL = 7.5 cm and LE = 6 cm.

Q192Short answer

Construct a quadrilateral BEAR in which BE = 6 cm, EA = 7 cm, RB = RE = 5 cm and BA = 9 cm. Measure its fourth side.

Q193Short answer

Construct a parallelogram POUR in which, PO=5.5 cm, OU = 7.2 cm and ∠O = 70°.

Q194Short answer

Draw a circle of radius 3 cm and draw its diameter and label it as AC. Construct its perpendicular bisector and let it intersect the circle at B and D. What type of quadrilateral is ABCD? Justify your answer.

Q195Short answer

Construct a parallelogram HOME with HO = 6 cm, HE = 4 cm and OE = 3 cm.

Q196Short answer

Is it possible to construct a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 5.4 cm, DA = 5.9 cm and diagonal AC = 8 cm? If not, why?

Q197Short answer

Is it possible to construct a quadrilateral ROAM in which RO=4 cm, OA = 5 cm, ∠O = 120°, ∠R = 105° and ∠A = 135°? If not, why?

Q198Short answer

Construct a square in which each diagonal is 5cm long.

Q199Short answer

Construct a quadrilateral NEWS in which NE = 7cm, EW = 6 cm, ∠N = 60°, ∠E = 110° and ∠S = 85°.

Q200Short answer

Construct a parallelogram when one of its side is 4cm and its two diagonals are 5.6 cm and 7cm. Measure the other side.

Q201Short answer

Find the measure of each angle of a regular polygon of 20 sides?

Q202Short answer

Construct a trapezium RISK in which RI||KS, RI = 7 cm, IS = 5 cm, RK=6.5 cm and ∠I = 60°.

Q203Multiple choice

Construct a trapezium ABCD where AB||CD, AD = BC = 3.2cm, AB = 6.4 cm and CD = 9.6 cm. Measure ∠B and ∠A. [Hint : Difference of two parallel sides gives an equilateral triangle.] 1 : Constructing a Tessellation Tessellation: A tessellation is created when a shape is repeated over and over again covering a plane surface without any gaps or overlaps. Regular Tesselations : It means a tessellation made up of congruent regular polygons. For example: A tessellation of triangles This arrangement can be extended to complete tiling of a floor (or tessellation). Rules for Regular Tessellation: (i) In tessellation there should be no overlappings/gaps between tiles. (ii) The tiles must be regular polygons. (iii) Design at each vertex must look the same. Caution Will pentagons work? The interior angle of a pentagon is 1080 . . . 1800 + 1080 + 1080 = 3240 degrees . . . No! Thus, since the regular polygons must fill the plane at each vertex, the interior angle must be an exact divisor of 360°. Now, find the regular polygon that can tessellate by trying a sample in table below. Polygon Tessellation 1. Triangle 2. Square 3. Regular Pentagon 4. Regular Hexagon 5. Regular Heptagon 6. Regular Octagon Conclusion Thus, only regular polygons that can tessellate are 1. ______________________ 2. ______________________ 3. ______________________ Assignment 1. You can construct a tessellation on computer using following steps: - Hold down a basic images and copy it to paintbrush. - Keep on moving and pasting by positioning each to see a tessellation. 2. Semi Regular Tessellation : These are made by using two or more different regular polygons. Every vertex must have the same configuration, e.g.: Y - yellow B - Blue G - Green R - Red Now discover same more tessellation of this type . 2 Constructing a TANGRAM Cut the pieces of given square as shown on next page and make different shapes as shown below. Different shapes can be made of Tangram Pieces Try to form a story using different shapes of animals. Required Square 3 Motivate the students to participate Read the following description of a square before the students and let them draw what you have described. Descriptions: My quadrilateral has opposite sides equal. Let students compare their drawings with each other and with your square. Let students discuss what all their drawings have in common (they are all parallelograms) and what additional information is necessary to guarantee that they all would draw a square. (e.g. All 4 sides equal and one right angle.) 4: Place ‘’ or ‘’ in the appropriate spaces according to the property of different quadrilaterals. Parallelogram Rectangle Rhombus Square Trapezium Trapezium Kite with non parallel sides equal Opposite sides parallel Opposite sides equal Opposite angles equal Diagonal forms congruent triangles Diagonals bisect each other Diagonals are perpen- dicular Diagonals are equal Diagonals bisect opposite angles All angles are right All sides are equal Use the quadrilateral chart at Page 167 to do the following activity and answer the following questions.

  • (a)How can you use the properties shown in the quadrilateral chart to make a statement that you believe is true about all parallelograms?
  • (b)How can you use the properties shown in the quadrilateral chart to make a statement that you believe is true about all rhombuses?
  • (c)How can you use the properties shown in the quadrilateral chart to make a statement that you believe is true about all rhombuses, but not parallelograms?
  • (d)How can you use the properties shown in the quadrilateral chart to make a statement that you believe is true about only rhombuses? (e) How are the properties of rhombuses like the properties of parallelograms in general? (f) How are the properties of rhombuses different from the properties of parallelograms? (g) Which quadrilaterals have exactly one line of symmetry? Exactly two? Exactly three? Exactly four? (h) Make a ‘Family Tree’ to show the relationship among the quadrilaterals you have been investigating. 5: Have students take each of the quadrilateral named below, join, in order, the mid points of the sides and describe the special kind of quadrilaterals they get each time: (a) Rhombus. (b) Rectangle. (c) Trapezium with non-parallel sides equal. (d) Trapezium with non-parallel sides unequal. (e) Kite. 6: Crossword Puzzle Solve the given crossword and then fill up the given boxes (on the next page). Clues are given below for across as well as downward filling. Also, for across and down clues, clue number is written at the corner of the boxes. Answers of clues have to be filled up in their respective boxes. Clues Across 1. A quadrilateral with pair of parallel sides. 2. A simple closed curve made up of only line segments. 3. A quadrilateral which has exactly two distinct consecutive pairs of sides of equal length. 4. A line segment connecting two non-consecutive vertices of a polygon. 5. The diagonals of a rhombus are _________ bisectors of one another. 6. The ___________ sides of a parallelogram are of equal length. 7. The number of sides of a regular polygon whose each exterior angle has a measure of 450. 8. The sum of measure of the three angles of a _________________ is 1800. 9. A polygon which is both equiangular and equilateral is called a _________ polygon. 10. Number of sides of a nonagon. Down 11. Name of the figure 12. The ___________ angles of a parallelogram are supplementary. 13. A ______________ is a quadrilateral whose pair of opposite sides are parallel. 14. The diagonals of a rectangle are of _______________ length. 15. A five sided polygon. 16. The diagonals of a parallelogram _____________ each other. 17. A quadrilateral having all the properties of a parallelogram and also that of a kite. MATHEMATICS