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Chapters
1: Rational and Irrational Numbers
UNIT-II: COMMERCIAL MATHEMATICS
2: Compound Interest
UNIT-III: ALGEBRA
3: Expansions
4: Factorisation
5: Simultaneous Linear Equations
6: Indices
7: Logarithms
UNIT-IV: GEOMETRY
8: Triangles
9: Inequalities
10: Mid-point Theorem
11: Pythagoras Theorem
▶ 12: Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)
13: Theorems on Area
14: Circles (Chord and Arc Properties)
UNIT-V: STATISTICS
15: Statistics
16: Graphical Representation of Statistical Data
UNIT-VI: MENSURATION
17: Mensuration
18: Surface Area and Volume of Solids
UNIT-VII: TRIGONOMETRY
19: Trigonometry
20: Simple 2-D Problems in Right Triangle
UNIT-VIII: COORDINATE GEOMETRY
21: Coordinate Geometry
![B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE chapter 12 - Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons) - Shaalaa.com](/images/mathematics-english-class-9-icse_6:a927b361d63845f4b2afea4ec6bbe35a.jpg "B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE chapter 12 - Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)")
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Solutions for Chapter 12: Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)
Below listed, you can find solutions for Chapter 12 of CISCE B Nirmala Shastry for Mathematics [English] Class 9 ICSE.
B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE 12 Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons) EXERCISE 12A [Pages 139 - 141]
PQRS is a rectangle. Find the angles marked x and y.
In parallelogram PQRS, ∠P = (9x – 5)° and ∠Q = (4x – 10)°. Find the values of x and ∠R.
In square ABCD, E is a point on CD such that ∠BEC = 60°. Find ∠DBE.
In square ABCD, ΔPAB is equilateral. Find the value of x in the figure.
ABCD is a rhombus, ΔPAB is equilateral, ∠D = 68°. Find the values of x and y.
PQRS is a rectangle. The diagonals intersect at O. Diagonal RP is produced to T. ∠QPT = 142°. Find the angles of ΔSOR.
In a rhombus ABCD, the diagonals AC and BD intersect at O. BD is produced to E. ∠ADO : ∠ADE = 1 : 4. Find the angles of ΔAOB.
In the parallelogram ABCD, ∠D = 5x, ∠DAC = 4x and ∠ACD = 3x. Find the angles of ΔABC.
In the parallelogram ABCD, ∠A = (3x – 15)°, ∠B = (2x + 5)°. Find the values of x and ∠D.
In a rhombus PQRS, ∠P = (4x – 10)° and ∠R = (3x + 20)°. Find the values of x and ∠Q.
In the square PQRS, the diagonals intersect at O. M is a point on PS such that PM = PO. Show that ∠POM = 3∠MOS.
In parallelogram ABCD, PA and QC are bisectors of ∠A and ∠C respectively. Prove that APCQ is a parallelogram.
P and Q are mid-points of sides AB and CD of a parallelogram ABCD. Prove that APCQ is a parallelogram.
In parallelogram PQRS, A and B are points on PR such that PA = AB = BR. Prove that ASBQ is a parallelogram.
P and Q are mid-points of sides AB and CD of a parallelogram ABCD. AQ and DP intersect at Rand BQ and PC intersect at S. Prove that PRQS is a parallelogram.
Prove that the bisectors of the angles of a parallelogram form a rectangle.
ABCD is a parallelogram. AP and CQ are perpendicular to diagonal DB. Prove that AP = CQ.
ABCD is a rectangle. M is the mid-point of AC and CPMN is a rectangle. Prove that:
- P is the mid-point of CD
- PN = `1/2` AC
ABCD is a parallelogram. AB is produced to E so that BE = AB. A line through E drawn parallel to AC meets CB produced at F. Show that ACEF is a parallelogram.
[Hint: Show that ΔABC ≅ ΔEBF ∴ BC = BF]
PQRS is a parallelogram. M is the mid-point of QR. PM is produced to meet SR produced at N. Prove that SN = 2SR.
ABCD is a square. PAB is an equilateral triangle. Find x and y.
Calculate the measures of x and y in the parallelogram ABCD.
In the parallelogram PQRS, ∠P = 128° and ∠QSR = 36°. Find x.
PQRS is a parallelogram, QP is extended to T so that ∠STP = 90°. Find x and y.
ABCD is a rhombus and ABEF is a square. ∠C = 62°. Find (a) ∠AFD, (b) ∠CDF.
ABCD is a rectangle. Find y and ∠ABD.
ABCD and CDEF are parallelograms. ∠A = 3x, ∠E = 5x and ∠BCF = 112°. Find x.
ABCD is a square. Diagonal DB and CP intersect at O. ∠BOP = 78°. Find x, y and z.
Find x and y in the parallelogram PQRS.
One angle of a quadrilateral is 48°. The other three angles are in ratio 6 : 9 : 11. Find these angles. What type of quadrilateral is this?
ABCD is a rhombus. ΔPAD is equilateral. If ∠ABC = 100°, then find the measure of ∠PCD.
In the parallelogram PQRS, PS = TS. ∠SQR = 88°, ∠R = 56°. Find angles a, b, c.
B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE 12 Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons) EXERCISE 12B [Page 149]
Construct a quadrilateral ABCD, when AB = 4.2 cm, BC = 5.5 cm, CD = 4.8 cm, AD = 5 cm and ∠B = 45°.
Construct a quadrilateral ABCD, when AB = 5.2 cm, BC = 4.5, CD = 4 cm, AD = 5 cm and ∠A = 75°
Construct a quadrilateral ABCD, when AB = 6 cm, BC = 5.5 cm, CD = 4.1 cm, ∠A = 120°, ∠B = 45°
Construct a quadrilateral ABCD, when AB = 4 cm, ∠B = 60°, BC = 5.5 cm, ∠C = 90°, CD = 3.5 cm.
Construct a quadrilateral ABCD, when AB = 4 cm, BC = 5 cm, CD = 4.5 cm, AD = 4 cm, AC = 6 cm
Construct a quadrilateral ABCD, when AB = 5 cm, BC = 4.5 cm, CD = 6 cm, AD = 4.2 cm, AC = 5.6 cm
Construct a quadrilateral ABCD, when AB = 3.5 cm, BC = 4 cm, CD = 4.5 cm, AC = BD = 6 cm
Construct a quadrilateral ABCD, when AB = 3 cm, BC = 5 cm, CD = 4 cm, AC = 6.8 cm, BD = 4 cm
Construct a parallelogram ABCD, when AB = 4.5 cm, BC = 6 cm and ∠B = 60°.
Construct a parallelogram ABCD, when AB = 3.5 cm, BC = 5.5 cm and ∠B = 75°.
Construct a parallelogram ABCD, when AB = 5 cm, BC = 6 cm and AC = 6.5 cm.
Construct a parallelogram ABCD, when AB = 4.5 cm, BC = 5.8 cm and AC = 7 cm.
Construct a parallelogram ABCD, when AB = 5 cm, AC = 8 cm and BD = 9 cm.
Construct a parallelogram ABCD, when AB = 6 cm, AC = 7 cm and BD = 9.4 cm.
Construct a parallelogram ABCD, when AC = 9 cm, BD = 8 cm and ∠BOC = 45°.
Construct a parallelogram ABCD, when AC = 10 cm, BD = 8.6 cm and ∠BOC = 30°.
Construct a parallelogram ABCD, when AB = 4.5 cm, BC = 5.5 cm and altitude on BC = 3.5 cm.
Construct a parallelogram ABCD, when AB = 5 cm, BC = 6.2 cm and altitude on AB = 3 cm.
Construct a parallelogram ABCD, when BC = 5 cm, BD = 4 cm and altitude on BC = 2 cm.
Construct a rhombus ABCD, when AB = 5 cm and ∠A = 120°.
Construct a rhombus ABCD, when BC = 5.5 cm and ∠B = 60°.
Construct a rhombus ABCD, when AB = 4.5 cm and AC = 5.5 cm.
Construct a rhombus ABCD, when AB = 5 cm and AC = 6.5 cm.
Construct a rhombus ABCD, when AC = 6 cm and BD = 8 cm. Measure AB.
Construct a rhombus ABCD, when AC = 12 cm and BD = 9 cm. Measure AB.
Construct a rhombus ABCD, when AC = 6.5 cm, ∠DAB = 60°.
Construct a rhombus ABCD, when BD = 6 cm, ∠ABC = 120°.
Construct a regular hexagon of side 3 cm.
Construct a regular hexagon of side 3.5 cm and draw all its lines of symmetry.
B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE 12 Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons) MULTIPLE CHOICE QUESTIONS [Pages 150 - 152]
In a quadrilateral the ratio of angles is 1 : 2 : 3 : 4. ∴ The largest angle is ______.
120°
150°
140°
144°
In the parallelogram ABCD, the bisectors of angles A and B meet at P. ∴ ∠APB is ______.
60°
45°
90°
120°
In the parallelogram ABCD, ∠A = 5x and ∠B = 4x. ∴ The measure of x is ______.
40°
20°
45°
30°
∠EDC = 2y + 10°, ∠C = 110°. ∴ The value of y is ______.
40°
50°
30°
45°
ABCD is a rhombus. ΔPAD is equilateral, ∠B = 70°. ∴ The measure of angle x is ______.
20°
25°
30°
35°
PQRS is a square. ΔOPQ is equilateral. The measure of angle a is ______.
80°
70°
75°
60°
ABCD is a rectangle. DB is extended to point P. ∠PBA = 140°. ∴ find ∠DBC.
40°
50°
60°
45°
In the rectangle ABCD, ∠CAB = 25°, then ∠AOB is ______.
100°
110°
120°
130°
In the parallelogram PQRS ∠P = 3x – 10°, ∠Q = 2x + 20°. ∴ The measure of x is ______.
30°
34°
28°
36°
E is a point on side BC of the square ABCD, ∠AEB = 55°. find ∠EAC.
35°
25°
10°
15°
In the rectangle ABCD, ∠CAB = 28°
The measure of angle x is:
62°
72°
28°
52°
In the rectangle ABCD, ∠CAB = 28°
The measure of angle y is:
144°
124°
120°
130°
In a quadrilateral ABCD, ∠A + ∠B = 180°. ∴ It is a ______.
Parallelogram
Rectangle
Trapezium
Kite
In a parallelogram the opposite angles are ______.
supplementary
equal
complementary
right angles
Both diagonals of the following quadrilateral are equal.
Trapezium
Rhombus
Rectangle
Parallelogram
The diagonals of the following quadrilateral bisect each other at right angles.
Parallelogram
Rectangle
Kite
Rhombus
If one pair of opposite sides of a quadrilateral are equal and parallel then it is a ______.
isosceles trapezium
rhombus
kite
parallelogram
If the diagonals of a quadrilateral are equal and bisect each other at right angles it is a ______.
Parallelogram
Square
Rectangle
Rhombus
If one diagonal bisects the angles at the opposite vertices it is a ______.
parallelogram
rhombus
kite
square
Diagonals of a rhombus
- are equal
- bisect each other at right angles
- bisect the angles at the vertices.
Which of the above statements is true?
only 1
only 2
1 and 2
2 and 3
In the rhombus ABCD ∠B = 60°, AB = 3x – 7, BC = x + 5, AC = y + 3, x, y are = ______.
x = 5, y = 10
x = 6, y = 8
x = 3, y = 5
x = 5, y = 7
In the given figure AB || DC, AD || BC ∠D = 52° and ∠BCE = 28°. ∴ ∠E is ______.
30°
28°
24°
38°
In the quadrilateral ABCD, ∠A = ∠C = 110°, ∠B = 60°. DE = DF. The measure of angle x is ______.
120°
130°
100°
80°
Direction for Questions 23 to 29: In each of the following questions, a statement of assertion (A) is given and a statement of Reason (R) given below it choose the correct option for each question.
Assertion: A triangle is not a polygon.
Reason: A closed plane figure bounded by three or more line segments is called a polygon.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: If the opposite angles of a parallelogram are (3x – 6)° and (50 – x)°, then measure of the angles is 36°.
Reason: Opposite angles of a parallelogram are equal.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: In the quadrilateral ABCD, the bisectors of angles B and C meet at P. If ∠BPC = 110°, ∠A = 120°, then x = 100°.
Reason: If ∠B = 2a, ∠C = 2b, a + b = 70° and sum of the angles of a quadrilateral is 360°.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: In the quadrilateral ABCD, AB = BC = CD. ∠B = 100°, ∠C = 60°, then ∠A = 70°.
Reason: Join BD, ΔBCD is equilateral and ΔABD isosceles.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: The diagonals of a square PQRS intersect at O. Then ΔPOQ is an isosceles right-angled triangle.
Reason: The diagonals of a square are equal and bisect each other at 90°.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: ABCD is a trapezium if ∠A + ∠C = 180°.
Reason: If one pair of opposite sides of a quadrilateral are parallel then it is a trapezium.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
Assertion: In the figure ∠ABC = 150°, ∠C = 35°, ∠ADE = 115°. ∴ ∠A = 60°.
Reason: Sum of the angles of a quadrilateral is 360°.
Both A and R are true and R is the correct reason for A.
Both A and R are true but R is the incorrect reason for A.
A is true but R is false.
A is false but R is true.
B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE 12 Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons) MISCELLANEOUS EXERCISE [Page 152]
ABCD is a parallelogram. ∠C = 110°. E is a point on DC so that AD = ED. Find ∠AEC.
CDEF is a rectangle. Find the angles a and b, if ∠DCE = 32°.
In the rhombus PQRS, ∠QRS = 104°. Find x, y and z.
ABCD is a rhombus and ΔCDE is equilateral. ∠BCD = 98°. Find x, y and z.
In the figure, PQRS and PQXY are parallelograms.
- Prove that SX and RY bisect each other.
- If SX = RY, prove that ∠RSY = 90°
Construct a parallelogram PQRS, PQ = 4 cm, PR = 6 cm and QS = 8 cm.
Construct a parallelogram ABCD, AC = 8 cm, BD = 6.5 cm, ∠AOB = 60°.
Construct a rhombus PQRS, PR = 5 cm and QS = 8 cm.
Construct a rhombus ABCD, BC = 6 cm and ∠B = 120°.
Construct a rhombus PQRS, PR = 5.5 cm and ∠QPR = 60°.
Construct a hexagon of side 4.5 cm and draw all its lines of symmetry.
Solutions for 12: Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)
B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE chapter 12 - Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)
Shaalaa.com has the CISCE Mathematics Mathematics [English] Class 9 ICSE CISCE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. B Nirmala Shastry solutions for Mathematics Mathematics [English] Class 9 ICSE CISCE 12 (Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons)) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
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Concepts covered in Mathematics [English] Class 9 ICSE chapter 12 Rectilinear Figures (Theorems on Parallelograms and Construction of Polygons) are Introduction of Rectilinear Figures, Diagonal Properties of Different Kinds of Parallelograms, Property: The Diagonals of a Rectangle Are of Equal Length., Property: The diagonals of a square are perpendicular bisectors of each other., Property: The Opposite Sides of a Parallelogram Are of Equal Length., Property: The Opposite Angles of a Parallelogram Are of Equal Measure., Property: The diagonals of a rhombus are perpendicular bisectors of one another., Classification of Polygons, Construction of Parallelograms, Construction of Trapezium, Construction of a Rectangle When Its Length and Breadth Are Given., Construction of Rhombus, To Construct a Regular Hexagon, Constructing a Quadrilateral, Square: Properties and Construction.
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