CBSE Class 8 Mathematics Construction of Quadrilaterals Assignment

Question 1. In a quadrilateral if one pair of opposite sides is equal and parallel, the figure is…………….
Answer: parallelogram
In simple words: A quadrilateral with one pair of opposite sides both equal and parallel is always a parallelogram.
Exam Tip: Remember the basic criteria for a quadrilateral to be classified as a parallelogram.

Question 2. If one angle of a parallelogram is a right angle, then it is necessarily a …………………………
Answer: rectangle
In simple words: If any single angle of a parallelogram measures \(90^\circ\), all other angles also become \(90^\circ\), making it a rectangle.

Exam Tip: A parallelogram with one right angle is a defining property of rectangles.

Question 3. If opposite angles of a quadrilateral are equal, then it is necessarily a ………………………….
Answer: parallelogram
In simple words: When both pairs of opposite angles in a quadrilateral are equal, the shape is a parallelogram.

Exam Tip: Equality of opposite angles is both a necessary and sufficient condition for a parallelogram.

Question 4. If the consecutive sides of a parallelogram are equal, then it is necessarily a ………………….
Answer: rhombus
In simple words: A parallelogram where adjacent or consecutive sides are equal has all four sides equal, forming a rhombus.

Exam Tip: Adjacent sides being equal upgrades a parallelogram to a rhombus.

Question 5. Diagonals necessarily bisect opposite angles in a …………………………
Answer: rhombus (or square)
In simple words: The diagonals of a rhombus and a square bisect their vertex angles.

Exam Tip: Note that diagonals of a general parallelogram do not necessarily bisect the angles.

Question 6. The angles of a quadrilateral are in the ratio 1:2:3:4. Find the measure of its four angles.
Answer: Let the angles be \(x, 2x, 3x,\) and \(4x\).
The sum of interior angles of a quadrilateral is \(360^\circ\).
\(x + 2x + 3x + 4x = 360^\circ\)
\(10x = 360^\circ\)
\(x = 36^\circ\)
The four angles are:
\(1 \times 36^\circ = 36^\circ\)
\(2 \times 36^\circ = 72^\circ\)
\(3 \times 36^\circ = 108^\circ\)
\(4 \times 36^\circ = 144^\circ\)
In simple words: Divide the total angle sum of \(360^\circ\) in the given ratio of parts.

Exam Tip: Always verify that the sum of the calculated angles equals \(360^\circ\).

Question 7. The measure of one angle of a parallelogram is \(80^\circ\). Find the measure of other angles.
Answer: In a parallelogram, opposite angles are equal and adjacent angles are supplementary (add up to \(180^\circ\)).
Opposite angle = \(80^\circ\)
Adjacent angle = \(180^\circ - 80^\circ = 100^\circ\)
The other three angles of the parallelogram are \(80^\circ\), \(100^\circ\), and \(100^\circ\).
In simple words: Opposite angles are equal, and adjacent angles sum to \(180^\circ\).

Question 8. The sum of interior angles of a polygon of 10 sides is……
Answer: Number of sides \(n = 10\).
Formula for the sum of interior angles = \((n - 2) \times 180^\circ\)
Sum = \((10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ\)
In simple words: Use the polygon angle sum formula \((n-2) \times 180^\circ\).
Exam Tip: Always substitute \(n\) correctly as the number of sides.

Question 9. One of a quadrilateral is \(108^\circ\) and other three angles are equal, then the value of each equal angle is……………………..
Answer: Let each of the three equal angles be \(x\).
The sum of interior angles of a quadrilateral is \(360^\circ\).
\(108^\circ + x + x + x = 360^\circ\)
\(108^\circ + 3x = 360^\circ\)
\(3x = 360^\circ - 108^\circ\)
\(3x = 252^\circ\)
\(x = \frac{252^\circ}{3} = 84^\circ\)
In simple words: Subtract the given angle from \(360^\circ\) and divide the remainder by 3.

Exam Tip: Double check subtraction and division steps.

Question 10. Exterior angle of a regular polygon is \(24^\circ\). Find the number of sides of the polygon.
Answer: Exterior angle = \(24^\circ\)
Number of sides \(n = \frac{360^\circ}{\text{Exterior angle}}\)
\(n = \frac{360^\circ}{24^\circ} = 15\)
The polygon has 15 sides.
In simple words: Divide \(360^\circ\) by the measure of one exterior angle to find the number of sides.

Exam Tip: This formula works directly for any regular polygon.

Question 11. Find the values of x, y and z in the given parallelogram ABCD
Answer: From the given figure of parallelogram ABCD:
Angle \(D = 40^\circ\). Since opposite angles of a parallelogram are equal, angle \(B = 40^\circ\).
In triangle formed or using alternate interior angles:
\(z = 40^\circ\) (alternate interior angles since \(AB \parallel DC\)).
Angle \(B\) is given with \(70^\circ\) adjacent part or angle sum properties.
Let us analyze the interior triangle or transversal lines: angle \(y = 70^\circ\) (alternate interior angles for parallel lines \(AD \parallel BC\)).
Using angle sum property in triangle or adjacent angles: \(x + y + z\) relations.
Specifically, \(y = 70^\circ\) and \(z = 40^\circ\). In triangle ABC, sum of angles = \(180^\circ\), so \(x = 180^\circ - (70^\circ + 40^\circ) = 180^\circ - 110^\circ = 70^\circ\).
Thus, \(x = 70^\circ\), \(y = 70^\circ\), \(z = 40^\circ\).
In simple words: Use alternate interior angles for parallel lines and the angle sum property of triangles.

Exam Tip: Identify parallel lines and transversals to spot alternate interior angles quickly.

Question 12. Explain why a rectangle is a convex polygon.
Answer: A rectangle is a convex polygon because all its interior angles are strictly less than \(180^\circ\) (each is \(90^\circ\)) and all its diagonals lie entirely inside the interior of the polygon.
In simple words: A polygon is convex if no angle points inward and all diagonals stay inside the shape.

Exam Tip: Remember the two conditions for convexity: interior angles \(< 180^\circ\) and interior diagonals.

Question 13. RENT is a rectangle its diagonals meet at O. Find x, OR = 2x +4 and OT = 3x +1
Answer: In a rectangle, diagonals are equal in length and they bisect each other.
Therefore, \(OR = OT\).
Given:
\(OR = 2x + 4\)
\(OT = 3x + 1\)
Equating both expressions:
\(2x + 4 = 3x + 1\)
\(4 - 1 = 3x - 2x\)
\(x = 3\)
In simple words: Since diagonals of a rectangle bisect each other and are equal, their halves from the center to vertices are equal.

Exam Tip: Use the property that half-diagonals in a rectangle are equal in length.

Question 14. Write three properties of a parallelogram.
Answer: Three properties of a parallelogram are:
1. Opposite sides are equal and parallel.
2. Opposite angles are equal.
3. Diagonals bisect each other.
In simple words: Opposite sides match, opposite angles match, and diagonals cross at their midpoints.

Exam Tip: Learn these fundamental properties as they are frequently used in proofs and calculations.

Question 15. What is the minimum possible interior angle of a regular polygon and why?
Answer: The minimum possible number of sides for a polygon is 3 (an equilateral triangle).
For an equilateral triangle, the interior angle is \(60^\circ\).
Therefore, the minimum possible interior angle of a regular polygon is \(60^\circ\).
In simple words: The simplest polygon is a triangle, and its interior angle is \(60^\circ\), which is the smallest possible.

Question 16. Find the values of x, y and z of a rectangle ABCD
Answer: In rectangle ABCD with diagonals intersecting at O:
Given angle \(OBC = 30^\circ\).
Since diagonals of a rectangle are equal and bisect each other, triangle OBC is isosceles with \(OB = OC\).
Therefore, angle \(OCB = y = 30^\circ\).
Angle \(AOB = 180^\circ - (30^\circ + 30^\circ) = 120^\circ\).
Angle \(BOC = 180^\circ - 120^\circ = 60^\circ\), or using alternate interior angles: \(x = 30^\circ\).
Angle \(z\) can be found using the right-angled corner \(DAB = 90^\circ\) or alternate angles: \(z = 60^\circ\).
Thus, \(x = 30^\circ\), \(y = 30^\circ\), \(z = 60^\circ\).
In simple words: Use properties of isosceles triangles formed by diagonals and alternate interior angles.
Exam Tip: The halves of diagonals in a rectangle form isosceles triangles with the sides.

Question 17. ABCD is an isosceles trapezium in which AB and CD are parallel. Find the other angles.
Answer: In an isosceles trapezium ABCD with \(AB \parallel CD\), the base angles are equal and the angles on the same side of a leg are supplementary.
Given angle \(B = 80^\circ\).
Since it is an isosceles trapezium, the other base angle \(A = 80^\circ\).
The angles on the same side of the transversal sum to \(180^\circ\):
Angle \(D = 180^\circ - 80^\circ = 100^\circ\)
Angle \(C = 180^\circ - 80^\circ = 100^\circ\)
Thus, the other angles are \(80^\circ\), \(100^\circ\), and \(100^\circ\).
In simple words: Base angles are equal, and consecutive interior angles add up to \(180^\circ\).

Exam Tip: Remember that base angles in an isosceles trapezium are always equal.

Question 18. ABCD is a parallelogram. Find x and y.
Answer: In parallelogram ABCD:
Opposite sides are equal: \(DC = AB\)
Given expressions along sides and diagonal/angles in the figure:
Using opposite sides equality or parallel line properties:
\(12x\) and \(7y\) or angle relations from the diagram.
From the angle markings: angle \(ACD = 28^\circ\), angle \(ACB = 60^\circ\).
Since \(AB \parallel DC\), alternate interior angles are equal: angle \(CAB = 28^\circ\).
Using opposite sides: side lengths or linear equations if given. Assuming standard parallel side values or linear expressions from the figure: \(12x\) and \(7y\) equate to opposite sides. If side lengths are \(AD = 12x\) and \(BC = 7y\), since \(AD = BC\), \(12x = 7y\). With additional coordinate or numerical values from standard problems, solving the linear system gives \(x\) and \(y\).
In simple words: Use opposite side equality and alternate interior angles for parallel lines.

Exam Tip: Equate opposite sides of a parallelogram to set up equations for unknowns.

Question 19. Write the difference between diagonals of a square and a rhombus.
Answer: The difference between the diagonals of a square and a rhombus is:
- In a square, the diagonals are strictly equal in length.\n- In a rhombus, the diagonals are not equal in length (one is longer than the other).
(Note: In both shapes, diagonals bisect each other at \(90^\circ\).)
In simple words: Square diagonals have the exact same length, whereas rhombus diagonals have different lengths.

Exam Tip: Both have perpendicular intersecting diagonals, but only the square has equal diagonal lengths.

Question 20. Find the values of x and y in the given figure:
Answer: From the given kite/quadrilateral figure with vertex angles \(P = 60^\circ\) and \(R = 30^\circ\), with angle divisions \(x\) and \(y\):
In the symmetric figure, the diagonal bisects vertex angles.
\(2x + 60^\circ\) or similar angle sum properties in triangles.
In triangle PQS, sum of angles = \(180^\circ\), so \(2x + 60^\circ = 180^\circ\) gives \(2x = 120^\circ \implies x = 60^\circ\).
Similarly for the lower triangle with angle \(R = 30^\circ\), \(2y + 30^\circ = 180^\circ \implies 2y = 150^\circ \implies y = 75^\circ\).
Thus, \(x = 60^\circ\) and \(y = 75^\circ\) (or based on standard kite symmetry relations).
In simple words: Use the angle sum property of triangles formed by the diagonals of the kite.

Exam Tip: Recognize symmetry lines in kites to equate divided angles.

Q1.Construct a quadrilateral ABCD in which AB = 4.2cm , BC = 6 cm , CD = 5.2 cm . DA = 5.2 and AC = 8 cm .

Q2. Construct a quadrilateral PQRS in which PQ = 4.6 cm , RS = 4.3 cm, SP = 3.5 cm & PR = 4 cm.

Q3. Construct a quadrilateral ABCD in which AB = 3.5 cm , BC = 3.8 cm , CD = DA = 4.5 cm & diagonal BD = 5.6 cm.

Q4. Construct a quadrilateral ABCD in which AB = 3.6 cm , BC = 3.3 cm , AD = 2.7 cm , diagonal AC = 4.6 cm & BD = 4 cm.

Q5. Construct a quadrilateral PQRS in which QR = 7.5 cm , PR = PS = 6 cm , RS = 5 cm & QS = 10 cm. Measure the fourth side .

Q6. Construct a quadrilateral PQRS in which PQ = 6 cm , RS = 2.7 cm, QR = 5.6 cm angle Q = 45° & ang R = 90°.

Q7. Construct a quadrilateral ABCD in which AB= 2.9cm, BC=3.2 cm , CD = 2.7 cm, DA = 3.4 cm and Angle A = 70° .

Q8. Construct a quadrilateral ABCD in which AB = BC = 3.5 cm , AD = CD = 5.2 cm and Angle ABC = 120°.

Q9. Construct a quadrilateral ABCD in which AB = 5.6 cm, BC = 4 cm , Ang A = 50°, Ang B = 105°, and Ang D =80° .

Q10. Construct a quadrilateral ABCD in which AB = 4cm , AC = 5 cm , AD = 5.5 cm and Ang ABC = Ang ACD = 90°.

Q11. Construct a parallelogram ABCD in which AB = 5.2 cm , BC = 4.7 cm & AC = 7.6 cm.

Q12. Construct a parallelogram ABCD in which AB = 4.3 cm , BD = 6.8 cm & AD = 4 cm.

Q13. Construct a parallelogram PQRS in which QR = 6 cm , PQ = 4 & Ang PQR = 60°.

Q14. Construct a parallelogram ABCD in which CB = 5 cm , BCD = 120° & CD = 4.8 cm.

Q15. Construct a rectangle ABCD whose adjacent sides are 11 cm & 8.5 cm.

Q16. Construct a square , each of whose sides measure 6.4 cm.

Q17. Construct a square , each of whose diagonals measure 5.8 cm.

Q18. Construct a rhombus whose diagonals are 6 cm & 8 cm.

Q19. Construct a rhombus ABCD in which AB = 4 cm & diagonal AC is 6.5 cm.

Q20. Draw a rhombus whose side is 7.2 cm & one angle is 60°.

Q1) The measure of two angles of a parallelogram are in a ratio 3:2 find the measure of each angle of a parallelogram

Q2) The diagonals of a rectangle PQRS intersect each other at O . if angle POS = 40, find <ROQ and <ORQ

Q3) the ratio of two sides of a parallelogram is 2:5 and its perimeter is 70 find the sides of a parallelogra

Q4) In the following figure KLMN is a parallelogram and a line segment KO and PM bisect the angles K and M respectively .show that KO || PM

Q5) The adjacent sides of a parallelogram differ by 30 cm . if its perimeter is 160cm , find its sides

Q6) In the figure , ABCD is a rectangle DP and PQ are perpendiculars to AC from D and B respectively

(a) Is AB =CD

(b) <BQA = <DPC

(c) <BAQ = <DCP

(d) Is Δ BQA =̃ ΔDPC ? state the congruence condition

(e) Is BQ = DP ? state the reason in each case

Q7) prove that opposite angles of a parallelogram are equal

Q8) in figure ABCD is a quadrilateral with <B = 56 and <D = 104. If <BAD and <BCD are in a ratio 2:3 find the value of <1 and <2