NCERT Solutions Class 8 Maths Chapter 3: Exercise 3.3
NCERT Solutions for Class 8 Maths Chapter 3 – Understanding Quadrilaterals Ex 3.3 are given here in simple, step-by-step form. NCERT stands for National Council of Educational Research and Training, the body that makes your textbook. A quadrilateral is a closed shape with four sides. This chapter looks at special quadrilaterals like the parallelogram, the rhombus, the rectangle and the square. Exercise 3.3 is all about the parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel.
This exercise has 12 questions. They use the main properties of a parallelogram again and again, so once you learn those properties, every question becomes easy. You will find missing angles and missing sides, work with the diagonals, and check whether a shape can be a parallelogram. Every question below is solved in short steps, with the final answer in bold. Students can use this page for homework and revision, parents can use it to check the work at home, and teachers can use it in class. A free printable PDF of these solutions is also available for offline study.
NCERT Solutions Class 8 Maths Chapter 3: Exercise 3.3
Understanding Quadrilaterals
Medium
1. AD = ......
2. ∠ DCB = ......
3. OC = ......
4. m ∠DAB + m ∠CDA = .....
Understanding Quadrilaterals
Difficult
Understanding Quadrilaterals
Difficult
Understanding Quadrilaterals
Medium
Understanding Quadrilaterals
Medium
Understanding Quadrilaterals
Medium
Understanding Quadrilaterals
Medium
Understanding Quadrilaterals
Difficult
Understanding Quadrilaterals
Medium
Understanding Quadrilaterals
Difficult
Properties of a Parallelogram
- Opposite sides are equal.
- Opposite angles are equal.
- Adjacent angles add up to 180°. Two angles that are next to each other are called adjacent angles, and together they make a straight angle.
- The diagonals bisect each other. That means each diagonal cuts the other into two equal parts.
- The sum of all four angles of any quadrilateral is 360°.
Exercise 3.3 – Questions and Answers
Exercise 3.3 – Questions and Answers
Q1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD = ......
(ii) ∠DCB = ......
(iii) OC = ......
(iv) m∠DAB + m∠CDA = .....
Answer:
(i) AD = BC
(Opposite sides are of equal length in a parallelogram.)
(ii) ∠DCB = ∠DAB
(Opposite angles are equal in measure.)
(iii) OC = OA
(In a parallelogram, diagonals bisect each other.)
(iv) m∠DAB + m∠CDA = 180°
(In a parallelogram, adjacent angles are supplementary to each other.)
Q2. Consider the following parallelograms. Find the values of the unknowns x, y, z.
Answer:
(i)
x + 100° = 180° (Adjacent angles of a parallelogram are supplementary)
⇒ x = 80°
Since opposite angles of a parallelogram are equal,
∴ ∠A = ∠C
⇒ z = x = 80°
Also, ∠B = ∠D
Hence, y = 100°
∴ x = 80°, y = 100° and z = 80°
(ii)
[Image: Q2_ii.png]
Since adjacent angles of a parallelogram are supplementary, therefore,
50° + y = 180°
⇒ y = 130°
Also, opposite angles of a parallelogram are equal.
∴ x = y = 130°
Now, 'z' and 'x' are the corresponding angles.
∴ z = x = 130°
∴ x = 130°, y = 130° and z = 130°
(iii)
[Image: Q2_iii.png]
x = 90° (Vertically opposite angles)
Also,
x + y + 30° = 180° (By angle sum property of triangles)
⇒ 120° + y = 180°
⇒ y = 60°
Now, z = y = 60° (Alternate interior angles)
∴ x = 90°, y = 60° and z = 60°
(iv)
[Image: Q2_iv.png]
In the given figure,
z = 80° (Corresponding angles)
y = 80° (Opposite angles are equal in a parallelogram)
Also,
x + y = 180° (Adjacent angles are supplementary)
⇒ x = 180° − 80°
⇒ x = 100°
∴ x = 100°, y = 80° and z = 80°
(v)
[Image: Q2_v.png]
y = 112° (Opposite angles are equal)
In a triangle,
x + y + 40° = 180° (By angle sum property)
⇒ x + 112° + 40° = 180°
⇒ x + 152° = 180°
⇒ x = 28°
Also,
z = x = 28° (Alternate interior angles)
∴ x = 28°, y = 112° and z = 28°
Q3. Can a quadrilateral ABCD be a parallelogram if
(i) ∠D + ∠B = 180°?
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) ∠A = 70° and ∠C = 65°?
[Image: Q3_figure.png]
Answer:
A parallelogram has opposite sides equal and parallel. The opposite angles are also equal. The adjacent angles are supplementary.
(i) So, if ∠D + ∠B = 180°, a quadrilateral ABCD may or may not be a parallelogram, as all the other conditions of the parallelogram should also be fulfilled by a quadrilateral ABCD.
(ii) In a parallelogram the opposite sides should be of equal length. Here, AB is equal to CD but AD is not equal to BC. Hence, ABCD is not a parallelogram.
(iii) In a parallelogram the opposite angles are of equal measure. Here, ∠A is not equal to ∠C. Hence, ABCD is not a parallelogram.
Q4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Answer:
[Image: Q4_figure.png]
Here, ABCD is a quadrilateral that is not a parallelogram but ∠B = ∠D.
(This figure is a kite. In a kite, two pairs of adjacent sides are equal and only one pair of opposite angles is equal, so it is not a parallelogram.)
Q5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.
[Image: Q5_figure.png]
Answer:
The two angles are in the ratio 3 : 2.
Let ∠A = 3x and ∠B = 2x
We know that the adjacent angles are supplementary in a parallelogram.
∴ ∠A + ∠B = 180°
⇒ 3x + 2x = 180°
⇒ 5x = 180°
⇒ x = 180°/5 = 36°
Hence,
∠A = 3x = 3 × 36° = 108°
∠B = 2x = 2 × 36° = 72°
Since opposite angles of a parallelogram are equal, therefore,
∠A = ∠C and ∠B = ∠D
∴ ∠A = ∠C = 108° and ∠B = ∠D = 72°
Check: 108° + 72° + 108° + 72° = 360°. ✔
Thus, the measures of the angles of the parallelogram are 108°, 72°, 108° and 72°.
Q6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
[Image: Q6_figure.png]
Answer:
In a parallelogram the adjacent angles are supplementary.
∴ ∠A + ∠B = 180°
Since ∠A = ∠B,
⇒ 2∠A = 180°
⇒ ∠A = 90°
⇒ ∠B = 90°
Now, ∠C = ∠A (Opposite angles)
⇒ ∠C = 90°
∠D = ∠B (Opposite angles)
⇒ ∠D = 90°
Hence, all the angles of the parallelogram are 90°.
Q7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
[Image: Q7_figure.png]
Answer:
70° = z + 40° (Corresponding angles of a parallelogram)
⇒ 70° − 40° = z
⇒ z = 30°
Also,
y = 40° (Alternate interior angles)
Now, ∠E + ∠H = 180° (Adjacent angles are supplementary)
⇒ x + (z + 40°) = 180°
⇒ x + (30° + 40°) = 180°
⇒ x + 70° = 180°
⇒ x = 110°
∴ x = 110°, y = 40° and z = 30°
Q8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
[Image: Q8_figure.png]
Answer:
(i) We know that the lengths of opposite sides of a parallelogram are equal to each other.
Therefore, GU = SN and SG = NU
⇒ 3y − 1 = 26
⇒ 3y = 27
⇒ y = 9
Also, SG = NU
⇒ 3x = 18
⇒ x = 6
Hence, the measures of x and y are 6 cm and 9 cm.
(ii) We know that the diagonals of a parallelogram bisect each other.
y + 7 = 20 and x + y = 16
Now, y + 7 = 20
⇒ y = 13
Also, x + y = 16
⇒ x + 13 = 16
⇒ x = 3
Hence, the measures of x and y are 3 cm and 13 cm.
Q9. In the given figure both RISK and CLUE are parallelograms. Find the value of x.
[Image: Q9_figure.png]
Answer:
In parallelogram RISK,
∠RKS + ∠ISK = 180° [Adjacent angles of a parallelogram are supplementary.]
⇒ 120° + ∠ISK = 180°
⇒ ∠ISK = 60°
In parallelogram CLUE,
∠ULC = ∠CEU = 70° (Opposite angles of a parallelogram are equal.)
Now, in a triangle,
x + 60° + 70° = 180° (By angle sum property)
⇒ x = 50°
Q10. Explain how this figure is a trapezium. Which of its two sides are parallel?
[Image: Q10_figure.png]
Answer:
Here, ∠NML + ∠MLK = 180°
(100° + 80° = 180°)
Hence, NM ∥ LK and ML is a transversal. (If a transversal intersects the two given lines such that the sum of the angles on the same side of the transversal is 180°, then the given two lines will be parallel to each other.)
Therefore, KLMN is a trapezium as it has one pair of parallel lines, namely NM and LK.
Q11. Find m∠C in the given figure if AB ∥ DC.
[Image: Q11_figure.png]
Answer:
Given: AB ∥ DC
⇒ ∠B + ∠C = 180° [Angles on the same side of a transversal are supplementary.]
⇒ 120° + ∠C = 180°
⇒ ∠C = 60°
Q12. Find the measure of ∠P and ∠S if SP ∥ RQ in the given figure. (If you find m∠R, is there more than one method to find m∠P?)
[Image: Q12_figure.png]
Answer:
Given: SP ∥ RQ
∴ ∠P + ∠Q = 180° (Angles on the same side of a transversal)
⇒ ∠P + 130° = 180°
⇒ ∠P = 50°
Also,
∠R + ∠S = 180° (Angles on the same side of a transversal)
⇒ 90° + ∠S = 180°
⇒ ∠S = 90°
Yes, there is one more method to find the measure of ∠P.
We can apply the angle sum property of a quadrilateral to find m∠P.
∠P + ∠Q + ∠R + ∠S = 360°
⇒ ∠P + 130° + 90° + 90° = 360°
⇒ ∠P = 360° − 310° = 50°
∴ ∠P = 50° and ∠S = 90°