CBSE Class 8 Mathematics Understanding Quadrilaterals Worksheet Set 05

WORK SHEET - SUBJECT: Maths - Chapter 3: Quadrilateral

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 that proving one pair of opposite sides equal and parallel is a standard sufficient condition for a quadrilateral to be 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: Consecutive angles of a parallelogram are supplementary, so if one is \(90^\circ\), the adjacent one is \(180^\circ - 90^\circ = 90^\circ\).

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: This is the converse of the property that opposite angles of a parallelogram are equal.

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

Question 5. Diagonals necessarily bisect opposite angles in a …………………………
Answer: rhombus (or square)
In simple words: In a rhombus and a square, the diagonals split the vertex angles into two equal halves.

Exam Tip: Note that diagonals of a general parallelogram do not bisect the interior 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 \(1x\), \(2x\), \(3x\), and \(4x\).
We know that the sum of all interior angles of a quadrilateral is \(360^\circ\).
\(1x + 2x + 3x + 4x = 360^\circ\)
\(10x = 360^\circ\)
\(x = \frac{360^\circ}{10} = 36^\circ\)
Therefore, 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 sum of \(360^\circ\) in the given ratio parts to find each angle.

Exam Tip: Always verify your answer by adding the calculated angles: \(36^\circ + 72^\circ + 108^\circ + 144^\circ = 360^\circ\).

Question 7. The measure of one angle of a parallelogram is \(80^\circ\). Find the measure of other angles.
Answer: Let the given angle be \(\angle A = 80^\circ\).
In a parallelogram, opposite angles are equal:
\(\angle C = \angle A = 80^\circ\)
Consecutive angles are supplementary (their sum is \(180^\circ\)):
\(\angle B = 180^\circ - 80^\circ = 100^\circ\)
The opposite angle to \(\angle B\) is also equal:
\(\angle D = \angle B = 100^\circ\)
Therefore, the other three angles are \(100^\circ\), \(80^\circ\), and \(100^\circ\).
In simple words: Opposite angles are equal, and adjacent angles add up to \(180^\circ\).

Exam Tip: Subtract the given angle from \(180^\circ\) to find the adjacent angle directly.

Question 8. The sum of interior angles of a polygon of 10 sides is……
Answer: The formula for the sum of interior angles of an \(n\)-sided polygon is given by \((n - 2) \times 180^\circ\).
Here, number of sides \(n = 10\).
Sum = \((10 - 2) \times 180^\circ\)
Sum = \(8 \times 180^\circ = 1440^\circ\)
In simple words: Subtract 2 from the number of sides and multiply the result by \(180^\circ\).

Exam Tip: Memorize the formula \((n - 2) \times 180^\circ\) as it applies to all convex polygons.

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 \(x\).
Given one angle = \(108^\circ\).
The sum of all four interior angles of a quadrilateral is \(360^\circ\).
\(108^\circ + x + x + x = 360^\circ\)
\(108^\circ + 3x = 360^\circ\)
\(3x = 360^\circ - 108^\circ\)
\(3x = 252^\circ\)
\(x = \frac{252^\circ}{3} = 84^\circ\)
Therefore, the value of each equal angle is \(84^\circ\).
In simple words: Subtract the known angle from \(360^\circ\) and divide the remaining sum by 3.

Exam Tip: Double-check subtraction: \(360 - 108 = 252\), and division: \(252 \div 3 = 84\).

Question 10. Exterior angle of a regular polygon is \(24^\circ\). Find the number of sides of the polygon.
Answer: The sum of all exterior angles of any convex polygon is \(360^\circ\).
For a regular polygon with \(n\) sides, each exterior angle is equal.
Number of sides \(n = \frac{360^\circ}{\text{Exterior Angle}}\)
\(n = \frac{360^\circ}{24^\circ} = 15\)
Therefore, the polygon has 15 sides.
In simple words: Divide \(360^\circ\) by the measure of one exterior angle to get the number of sides.

Question 11. Find the values of x, y and z in the given parallelogram ABCD
Answer: In parallelogram ABCD, line \(AB \parallel CD\) and line \(AD \parallel BC\).
1. Since \(AB \parallel CD\) and diagonal \(AC\) acts as a transversal, alternate interior angles are equal:
\(y = 40^\circ\)
2. Since \(AD \parallel BC\) and diagonal \(AC\) acts as a transversal, alternate interior angles are equal:
\(x = 70^\circ\)
3. To find \(z\), consider triangle ABC or use angle sum properties. In \(\triangle ABC\), the angles are \(\angle CAB = z\), \(\angle ACB = y = 40^\circ\), and \(\angle B\) can be found. Alternatively, opposite angles of the parallelogram are equal:
\(\angle D = \angle B = 70^\circ + 40^\circ = 110^\circ\).
Using consecutive angles property (\(\angle A + \angle B = 180^\circ\)):
\(z + 40^\circ + 70^\circ + 40^\circ = 180^\circ\) (Sum of angles in \(\triangle ABC\) is \(180^\circ\), where \(\angle B = 70^\circ\)):
\(z + 70^\circ + 40^\circ = 180^\circ\)
\(z + 110^\circ = 180^\circ\)
\(z = 180^\circ - 110^\circ = 70^\circ\)
Therefore, \(x = 70^\circ\), \(y = 40^\circ\), and \(z = 70^\circ\).
In simple words: Use alternate interior angles formed by parallel lines and the transversal diagonal.

Exam Tip: Look for "Z" shapes (alternate interior angles) when parallel lines are cut by a transversal.

Question 12. Explain why a rectangle is a convex polygon.क्की
Answer: A rectangle is a convex polygon because all of its interior angles are strictly less than \(180^\circ\) (each measures \(90^\circ\)), and if you take any two points inside the rectangle, the line segment joining them lies entirely inside the polygon. Furthermore, no portion of any diagonal or side extends outside the region (all diagonals lie completely in the interior).
In simple words: A polygon is convex if all its interior angles are less than \(180^\circ\) and no vertex points inward.

Exam Tip: Contrast this with a concave polygon, which has at least one interior angle greater than \(180^\circ\) (a reflex angle).

Question 13. RENT is a rectangle its diagonals meet at O. Find x, OR = 2x +4 and OT = 3x +1
Answer: In a rectangle, the diagonals are equal in length and they bisect each other.
Therefore, all four segments from the intersection point \(O\) to the vertices are equal:
\(OR = OT = ON = OE\)
Given:
\(OR = 2x + 4\)
\(OT = 3x + 1\)
Since \(OR = OT\):
\(2x + 4 = 3x + 1\)
\(4 - 1 = 3x - 2x\)
\(x = 3\)
In simple words: Diagonals of a rectangle are equal and bisect each other, so the half-diagonal segments are equal.

Exam Tip: Always equate the two half-diagonal expressions directly when dealing with rectangle diagonal properties.

Question 14. Write three properties of a parallelogram.
Answer: Three important properties of a parallelogram are:
1. Opposite sides are equal in length (\(AB = DC\) and \(AD = BC\)).
2. Opposite angles are equal in measure (\(\angle A = \angle C\) and \(\angle B = \angle D\)).
3. Diagonals bisect each other.
In simple words: Opposite sides are equal, opposite angles are equal, and diagonals cut each other in half.

Exam Tip: You can also mention that consecutive angles are supplementary as an alternative property.

Question 15. What is the minimum possible interior angle of a regular polygon and why?
Answer: The minimum possible interior angle of a regular polygon is \(60^\circ\).
Reason: The polygon with the least number of sides is an equilateral triangle, which has 3 sides (\(n = 3\)).
The formula for each interior angle of a regular polygon is:
\(\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}\)
For \(n = 3\):
\(\text{Interior Angle} = \frac{(3 - 2) \times 180^\circ}{3} = \frac{1 \times 180^\circ}{3} = 60^\circ\)
As the number of sides increases, the interior angle increases, so \(60^\circ\) is the minimum possible value.
In simple words: An equilateral triangle has 3 sides and the smallest possible interior angle of \(60^\circ\).

Question 16. Find the values of x, y and z of a rectangle ABCD
Answer: In rectangle ABCD, diagonals are equal and bisect each other, and each corner angle is \(90^\circ\).
1. Triangle OBC is an isosceles triangle because \(OB = OC\) (since diagonals are equal and bisect each other, all half-diagonal lengths from the center are equal).
Therefore, base angles are equal: \(\angle OBC = \angle OCB = 30^\circ\).
In \(\triangle OBC\), the sum of angles is \(180^\circ\):
\(x + 30^\circ + 30^\circ = 180^\circ\)
\(x + 60^\circ = 180^\circ\)
\(x = 180^\circ - 60^\circ = 120^\circ\)
2. Since \(\angle OAB\) and \(\angle OBC\) or alternate interior angles/isosceles properties apply, let us find \(y\):
In \(\triangle OAB\), \(OA = OB\), so \(\angle OAB = \angle OBA\). Also, the corner angle \(\angle ABC = 90^\circ\), so \(\angle OAB = 90^\circ - 30^\circ = 60^\circ\).
Thus, \(y = 60^\circ\).
3. Similarly, \(z = 60^\circ\) (or by alternate interior angles / isosceles triangle ODA).
Therefore, \(x = 120^\circ\), \(y = 60^\circ\), and \(z = 60^\circ\).
In simple words: Use the property that half-diagonals of a rectangle form isosceles triangles with the sides.

Exam Tip: The four triangles formed by the intersecting diagonals of a rectangle are two pairs of congruent isosceles triangles.

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 \(AB \parallel CD\):
1. Base angles on the same base are equal. Given \(\angle B = 80^\circ\), the other base angle at the same base is:
\(\angle A = \angle B = 80^\circ\)
2. Angles on the same side of a transversal between parallel lines (\(AB \parallel CD\)) are supplementary:
\(\angle A + \angle D = 180^\circ\)
\(80^\circ + \angle D = 180^\circ\)
\(\angle D = 180^\circ - 80^\circ = 100^\circ\)
3. Similarly, for the other base angles:
\(\angle C = \angle D = 100^\circ\)
Therefore, the other angles are \(80^\circ\), \(100^\circ\), and \(100^\circ\).
In simple words: Base angles of an isosceles trapezium are equal, and adjacent angles between parallel sides add up to \(180^\circ\).

Exam Tip: Base angles at the same parallel line are equal (\(\angle A = \angle B\) and \(\angle D = \angle C\)).

Question 18. ABCD is a parallelogram. Find x and y.
Answer: In parallelogram ABCD with diagonal AC:
1. Since \(AD \parallel BC\) and diagonal AC is a transversal, alternate interior angles are equal:
\(12x = 60^\circ\)
\(x = \frac{60^\circ}{12} = 5^\circ\)
2. Similarly, considering alternate interior angles for the other pair of parallel sides (\(AB \parallel CD\)):
\(7y = 28^\circ\)
\(y = \frac{28^\circ}{7} = 4\)
Therefore, \(x = 5\) and \(y = 4\).
In simple words: Equate alternate interior angles formed by the diagonal and the parallel sides.

Exam Tip: Always identify the parallel lines and the transversal line carefully before setting up alternate interior angle equations.

Question 19. Write the difference between diagonals of a square and a rhombus.
Answer: The primary difference between the diagonals of a square and a rhombus is:
- In a square, the diagonals are strictly **equal** in length (\(AC = BD\)).
- In a general rhombus, the diagonals are **not equal** in length (one diagonal is longer than the other).
*(Note: Both have diagonals that bisect each other at right angles, but equality of length distinguishes a square from a rhombus.)*
In simple words: Square diagonals are equal in length, whereas rhombus diagonals are of different lengths.

Question 20. Find the values of x and y in the given figure:
Answer: In the given kite figure (or symmetric quadrilateral where diagonals intersect at \(90^\circ\)):
1. The diagonals of a kite intersect at right angles (\(90^\circ\)).
Considering the top triangle formed by the intersection point and vertex P: the angle sum is \(180^\circ\), with one angle being \(90^\circ\) and another being \(60^\circ\):
\(x = 90^\circ - 60^\circ = 30^\circ\)
2. Similarly, using the bottom triangle or vertex R where the angle is \(30^\circ\):
\(y = 90^\circ - 30^\circ = 60^\circ\)
Therefore, \(x = 30^\circ\) and \(y = 60^\circ\).
In simple words: Since diagonals meet at \(90^\circ\), the acute angles in the right triangles add up to \(90^\circ\).

Exam Tip: In any right-angled triangle, the two acute angles are complementary (their sum is \(90^\circ\)).