Chapter 8 Quadrilaterals Figure it out Q & Answers
Class 8 Maths (Ganita Prakash) - Chapter 4: QUADRILATERALS
Study Material & Competency-Based Worksheet
1. Introduction to Quadrilaterals
A quadrilateral is a closed figure with four sides, four vertices, and four angles. The sum of the interior angles of any quadrilateral is 360°.
SUMMARY
A rectangle is a quadrilateral in which the angles are all 90°.
Properties of a rectangle —
Opposite sides of a rectangle are equal.
Opposite sides of a rectangle are parallel to each other.
Diagonals of a rectangle are of equal length and they bisect each other.
A square is a quadrilateral in which all the angles are 90°, and all the sides are of equal length. Properties of a square —
The opposite sides of a square are parallel to each other.
The diagonals of a square are of equal lengths and they bisect each other at 90°.
The diagonals of a square bisect the angles of the square.
A parallelogram is a quadrilateral in which opposite sides are parallel.
Properties of a parallelogram —
The opposite sides of a parallelogram are equal.
In a parallelogram, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a parallelogram bisect each other.
A rhombus is a quadrilateral in which all the sides have the same length.
Properties of a rhombus —
The opposite sides of a rhombus are parallel to each other.
In a rhombus, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a rhombus bisect each other at right angles.
The diagonals of a rhombus bisect its angles.
A kite is a quadrilateral with two non-overlapping adjacent pairs of sides having the same length.
A trapezium is a quadrilateral having at least one pair of parallel opposite sides.
The sum of the angle measures in a quadrilateral is 360°
2. Types of Quadrilaterals
Opposite sides of a rectangle are equal.
Opposite sides of a rectangle are parallel to each other.
Diagonals of a rectangle are of equal length and they bisect each other.
The opposite sides of a square are parallel to each other.
The diagonals of a square are of equal lengths and they bisect each other at 90°.
The diagonals of a square bisect the angles of the square.
The opposite sides of a parallelogram are equal.
In a parallelogram, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a parallelogram bisect each other.
The opposite sides of a rhombus are parallel to each other.
In a rhombus, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a rhombus bisect each other at right angles.
The diagonals of a rhombus bisect its angles.
Trapezium
A quadrilateral with one pair of parallel sides.
A quadrilateral with one pair of parallel sides.
Kite
A quadrilateral with two distinct pairs of adjacent sides that are equal.
Diagonals are perpendicular to each other.
A quadrilateral with two distinct pairs of adjacent sides that are equal.
Diagonals are perpendicular to each other.
Parallelogram
A quadrilateral with both pairs of opposite sides parallel.
Key Properties:
Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.
Consecutive angles are supplementary (sum = 180°).
A quadrilateral with both pairs of opposite sides parallel.
Key Properties:
Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.
Consecutive angles are supplementary (sum = 180°).
Rhombus
A parallelogram with all four sides equal.
Key Properties (inherits all parallelogram properties plus):
Diagonals are perpendicular bisectors of each other.
Diagonals bisect the interior angles.
A parallelogram with all four sides equal.
Key Properties (inherits all parallelogram properties plus):
Diagonals are perpendicular bisectors of each other.
Diagonals bisect the interior angles.
Rectangle
A parallelogram with all angles equal to 90°.
Key Properties (inherits all parallelogram properties plus):
Each angle is 90°.
Diagonals are equal in length.
A parallelogram with all angles equal to 90°.
Key Properties (inherits all parallelogram properties plus):
Each angle is 90°.
Diagonals are equal in length.
Square ☐
A parallelogram that is both a rectangle and a rhombus.
Key Properties (inherits all properties of rectangle and rhombus):
All sides are equal.
All angles equal to 90°.
Diagonals are equal, perpendicular, and bisect each other.
A parallelogram that is both a rectangle and a rhombus.
Key Properties (inherits all properties of rectangle and rhombus):
All sides are equal.
All angles equal to 90°.
Diagonals are equal, perpendicular, and bisect each other.
3. Important Theorems & Midpoint Theorem
The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of it.
The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of it.
Figure it Out Page number-107-109
Question 1
Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Show Answer
The quadrilateral is a rhombus.
Sides: All four sides are 4 cm.
Angles: Two angles are 60° and two angles are 120°.
Show Explanation
When you join two equilateral triangles along a common side, you create a quadrilateral.
The common side becomes a side of the quadrilateral. The other two sides from each triangle become adjacent sides.
Since all sides of an equilateral triangle are equal, the quadrilateral will have all four sides equal to 4 cm. This makes it a rhombus.
The angle at the vertex where the triangles are joined is the sum of two angles from the triangles. In an equilateral triangle, each angle is 60°. So, the joined angle is 60° + 60° = 120°.
The angle opposite to this 120° angle will also be 120° (property of a rhombus/parallelogram).
The other two angles will be equal to each other and supplementary to 120°. So, each of them is (360° - 120° - 120°)/2 = 60°.
Question 2
Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Show Answer
Construction completed.
Show Explanation
Steps of Construction:
Draw a vertical line segment (the longer diagonal) BD of length 8 cm.
Mark its midpoint O.
At point O, draw a perpendicular line (the shorter diagonal).
On this perpendicular line, on both sides of O, mark points A and C such that OA = OC = 3 cm (half of 6 cm).
Join points A, B, C, and D to form kite ABCD, where AB = AD and CB = CD.
Question 3
Find the remaining angles in the following trapeziums.
Trapezium 1: Angles given: 135°, 105°, 100°
Trapezium 2: Angles given: 100°, 60°, 120° (from image description)
Show Answer
Trapezium 1: The fourth angle is 20°.
Trapezium 2: The fourth angle is 80°.
Show Explanation
The sum of the interior angles of any quadrilateral is 360°.
For Trapezium 1: Let the fourth angle be .
For Trapezium 2: Let the fourth angle be .
Question 4
Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions:
(i) What is the quadrilateral that is both a kite and a parallelogram?
(ii) Can there be a quadrilateral that is both a kite and a rectangle?
(iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
Show Answer
(i) Rhombus
(ii) No
(iii) No. A rhombus is a special type of kite where all sides are equal.
Show Explanation
(i) A rhombus is a parallelogram with all sides equal. A rhombus also qualifies as a kite because it has two distinct pairs of adjacent sides that are equal (in fact, all four sides are equal). It is the only parallelogram that is also a kite.
(ii) For a quadrilateral to be a kite, two pairs of adjacent sides must be equal. For it to be a rectangle, all angles must be 90°. A rectangle that is not a square has only opposite sides equal, not adjacent sides. Therefore, the only quadrilateral that is both a kite and a rectangle is a square. A general rectangle is not a kite.
(iii) No, not every kite is a rhombus. A kite only requires two pairs of adjacent sides to be equal. A rhombus requires all four sides to be equal. Therefore, a rhombus is a special type of kite where the two pairs of equal adjacent sides are, in fact, all equal.
Question 5
If PAIR and RODS are two rectangles, find ∠IOD.
Show Answer
∠IOD = 90°
Show Explanation
PAIR and RODS are rectangles. In a rectangle, all angles are 90°.
The angle in question, ∠IOD, is part of the larger angle in one of the rectangles.
Given the naming, point O is likely the intersection of the diagonals. In a rectangle, the diagonals are equal and bisect each other, creating isosceles triangles.
Without a specific diagram, the most logical and common answer for an angle formed by intersecting lines in a rectangle (especially one involving the center) is 90°, as the lines are often perpendicular or form 90° angles with the sides.
Question 6
Construct a square with diagonal 6 cm without using a protractor.
Show Answer
Construction completed.
Show Explanation
Steps of Construction:
Draw a line segment AC of length 6 cm (this will be the diagonal).
Draw the perpendicular bisector of AC. Let it intersect AC at point O.
With O as center and radius equal to half the diagonal (3 cm), draw arcs to intersect the perpendicular bisector at points B and D.
Join points A, B, C, and D to form square ABCD.
Question 7
CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. 
Show Answer
UVWX is a Square.
Show Explanation
Geometric Reasoning:
Consider square CASE. By the Midpoint Theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side.
This can be used to show that all sides of UVWX are equal and all its angles are 90°.
The sides of UVWX are parallel to the diagonals of the original square CASE. Since the diagonals of a square are equal and perpendicular, the sides of UVWX are all equal and perpendicular to each other.
Therefore, UVWX is a square.
Question 8
If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
Show Answer
Yes.
Show Explanation
Geometric Reasoning:
A quadrilateral with four equal sides is a rhombus.
In a rhombus, opposite angles are equal, and consecutive angles are supplementary.
If one angle is 90°, its opposite angle is also 90°.
The consecutive angles to this 90° angle must be supplementary, so they are 180° - 90° = 90° each.
Therefore, all angles are 90°.
A rhombus with all angles 90° is a square.
Question 9
What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Hint: Draw a diagonal and check for congruent triangles.
Show Answer
It is a Parallelogram.
Show Explanation
Justification:
Let the quadrilateral be ABCD where AB = CD and BC = AD.
Draw diagonal AC.
Consider ΞABC and ΞCDA.
AB = CD (Given)
BC = DA (Given)
AC = CA (Common side)
Therefore, ΞABC ≅ ΞCDA by the SSS congruence rule.
By CPCT, ∠BCA = ∠DAC and ∠BAC = ∠DCA.
But these are alternate interior angles for lines BC & AD and AB & CD respectively.
Since alternate interior angles are equal, BC ∥ AD and AB ∥ CD.
A quadrilateral with both pairs of opposite sides parallel is a parallelogram.
Question 10
Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
(Note: The "following one" likely refers to a concave quadrilateral)
Show Answer
Yes, the sum is still 360°.
Show Explanation
Geometric Reasoning:
The sum of the interior angles of any quadrilateral, whether convex or concave, is always 360°. This can be proven by dividing the quadrilateral into two triangles by drawing a diagonal. The sum of angles in each triangle is 180°, so for two triangles, it is 360°. This holds true for concave quadrilaterals as well.
Question 11
State whether the following statements are true or false. Justify your answers.
(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
Show Answer
(i) False
Show Explanation
Justification:
A quadrilateral whose diagonals bisect each other is a parallelogram. A parallelogram with equal diagonals is a rectangle. This rectangle could be a square, but it does not have to be. It is only a square if the diagonals are also perpendicular. Therefore, the statement is false; the quadrilateral is a rectangle, not necessarily a square.
Figure it Out (Page 102)
Question 1
Find the remaining angles in the following quadrilaterals.
Show Answer
The sum of the remaining angles will depend on the three given angles in each quadrilateral. The general solution is:
Remaining Angle = 360° - (Sum of three given angles)
Show Explanation
The sum of the interior angles of any quadrilateral is 360°. To find one missing angle, subtract the sum of the three known angles from 360°.
Example:
If the three given angles are 80°, 90°, and 110°:
Sum of known angles = 80° + 90° + 110° = 280°
Remaining angle = 360° - 280° = 80°
Question 2
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Show Answer
Construction completed.
Show Explanation
Steps of Construction:
Draw a horizontal line segment AC of length 7 cm.
Find its midpoint O.
At point O, construct a line making an angle of 140° with AC.
On this line, on both sides of O, mark points B and D such that OB = OD = 2.5 cm (half of 5 cm).
Join points A, B, C, and D to form parallelogram ABCD.
Question 3
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Show Answer
Construction completed.
Show Explanation
Steps of Construction:
Draw a vertical line segment BD of length 4 cm.
Find its midpoint O.
At point O, draw a perpendicular line to BD.
On this perpendicular line, on both sides of O, mark points A and C such that OA = OC = 2.5 cm (half of 5 cm).
Join points A, B, C, and D to form rhombus ABCD.
Figure it Out 94
Question 1
Find all the other angles inside the following rectangles.
Show Answer
In any rectangle, all four angles are 90°.
Show Explanation
By definition, a rectangle is a quadrilateral with all four interior angles equal to 90°.
Question 2
Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of (i) 30° (ii) 40° (iii) 90° (iv) 140°
Show Answer
These will all be rectangles that are "tilted" or sheared, except for (iii) which is a square.
Show Explanation
A quadrilateral whose diagonals bisect each other is a parallelogram.
A parallelogram with equal diagonals is a rectangle.
Therefore, all these quadrilaterals are rectangles.
The angle at which the diagonals intersect changes the shape's appearance (making it a "rhomboid" rectangle), but it remains a rectangle. When the intersection angle is 90°, the rectangle is a square.
Construction Steps (for any angle):
Draw a line segment of 8 cm (diagonal 1) and mark its midpoint O.
At O, draw another line segment of 8 cm (diagonal 2) making the given angle (30°, 40°, etc.) with the first diagonal.
Ensure the second diagonal is also bisected at O.
Join the endpoints of the diagonals to form the quadrilateral.
Question 3
Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
Show Answer
APML is a Square.
Show Explanation
PL and AM are perpendicular diameters. So, they intersect at right angles at the center O.
All radii of a circle are equal: OA = OP = OM = OL.
The diameters bisect each other at O, so OA=OM and OP=OL.
In quadrilateral APML, the diagonals (PL and AM) are equal (both are diameters), they bisect each other, and they are perpendicular.
A quadrilateral with diagonals that are equal, bisect each other, and are perpendicular is a square.
Therefore, APML is a square.
Question 4
We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
Show Answer
Use the 3-4-5 Pythagorean triple method or the method of constructing a perpendicular bisector.
Show Explanation
Method 1 (Using a thread as a compass):
Place one stick horizontally. This is the base line.
Take the thread and tie it to both ends of the second stick, making a long loop.
Use the thread to mark a point exactly 3 units from one end of the base stick and 4 units from the other end (using the stick itself or knots in the thread as a unit measure).
The angle between the two sticks at the point where the 3-unit and 4-unit marks meet will be 90° if the distance between the two ends is 5 units (since 3² + 4² = 5²).
Method 2 (Perpendicular Bisector):
Place the two sticks of equal length so that they bisect each other. (Use the thread to find the midpoints).
When two equal line segments bisect each other, they are the diagonals of a rectangle.
Since the sticks are equal, they are the diagonals of a square, and thus they are perpendicular to each other, creating a 90° angle.
Show Answer
No.
Show Explanation
A quadrilateral with both pairs of opposite sides parallel and equal is a parallelogram.
A rectangle is a special type of parallelogram where all angles are 90°.
However, not all parallelograms are rectangles. For example, a rhombus has opposite sides parallel and equal, but its angles are not 90°.
Therefore, "opposite sides parallel and equal" is the definition of a parallelogram, not a rectangle. The correct definition of a rectangle is "a parallelogram with one right angle" (or "a parallelogram with all angles equal").
