Question 5
We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
A rectangle is a special type of quadrilateral with all angles equal to 90 degrees.
Step 1 — Understanding the given properties
Let us consider a shape called a quadrilateral. A quadrilateral has four sides. We are given that its opposite sides are parallel. This means side AB is parallel to side DC. Also, side AD is parallel to side BC. We are also given that its opposite sides are equal in length. This means side AB has the same length as side DC. Also, side AD has the same length as side BC. A quadrilateral with these properties is called a parallelogram.
Step 2 — What a rectangle needs
A rectangle is a special type of quadrilateral. For a quadrilateral to be a rectangle, it must have all four angles equal to 90 degrees. This means angle A must be 90 degrees. Angle B must be 90 degrees. Angle C must be 90 degrees. Angle D must be 90 degrees. The properties of opposite sides being parallel and equal define a parallelogram. A parallelogram does not always have 90-degree angles.
Step 3 — Comparing definitions
Let's think about a parallelogram that is not a rectangle. Imagine pushing on the top side of a rectangle. It will tilt and become a parallelogram. Its opposite sides are still parallel. Its opposite sides are still equal in length. But its angles are no longer 90 degrees. Some angles become acute (less than 90 degrees). Other angles become obtuse (more than 90 degrees). So, a parallelogram has opposite sides parallel and equal. But it is not always a rectangle. Therefore, having opposite sides parallel and equal is not enough. We need the extra condition that all angles are 90 degrees. This means the given properties are not a complete definition for a rectangle.
Answer
Let ABCD be a quadrilateral in which opposite sides are parallel and equal. Here AB || DC and AD || BC. Also, AB = DC and AD = BC. In the quadrilateral ABCD, opposite sides are equal. For ABCD to be a rectangle, we require each angle to be 90 degrees. Given information AB || DC and AD || BC cannot help us to prove that each angle of ABCD is 90 degrees. Therefore, ABCD may not be a rectangle. Therefore, a rectangle cannot be defined as a quadrilateral with equal and parallel opposite sides.
More questions in FIO
Find all the other angles inside the following rectangles.
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°
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.
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?
We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
Find the remaining angles in the following quadrilaterals.
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Find the remaining angles in the following trapeziums—
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?
If PAIR and RODS are two rectangles, find .
Construct a square with diagonal 6 cm without using a protractor.
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. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).
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.
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.
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.
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.
(ii) A quadrilateral having three right angles must be a rectangle.
(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.
(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.
(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.
(vi) A quadrilateral in which all the angles are equal is a rectangle.
(vii) Isosceles trapeziums are parallelograms.