Quadrilaterals

Are squares the only quadrilaterals that have equal sidelengths? Let us explore this question through construction.

Draw two equal sides AD and AB, that are not perpendicular to each other.

Can we complete this quadrilateral so that all its sides are of the same length?

Mark a point C whose distance from B and D is equal to AB (or AD). To do this, measure AB using a compass. Keeping this length as the radius, cut arcs from B and D.

Place two rubber bands perpendicular to each other, forming diagonals of equal length. Join the ends.

What is the quadrilateral that you get? Justify your answer.

Extend one of the diagonals on both sides by 2 cm.

What quadrilateral will you get now? Justify your answer.

Take two cardboard cutouts of an equilateral triangle of sidelength 8 cm.

• Can you join them to get a quadrilateral?
• What type of a quadrilateral is this? Justify your answer.

Take two cardboard cutouts of an isosceles triangle with sidelengths 8 cm, 8 cm, and 6 cm.

• What are the different ways they can be joined to get a quadrilateral?
• What quadrilaterals are these? Justify your answers.

Take two cardboard cutouts of a scalene triangle with sides 6 cm, 9 cm, and 12 cm.

• What are the different ways they can be joined to get a quadrilateral?
• Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.

Which Quad?

Gameplay

1. Fold a sheet into half.
2. Now, fold it once more into a quarter.
3. Make a triangular crease at the corner that is at the middle of the paper.
4. Open the sheet. What is the shape formed by the creases?
5. How would you fold the quarter paper to get the kinds of creases shown in the following image.
6. How would you fold the quarter paper such that a square is formed?

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 $\angle\text{IOD}$.

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.

Observe the following figures.

Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?

Are there other ways to define a rectangle?

A Carpenter's Problem

A carpenter needs to put together two thin strips of wood, as shown in Fig. 1, so that when a thread is passed through their endpoints, it forms a rectangle. She already has one 8 cm long strip. What should be the length of the other strip? Where should they both be joined?

Let us first model the structure that the carpenter has to make. The strips can be modelled as line segments. They are the diagonals of the quadrilateral formed by their endpoints. For the quadrilateral to be a rectangle, we need to answer the following questions —

1. What is the length of the other diagonal?
2. What is the point of intersection of the two diagonals?
3. What should the angle be between the diagonals?

Can the following equalities be used to establish that $\Delta AOD \cong \Delta COB$?

• $AO = CO$ (proved above)
• $\angle AOB = \angle COD$ (vertically opposite angles)
• $AD = CB$

Context: Let us check what quadrilateral we get if we draw the two diagonals such that their lengths are equal, they bisect each other and have an arbitrary angle, say $60^\circ$, between them as shown in the figure to the right.

Q. Can you find all the remaining angles?

Context: In $\Delta AOB$, since $OA = OB$, the angles opposite them are equal, say $a$.

Q. Can you find the value of $a$?

Can we now identify what type of quadrilateral ABCD is?

Notice that its angles all add up to 90° (30° + 60°).

What can we say about its sides?

Will ABCD remain a rectangle if the angles between the diagonals are changed? Can we generalise this?

Take one of the angles between the diagonals as $x$.

Context: We can compute the four angles between the diagonals to be $x, x, 180 - x,$ and $180 - x.$

Q. Can you find the other angles?

Context: Since we know that $\Delta AOB$ is isosceles, we can denote the measures of both of its base angles by $a$.

Q. What is the value of $a$ (in degrees) in terms of $x$?

Context: Thus, all four angles of the quadrilateral ABCD are 90°.

Q. What can we say about AB and CD, and AD and BC?

In the earlier definition, we stated that a rectangle has (a) opposite sides of equal length, and (b) all angles equal to 90°. Would we be wrong if we just define a rectangle as a quadrilateral in which all the angles are 90°?

If you think that this definition is incomplete, try constructing a quadrilateral in which the angles are all 90° but the opposite sides are not equal.

Are you able to construct such a quadrilateral?

Is it wrong to write ΔBAD ≅ ΔCDB? Why?

Can you similarly show that AB is parallel to DC (AB || DC)?

In the quadrilaterals below, are there any non-rectangles?

Let us consider the Carpenter's Problem again. If the wooden strips have to be placed such that the thread passing through their endpoints forms a square, what must be done?

What more needs to be done to get equal sidelengths as well? Can this be achieved by properly choosing the angle between the diagonals? See if you can reason and/or experiment to figure this out!

Can this be used to find the angles $\angle\text{BOA}$ and $\angle\text{BOC}$ formed by the diagonals?

Context: The diagonals of a square are of equal lengths and bisect each other at right angles.

Q. Using this fact, construct a square with a diagonal of length 8 cm.

Context: Since a square is a special type of rectangle, all the properties of a rectangle hold true for a square.

Q. Verify if this is true by going through geometric reasoning in Deduction 1 and Deduction 2, and see if they apply to a square as well.

Q. Similarly, find $\angle 2$ and $\angle 4$.

4.2 Angles in a Quadrilateral

Is it possible to construct a quadrilateral with three angles equal to 90° and the fourth angle not equal to 90°?

But why not?

Are there quadrilaterals that have parallel opposite sides that are not rectangles?

Construct such a figure by recalling how parallel lines can be constructed using a ruler and a set-square, or a compass and a ruler.

Context: Consider a parallelogram $ABCD$ with adjacent sides of lengths $4\text{ cm}$ and $5\text{ cm}$, and an angle of $30^\circ$ between them.

Q. What are the remaining angles of the parallelogram? What are the lengths of the remaining sides? See if you can reason out and/or experiment to figure these out.

Deduction 7— What can we say about the sides of a parallelogram?

By looking at a parallelogram, it appears that the opposite sides are equal. Can we again use congruence to show this? Which two triangles can be considered for this?

Is it wrong to write $\Delta\text{ABD} \cong \Delta\text{CBD}$? Why?

Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed.

Context: We see that the diagonals of a parallelogram need not be equal.

Q. Do they bisect each other (do they intersect at their midpoints)? Reason and/or experiment to figure this out.

Is it wrong to write $\Delta\text{AOE} \cong \Delta\text{SOY}$? Why?

Do the diagonals of a parallelogram intersect at a particular angle?

What are the other angles of the rhombus ABCD that we have constructed? Reason and/or experiment to figure this out.

It can be seen that $\Delta GAE \cong \Delta MAE$ (How?)

So a rhombus is a parallelogram, and a rectangle is also a parallelogram. How can this be represented using a Venn diagram?

Where will the set of squares occur in this diagram?

Are the diagonals of a rhombus equal?

Do the diagonals of a rhombus intersect at any particular angle? Reason out and/or experiment to figure this out!

In the rhombus GAME, we have $\Delta\text{GEO} \cong \Delta\text{MEO}$ (why?).

In the kite, show that the diagonal $BD$

(i) bisects $\angle ABC$ and $\angle ADC$,

(ii) bisects the diagonal $AC$, that is, $AO = OC$, and is perpendicular to it.

Hint: Is $\Delta AOB \cong \Delta COB$?

Construct a trapezium. Measure the base angles (marked in the figure).

Can you find the remaining angles without measuring them?

How do we construct an isosceles trapezium?

Construct an isosceles trapezium UVWX, with UV || XW. Measure ∠U.

Now, it can be shown that $\Delta\text{UXY} \cong \Delta\text{VWZ}$. (How?)