Quadrilaterals

Hello my dear students! Welcome to today's mathematics lesson. I am so happy to see you all here, ready to learn something new and exciting. Today, we are going to study a very interesting chapter — Chapter 4: Quadrilaterals. Now, before we begin, let me ask you a question. Can you look around your classroom and tell me what shapes you see that have four sides? Yes, the door, the blackboard, the windows, maybe the tiles on the floor — many of these are quadrilaterals. So, quadrilaterals are all around us, and today we will understand them in great detail.

Let me start by telling you what the word "quadrilateral" means. The word comes from Latin — "quadri" means four, and "latus" means side. So, a quadrilateral is simply a four-sided figure. Now, look at the figures shown in your book. Some of them are quadrilaterals and some are not. Can you tell why? A quadrilateral must have exactly four sides and four angles. The angles of a quadrilateral are the angles between its sides. So, figures (i), (ii), and (iii) in your book are quadrilaterals because they each have four sides and four angles, while the others do not satisfy this condition.

Now, let us begin with the most familiar quadrilaterals that you already know — rectangles and squares. We will explore them in great detail and discover some very interesting properties.

Let us define a rectangle. A rectangle is a quadrilateral in which first, all the angles are right angles, that is, 90 degrees each, and second, the opposite sides are of equal length. This definition tells us exactly what conditions a quadrilateral must satisfy to be called a rectangle. But wait, is there another way to define a rectangle? Let us think about this.

To understand this better, let us consider a very practical problem — a carpenter's problem. Imagine a carpenter who needs to put together two thin strips of wood, as shown in figure 1, so that when a thread is passed through their endpoints, it forms a rectangle. She already has one strip that is 8 centimeters long. What should be the length of the other strip? Where should they both be joined? This is a very interesting question, and by answering it, we will discover some amazing properties of rectangles.

Let us model this situation. The strips can be thought of as line segments. They are actually the diagonals of the quadrilateral formed by their endpoints. For the quadrilateral to be a rectangle, we need to answer three important questions. First, what is the length of the other diagonal? Second, what is the point of intersection of the two diagonals? And third, what should be the angle between the diagonals? Let us answer these questions one by one using geometric reasoning.

First, let us suppose that we have placed the diagonals such that their endpoints form the vertices of a rectangle, as shown in figure 2. Now, let us find the length of the other diagonal.

Since ABCD is a rectangle, we have AB equal to CD. Also, angle BAD and angle CDA are both 90 degrees. And AD is common to both triangles ADC and DAB. So, triangle ADC is congruent to triangle DAB by the SAS congruence condition. Therefore, AC equals BD, since they are corresponding parts of congruent triangles. This shows that the diagonals of a rectangle always have the same length. So, the other diagonal must also be 8 centimeters long. You can verify this property by constructing some rectangles and measuring their diagonals.

Now, let us find the point of intersection of the two diagonals. This can also be found using congruence. We need to know the relation between OA and OC, and between OB and OD. Look at the rectangle ABCD. The angles marked in blue are equal because they are vertically opposite angles. Now, to show congruence, consider angle 1 and angle 2. Are they equal? Since angle B is 90 degrees, angle 3 plus angle 1 equals 90 degrees. In triangle BCD, since angle 3 plus angle 2 plus 90 equals 180, we have angle 3 plus angle 2 equals 90 degrees. So, angle 1 equals angle 2, which is 90 degrees minus angle 3. Thus, by the AAS condition for congruence, triangle AOB is congruent to triangle COD. Hence, OA equals OC and OB equals OD, since they are corresponding parts of congruent triangles. So, O is the midpoint of both AC and BD. This shows that the diagonals of a rectangle always intersect at their midpoints. When the diagonals cross at their midpoints, we say that the diagonals bisect each other. Bisecting a quantity means dividing it into two equal parts. So, to get a rectangle, the diagonals must be drawn so that they are equal and intersect at their midpoints.

Now, let us think about the third question — what are the angles between the diagonals? 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 degrees, between them. Can you find all the remaining angles? We can find the remaining angles between the diagonals using our understanding of vertically opposite angles and linear pairs.

In triangle AOB, since OA equals OB, the angles opposite them are equal, say a. In triangle AOB, we have a plus a plus 60 equals 180, because the sum of interior angles of a triangle is 180 degrees. Therefore, 2a equals 120, so a equals 60 degrees. Similarly, we can find the values of all the other angles. Now, can we identify what type of quadrilateral ABCD is? Notice that its angles all add up to 90 degrees. What can we say about its sides? We can see that triangle AOB is congruent to triangle COD, and triangle AOD is congruent to triangle COB. Hence, AB equals CD, and AD equals CB, since they are corresponding parts of congruent triangles. Therefore, ABCD is a rectangle since it satisfies the definition of a rectangle.

Now, let us generalize this. Take one of the angles between the diagonals as x. We can compute the four angles between the diagonals to be x, x, 180 minus x, and 180 minus x. Let us find the other angles. Since we know that triangle AOB is isosceles, we can denote the measures of both of its base angles by a. What is the value of a in terms of x? We have a plus a plus x equals 180, which is the sum of the interior angles of a triangle. So, 2a equals 180 minus x, and a equals 90 minus x/2. Similarly, in the isosceles triangle AOD, let the base angles be b. Then b plus b plus 180 minus x equals 180. So, 2b equals x, and b equals x/2. All the angles of the quadrilateral are a plus b, which is 90 minus x/2 plus x/2, which equals 90. Thus, all four angles of the quadrilateral ABCD are 90 degrees. What can we say about AB and CD, and AD and BC? We have triangle AOB congruent to triangle COD, and triangle AOD congruent to triangle COB. Hence, AB equals CD, and AD equals CB. So, no matter what the angles between the diagonals are, if the diagonals are equal and they bisect each other, then the angles of the quadrilateral formed are 90 degrees each, and the opposite sides are equal. Thus, the quadrilateral is a rectangle.

So, now we know how the wooden strips have to be put together to form the vertices of a rectangle. They should be equal and connected at their midpoints. This method is actually used in practice to make rectangles. Carpenters in Europe use this method to get a rectangular frame. It is also known that farmers in Mozambique, a country in Africa, use this method while constructing houses to get the base of the house in a rectangular shape. Isn't it wonderful how geometry is used all over the world?

Now, let me tell you something very important. As we have been seeing from lower grades, properties of geometric objects such as parallel lines, angles, and triangles can be deduced through geometric reasoning. We will continue to deduce properties of special types of quadrilaterals in this chapter. Once you have deduced a property of a quadrilateral, it is good to verify it with a real-world quadrilateral, either the quadrilateral constructed on paper or simply a surface having the shape of the quadrilateral. If you are not able to figure out the property using deduction, you could experiment by taking real-world quadrilaterals and observing the property through measurement. Note that these observations give useful insights about the property, but with them, we can only form a conjecture, that is, a statement about which we are highly confident, but not yet sure if it always holds true. For example, constructing a few rectangles and observing through measurement that their diagonals bisect each other does not necessarily mean that this will always be the case. Can we be sure that the 1000th rectangle we construct will also have this property? The only way we can be sure of this property is by justifying or proving the statement, just as we did in our deduction.

Now, the carpenter's problem shows that rectangles can also be defined in another way. A rectangle is a quadrilateral whose diagonals are equal and bisect each other. Observe how different this definition is from the earlier one. Yet, both capture the same class of quadrilaterals. Further, it turns out that the first definition can be simplified.

In the earlier definition, we stated that a rectangle has first, opposite sides of equal length, and second, all angles equal to 90 degrees. Would we be wrong if we just define a rectangle as a quadrilateral in which all the angles are 90 degrees? If you think that this definition is incomplete, try constructing a quadrilateral in which the angles are all 90 degrees but the opposite sides are not equal. Are you able to construct such a quadrilateral? Let us prove why this is impossible.

Consider a quadrilateral ABCD with all angles measuring 90 degrees. What can we say about the opposite sides of such a quadrilateral? Join BD. Triangle BAD and triangle DCB seem congruent. Can we justify this claim? Two equalities can be directly seen in the triangles. What can we say about angle 1 and angle 2? Recall that we tackled a very similar problem earlier. We can use the same reasoning here. Since angle B is 90 degrees, angle 3 plus angle 1 equals 90 degrees. In triangle BCD, since angle 3 plus angle 2 plus 90 degrees equals 180 degrees, angle 3 plus angle 2 equals 90 degrees. So, angle 1 equals angle 2. Thus, by the AAS congruence condition, triangle BAD is congruent to triangle DCB. Therefore, AD equals CB, and DC equals BA, since these are corresponding sides of congruent triangles. Thus, we have established that if all the angles of a quadrilateral are right angles, then the opposite sides have equal lengths. Therefore, the quadrilateral is a rectangle. So, a rectangle can simply be defined as follows — a rectangle is a quadrilateral in which the angles are all 90 degrees.

Now, let us list the properties of a rectangle. Property 1: All the angles of a rectangle are 90 degrees. Property 2: The opposite sides of a rectangle are equal. Are the opposite sides of a rectangle parallel? They definitely seem so. This fact can be justified using one of the transversal properties. Notice that AB acts as a transversal to AD and BC, and that angle A plus angle B equals 90 degrees plus 90 degrees, which equals 180 degrees. When the sum of the internal angles on the same side of the transversal is 180 degrees, the lines are parallel. We can use this fact to conclude that the lines AD and BC are parallel, which we represent as AD is parallel to BC. Can you similarly show that AB is parallel to DC? Property 3: The opposite sides of a rectangle are parallel to each other. Property 4: The diagonals of a rectangle are of equal length and they bisect each other.

Now, let us talk about a special rectangle. In the quadrilaterals shown in your book, are there any non-rectangles? We have quadrilaterals with sides 2 cm, 5 cm, 2 cm, 5 cm with right angles; 3.6 cm, 6 cm, 3.6 cm, 6 cm with right angles; 1 cm, 5 cm, 1 cm, 5 cm with right angles; and 4 cm, 4 cm, 4 cm, 4 cm with right angles. All these quadrilaterals are rectangles, including the last one. The last quadrilateral is a rectangle because all its angles are 90 degrees. However, it is a special kind of rectangle with all sides of equal length. We know that this quadrilateral is also called a square.

So, a square is a quadrilateral in which all the angles are equal to 90 degrees, and all the sides are of equal length. Thus, every square is also a rectangle, but clearly every rectangle is not a square.

Let me give you an analogy to understand this relationship. Suppose I say "I am an Indian and I am a Malayali." Wait, which one of them am I? How can I be both? Well, I can be both because being Malayali means I am from Kerala, which is a part of India. So, every Malayali is an Indian, but not every Indian is a Malayali. Similarly, every square is a rectangle, but not every rectangle is a square. This relation can be pictorially represented using a Venn diagram. In a Venn diagram, a set of objects is represented as points inside a closed curve. Typically, these closed curves are ovals or circles. Since every square is a rectangle, the Venn diagram representation of these two sets would show the set of squares inside the set of rectangles.

Now, 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? As in the previous case, let us try to construct a square, one of whose diagonals is of length 8 centimeters. While solving the carpenter's problem for the case of a rectangle, we have seen that to get a quadrilateral with all angles 90 degrees and opposite sides of equal length, the diagonals have to be drawn such that they are of equal lengths and they bisect each other. What more needs to be done to get equal side lengths as well? Can this be achieved by properly choosing the angle between the diagonals?

Let us find the angle formed by the diagonals. The angle between the diagonals can be found using the notion of congruence. Suppose we join the equal diagonals such that they bisect each other and result in a square. Let us label the square ABCD. To find the angle formed by the diagonals, what are the two triangles we should consider for congruence? By the SSS condition for congruence, triangle BOA is congruent to triangle BOC. Can this be used to find the angles BOA and BOC formed by the diagonals? Since these angles are corresponding parts of congruent triangles, they are equal. Further, these angles together form a straight angle. So, angle BOA plus angle BOC equals 180 degrees. Thus, these angles have to be 90 degrees each.

This shows that the diagonals of a square bisect each other at right angles. This means that the diagonals have to be drawn such that they are of equal lengths and bisect each other at right angles. Since the endpoints of the diagonals uniquely determine the vertices of a quadrilateral, we will get a square when the diagonals are joined this way. Using this fact, you can construct a square with a diagonal of length 8 centimeters.

Now, let us list the properties of a square. Since a square is a special type of rectangle, all the properties of a rectangle hold true for a square. Property 1: All the sides of a square are equal to each other. Property 2: The opposite sides of a square are parallel to each other. Property 3: The angles of a square are all 90 degrees. Property 4: The diagonals of a square are of equal length and they bisect each other at 90 degrees. There is one more special property of a square. What are the measures of angle 1, angle 2, angle 3, and angle 4? In triangle ADC, we have angle 1 plus angle 3 plus 90 equals 180. Since AD equals DC, we have angle 1 equals angle 3. Thus, angle 1 equals angle 3 equals 45 degrees. Similarly, we can find angle 2 and angle 4. Thus, we have another property of a square — the diagonals of a square divide the angles of the square into equal halves. This can also be expressed as the diagonals of a square bisect the angles of the square.

Now, let us move on to a very important topic — angles in a quadrilateral. Is it possible to construct a quadrilateral with three angles equal to 90 degrees and the fourth angle not equal to 90 degrees? You might have observed through constructions that this may not be possible. But why not? This is due to a general property of quadrilaterals related to their angles.

We have seen that the sum of the angles of a triangle is 180 degrees. There is a similar regularity in the sum of the angles of a quadrilateral. Consider a quadrilateral SOME. Draw a diagonal SM. We get two triangles, triangle SEM and triangle SOM. In triangle SEM, we have angle 1 plus angle 2 plus angle 3 equals 180 degrees. And in triangle SOM, angle 4 plus angle 5 plus angle 6 equals 180 degrees. What do we get when we add all six angles? We will have angle 1 plus angle 2 plus angle 3 plus angle 4 plus angle 5 plus angle 6 equals 180 plus 180, which equals 360 degrees. Or, angle 1 plus angle 4, plus angle 3 plus angle 6, plus angle 2 plus angle 5 equals 360 degrees. Since angle 1 plus angle 4, angle 3 plus angle 6, angle 2, and angle 5 are the angles of this quadrilateral, we have the following result — the sum of all angles in any quadrilateral is 360 degrees. This explains why it is impossible for a quadrilateral to have three right angles with the fourth angle not being a right angle.

Now, let us study more quadrilaterals with parallel opposite sides. Rectangles and squares have parallel opposite sides. Are there quadrilaterals that have parallel opposite sides that are not rectangles? Let us try constructing one. This can be easily done by drawing two pairs of parallel lines, ensuring that they do not meet at right angles. Observe the quadrilateral ABCD. It has parallel opposite sides but is not a rectangle. Thus, a larger set of quadrilaterals exists in which the opposite sides are parallel. Such quadrilaterals are called parallelograms.

Is a rectangle a parallelogram? A rectangle has opposite sides parallel. So, it satisfies the parallelogram's definition. Hence, it is indeed a parallelogram. More specifically, a rectangle is a special kind of parallelogram with all its angles equal to 90 degrees.

To understand the relations between the sides and the angles of a parallelogram, let us construct the following figure. Draw a parallelogram with adjacent sides of lengths 4 centimeters and 5 centimeters, and an angle of 30 degrees between them. Step 1: Draw line segments AB equal to 4 centimeters and AD equal to 5 centimeters with an angle of 30 degrees between them. Step 2: Draw a line parallel to AB through the point D, and a line parallel to AD through B. Mark the point at which these lines intersect as C. ABCD is the required parallelogram. What are the remaining angles of the parallelogram? What are the lengths of the remaining sides?

Now, let us deduce what we can say about the angles of a parallelogram. In the parallelogram ABCD, AB is parallel to CD, and AD is a transversal to them. Angle A plus angle D equals 180 degrees, because these are the sum of the internal angles on the same side of a transversal. Therefore, angle D equals 180 minus angle A, which is 180 minus 30, which equals 150 degrees. Similarly, AD is parallel to BC, and AB and CD are transversals to them. So, angle A plus angle B equals 180 degrees. So, angle C plus angle D equals 180 degrees. Using these equations, we get angle B equals 150 degrees and angle C equals 30 degrees. We see that in this parallelogram, the adjacent pairs of angles add up to 180 degrees and opposite pairs of angles are equal. Thus, angle A plus angle B equals 180 degrees, angle A plus angle D equals 180 degrees, angle C plus angle D equals 180 degrees, and angle B plus angle C equals 180 degrees. And, angle A equals angle C, and angle B equals angle D. Since the adjacent angles are the interior angles on the same side of a transversal to a pair of parallel lines, they must add up to 180 degrees.

What about the opposite angles? Will they be equal in all parallelograms? If yes, how can we be sure? Let us take one of the angles to be x. What are the other angles? Since angle P plus angle R equals 180 degrees, angle R equals 180 minus angle P, which is 180 minus x. Similarly, since angle A plus angle R equals 180 degrees, angle A equals 180 minus angle R, which is 180 minus (180 minus x), which equals x. Thus, angle P equals angle A, which is x. Similarly, we can deduce that angle R equals angle E, which is 180 minus x. Therefore, this shows that the opposite angles of a parallelogram are always equal.

Now, let us see what we can 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? In triangle ABD and triangle CDB, the angles marked with a single arc are equal as they are the opposite angles of a parallelogram. Since AD is parallel to BC, and BD is a transversal to it, the angles marked with double arcs are equal as they are alternate angles. So, by the AAS condition, the triangles are congruent, that is, triangle ABD is congruent to triangle CDB. Therefore, AD equals CB, and AB equals CD. Thus, the opposite sides of a parallelogram are equal.

From these deductions, we can find the remaining sides and angles of the parallelogram. Let us list the properties of a parallelogram. Property 1: The opposite sides of a parallelogram are equal. Property 2: The opposite sides of a parallelogram are parallel. Property 3: In a parallelogram, the adjacent angles add up to 180 degrees, and the opposite angles are equal.

Now, let us think about the diagonals of a parallelogram. Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed. We see that the diagonals of a parallelogram need not be equal. Do they bisect each other? Let us find out.

As in the case of a rectangle, we can find out if the diagonals bisect each other by examining the congruence of triangle AOE and triangle YOS in the parallelogram EASY. AE equals YS, as they are the opposite sides of the parallelogram. The angles marked using a single arc are equal, and so are the angles marked using a double arc, since they are alternate angles of parallel lines. Thus, by the ASA condition, the triangles are congruent, that is, triangle AOE is congruent to triangle YOS. Therefore, OA equals OY, and OE equals OS, since they are corresponding parts of congruent triangles. Thus, O is the midpoint of both diagonals. So, Property 4: The diagonals of a parallelogram bisect each other.

Now, let us explore quadrilaterals with equal side lengths. Are squares the only quadrilaterals that have equal side lengths? 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. Now we have a quadrilateral with equal side lengths and one of its angles 50 degrees. Note that we could have constructed such a quadrilateral by taking any angle less than 180 degrees in place of 50 degrees. A quadrilateral in which all the sides have the same length is called a rhombus.

Now, what are the other angles of the rhombus ABCD that we have constructed? Let us deduce what we can say about the angles in a rhombus. Consider a rhombus GAME. In triangle GAE, since GE equals GA, angle a equals angle d. Similarly, in triangle MAE, since ME equals MA, angle b equals angle c. It can be seen that triangle GAE is congruent to triangle MAE. So, angle a equals angle b, angle c equals angle d, and angle G equals angle M, since they are corresponding parts of congruent triangles. Thus, we have angle a equals angle b equals angle c equals angle d. These facts hold for any rhombus. Let us apply them to the rhombus ABCD that we constructed earlier. Let the four equal angles formed by the diagonal be a, as shown in the figure. In triangle ADB, we have a plus a plus 50 equals 180 degrees. So, a equals 65 degrees. Thus, the angles of the rhombus ABCD are 50 degrees, 130 degrees, 50 degrees, and 130 degrees. So, in a rhombus, opposite angles are equal to each other.

Interestingly, there is one more way by which we could have figured out the other angles of the rhombus ABCD. We have shown that in a general rhombus GAME, the four angles formed by a diagonal are equal to each other. Consider the lines EM and GA and its transversal AE. Since the alternate angles are equal, EM is parallel to GA. Similarly, consider the lines GE and AM and its transversal AE. Since the alternate angles are equal, GE is parallel to AM. As opposite sides are parallel, GAME is also a parallelogram. Thus, every rhombus is a parallelogram, and the properties of a parallelogram hold true for a rhombus as well. Thus, the adjacent angles of a rhombus add up to 180 degrees, and the opposite angles are equal. So, in rhombus ABCD, angle A equals angle C equals 50 degrees, and angle D equals angle B equals 180 minus 50, which equals 130 degrees.

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? We know that a square is a rectangle. Since the opposite sides of a square are parallel, a square is also a parallelogram. Further, since all the sides of a square have the same length, a square is also a rhombus. Thus, the Venn diagram will show that the set of squares is the intersection of the set of rectangles and the set of rhombuses, and both rectangles and rhombuses are subsets of parallelograms.

Let us list the properties of a rhombus. Property 1: All the sides of a rhombus are equal to each other. Property 2: The opposite sides of a rhombus are parallel to each other. Property 3: In a rhombus, the adjacent angles add up to 180 degrees, and the opposite angles are equal. Are the diagonals of a rhombus equal? Property 4: The diagonals of a rhombus bisect each other. Property 5: The diagonals of a rhombus bisect its angles. Do the diagonals of a rhombus intersect at any particular angle? Let us find out.

What can we say about the angles formed by the diagonals of a rhombus at their point of intersection? In the rhombus GAME, we have triangle GEO is congruent to triangle MEO. So, angle GOE equals angle MOE, as they are corresponding parts of congruent triangles. As they add up to 180 degrees, they should be 90 degrees each. So, Property 6: Diagonals of a rhombus intersect each other at an angle of 90 degrees.

Now, let us learn about two more types of quadrilaterals — kite and trapezium.

First, let us talk about a kite. One of the ways two triangles of sides 6 cm, 9 cm, and 12 cm can be joined together is as follows. This quadrilateral looks like a kite. Observe that the adjacent sides are of the same length. A kite is a quadrilateral that can be labelled ABCD such that AB equals BC, and CD equals DA. Now, let us find the properties of a kite. In the kite, show that the diagonal BD first, bisects angle ABC and angle ADC, second, bisects the diagonal AC, that is, AO equals OC, and is perpendicular to it. To prove this, we can check if triangle AOB is congruent to triangle COB.

Now, let us talk about trapezium. Parallelograms are quadrilaterals that have parallel opposite sides. We get a new type of quadrilateral if we relax this condition. A trapezium is a quadrilateral with at least one pair of parallel opposite sides. Construct a trapezium. Measure the base angles marked in the figure. Can you find the remaining angles without measuring them? Since PQ is parallel to SR, we have Property 1: angle S plus angle P equals 180 degrees, and angle R plus angle Q equals 180 degrees. Using these facts, the remaining angles can easily be found.

When the non-parallel sides of a trapezium have the same lengths, the trapezium is called an isosceles trapezium. How do we construct an isosceles trapezium? Construct an isosceles trapezium UVWX, with UV parallel to XW. Measure angle U. Mark X and W such that UX equals VW. Can you find the remaining angles without measuring them? Does it appear that the angles opposite to the equal sides, angle U and angle V, are also equal? Can we find congruent triangles here? Consider line segments XY and WZ perpendicular to UV. What type of quadrilateral is XWZY? Since XW is parallel to UV, angle a equals 180 degrees minus angle XYZ, which equals 90 degrees, and angle b equals 180 degrees minus angle WZY, which equals 90 degrees. Hence, XWZY is a rectangle. Now, it can be shown that triangle UXY is congruent to triangle VWZ. Thus, angle U equals angle V. Using this fact, the remaining angles of the isosceles trapezium can be determined. Property 2: In an isosceles trapezium, the angles opposite to the equal sides are equal.

Now, let me give you a comprehensive summary of everything we have learned in this chapter.

A rectangle is a quadrilateral in which the angles are all 90 degrees. The properties of a rectangle are: first, opposite sides of a rectangle are equal; second, opposite sides of a rectangle are parallel to each other; and third, 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 degrees, and all the sides are of equal length. The properties of a square are: first, the opposite sides of a square are parallel to each other; second, the diagonals of a square are of equal lengths and they bisect each other at 90 degrees; and third, the diagonals of a square bisect the angles of the square.

A parallelogram is a quadrilateral in which opposite sides are parallel. The properties of a parallelogram are: first, the opposite sides of a parallelogram are equal; second, in a parallelogram, the adjacent angles add up to 180 degrees, and the opposite angles are equal; and third, the diagonals of a parallelogram bisect each other.

A rhombus is a quadrilateral in which all the sides have the same length. The properties of a rhombus are: first, the opposite sides of a rhombus are parallel to each other; second, in a rhombus, the adjacent angles add up to 180 degrees, and the opposite angles are equal; third, the diagonals of a rhombus bisect each other at right angles; and fourth, 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.

And finally, the sum of the angle measures in a quadrilateral is always 360 degrees.

Now, students, I want you to remember that geometry is not just about memorizing formulas and properties. It is about understanding why these properties are true and how they connect to each other. The relationship between different quadrilaterals is like a family tree — squares, rectangles, rhombuses, and parallelograms are all related to each other. A square is a rectangle, a rhombus, and a parallelogram all at the same time. A rectangle is a parallelogram but not a rhombus. A rhombus is a parallelogram but not necessarily a rectangle. And a parallelogram is the most general of these, with just the condition that opposite sides are parallel.

I hope you enjoyed this lesson as much as I enjoyed teaching it to you. Remember to practice the constructions and properties we discussed today. Thank you for being such wonderful students, and I will see you in the next lesson!