Parallelograms and Rectangles

Measurement and also Geometry : Module 20Years : 8-9

June 2011

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PDF version of module


Assumed knowledge

Introductory airplane geometry entailing points and also lines, parallel lines and also transversals, edge sums of triangles and also quadrilaterals, and general angle-chasing.The 4 standard congruence tests and also their application in problems and proofs.Properties of isosceles and equilateral triangles and also tests because that them.Experience through a logical debate in geometry being created as a sequence of steps, every justified by a reason.Ruler-and-compasses constructions.Informal endure with one-of-a-kind quadrilaterals.

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Motivation

There are just three essential categories of distinct triangles − isosceles triangles, it is intended triangles and also right-angled triangles. In contrast, there are countless categories of special quadrilaterals. This module will attend to two of lock − parallelograms and rectangles − leaving rhombuses, kites, squares, trapezia and also cyclic quadrilaterals to the module, Rhombuses, Kites, and Trapezia.

Apart from cyclic quadrilaterals, these special quadrilaterals and their properties have actually been introduced informally over numerous years, but without congruence, a rigorous discussion of lock was not possible. Every congruence proof supplies the diagonals to divide the quadrilateral right into triangles, ~ which we can use the approaches of congruent triangles arisen in the module, Congruence.

The present treatment has four purposes:

The parallelogram and rectangle are carefully defined.Their significant properties room proven, mostly using congruence.Tests for them are established that deserve to be supplied to check that a offered quadrilateral is a parallelogram or rectangle − again, congruence is mostly required.Some ruler-and-compasses build of them are arisen as an easy applications of the definitions and also tests.

The product in this module is suitable for Year 8 as additional applications the congruence and also constructions. Due to the fact that of its systematic development, it provides fantastic introduction to proof, converse statements, and sequences that theorems. Substantial guidance in such principles is normally forced in Year 8, i beg your pardon is consolidated by further discussion in later years.

The complementary principles of a ‘property’ of a figure, and also a ‘test’ for a figure, become specifically important in this module. Indeed, clarity around these ideas is one of the numerous reasons for teaching this material at school. Many of the tests that we meet are converses of properties that have already been proven. For example, the fact that the base angle of one isosceles triangle room equal is a residential or commercial property of isosceles triangles. This property have the right to be re-formulated together an ‘If …, climate … ’ statement:

If 2 sides that a triangle space equal, then the angle opposite those sides room equal.

Now the corresponding test because that a triangle to be isosceles is clearly the converse statement:

If 2 angles of a triangle are equal, then the sides opposite those angles room equal.

Remember the a statement might be true, but its converse false. That is true that ‘If a number is a multiple of 4, climate it is even’, but it is false the ‘If a number is even, climate it is a many of 4’.


Quadrilaterals

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In various other modules, we characterized a square to be a closed airplane figure bounded by four intervals, and also a convex quadrilateral to be a quadrilateral in i beg your pardon each internal angle is much less than 180°. We confirmed two essential theorems about the angle of a quadrilateral:

The sum of the inner angles of a quadrilateral is 360°.The amount of the exterior angle of a convex quadrilateral is 360°.

To prove the an initial result, we created in each instance a diagonal the lies totally inside the quadrilateral. This separated the quadrilateral into two triangles, each of whose angle amount is 180°.

To prove the second result, we created one next at every vertex the the convex quadrilateral. The amount of the 4 straight angle is 720° and also the sum of the four interior angles is 360°, so the sum of the 4 exterior angle is 360°.


Parallelograms

We start with parallelograms, due to the fact that we will be making use of the results around parallelograms when mentioning the various other figures.

Definition the a parallelogram

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A parallel is a square whose the contrary sides space parallel. Hence the square ABCD presented opposite is a parallelogram because abdominal || DC and also DA || CB.

The native ‘parallelogram’ comes from Greek words an interpretation ‘parallel lines’.

Constructing a parallelogram utilizing the definition

To build a parallelogram using the definition, we can use the copy-an-angle construction to kind parallel lines. For example, expect that us are provided the intervals ab and advertisement in the chart below. Us extend advertisement and abdominal and copy the angle at A to equivalent angles at B and also D to recognize C and complete the parallel ABCD. (See the module, Construction.)

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This is no the easiest way to build a parallelogram.

First building of a parallel − the opposite angles space equal

The three properties that a parallelogram emerged below issue first, the internal angles, secondly, the sides, and thirdly the diagonals. The first property is most conveniently proven using angle-chasing, but it can likewise be proven utilizing congruence.

Theorem

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The opposite angles of a parallelogram are equal.

Proof

Let ABCD it is in a parallelogram, with A = α and B = β.
Prove the C = α and also D = β.
α + β = 180°(co-interior angles, ad || BC),
soC = α(co-interior angles, abdominal || DC)
and D = β(co-interior angles, abdominal muscle || DC).

Second property of a parallel − the contrary sides space equal

As an example, this proof has actually been set out in full, through the congruence test fully developed. Many of the continuing to be proofs however, space presented as exercises, with an abbreviation version given as an answer.

Theorem

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The opposite political parties of a parallelogram room equal.

Proof

ABCD is a parallelogram.
To prove that abdominal muscle = CD and ad = BC.
Join the diagonal line AC.
In the triangles ABC and also CDA:
BAC = DCA (alternate angles, ab || DC)
BCA = DAC (alternate angles, ad || BC)
AC = CA (common)
so alphabet ≡ CDA (AAS)
Hence abdominal muscle = CD and BC = ad (matching sides of congruent triangles).

Third home of a parallelogram − The diagonals bisect each other

Theorem

The diagonals the a parallelogram bisect every other.


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EXERCISE 1

a Prove that ABM ≡ CDM.

b therefore prove that the diagonals bisect each other.


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As a an effect of this property, the intersection that the diagonals is the center of 2 concentric circles, one v each pair of the opposite vertices.

Notice that, in general, a parallelogram does not have actually a circumcircle with all 4 vertices.

First test because that a parallel − the opposite angles room equal

Besides the an interpretation itself, there space four helpful tests for a parallelogram. Our very first test is the converse of our very first property, the the opposite angles of a quadrilateral space equal.

Theorem

If the opposite angle of a quadrilateral room equal, climate the quadrilateral is a parallelogram.


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EXERCISE 2

Prove this an outcome using the figure below.

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Second test because that a parallel − the opposite sides space equal

This check is the converse that the property that the opposite sides of a parallelogram are equal.

Theorem

If the opposite political parties of a (convex) quadrilateral space equal, then the square is a parallelogram.


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EXERCISE 3

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Prove this an outcome using congruence in the figure to the right, whereby the diagonal line AC has actually been joined.


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This test offers a simple construction of a parallelogram offered two adjacent sides − ab and ad in the figure to the right. Draw a circle v centre B and radius AD, and another circle v centre D and radius AB. The circles intersect at two points − let C it is in the allude of intersection in ~ the non-reflex angle BAD. Climate ABCD is a parallelogram because its the contrary sides are equal.

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It additionally gives a technique of illustration the line parallel come a offered line v a given suggest P. Choose any kind of two point out A and B top top , and also complete the parallelogram PABQ.

Then PQ ||

Third test because that a parallel − One pair of the contrary sides space equal and parallel

This test transforms out come be really useful, because it offers only one pair of the opposite sides.

Theorem

If one pair that opposite political parties of a quadrilateral space equal and parallel, climate the quadrilateral is a parallelogram.


This test because that a parallelogram gives a quick and also easy method to build a parallelogram using a two-sided ruler. Attract a 6 centimeter interval on every side of the ruler. Joining up the endpoints gives a parallelogram.

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The check is an especially important in the later on theory of vectors. Mean that
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and
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space two directed intervals that are parallel and also have the same size − that is, they represent the exact same vector. Climate the number ABQP come the ideal is a parallelogram.

Even a basic vector property favor the commutativity of the addition of vectors relies on this construction. The parallelogram ABQP shows, for example, that

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+
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=
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=
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+
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Fourth test for a parallel − The diagonals bisect every other

This check is the converse of the residential property that the diagonals that a parallel bisect each other.

Theorem

If the diagonals that a quadrilateral bisect every other, then the quadrilateral is a parallelogram:


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This test offers a very straightforward construction of a parallelogram. Draw two intersecting lines, then attract two circles with various radii centred on your intersection. Join the clues where alternating circles cut the lines. This is a parallelogram because the diagonals bisect each other.

It also allows yet another method of perfect an angle bad to a parallelogram, as displayed in the following exercise.


EXERCISE 6

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Given two intervals abdominal muscle and ad meeting at a usual vertex A, build the midpoint M the BD. Complete this come a construction of the parallel ABCD, justifying your answer.


Parallelograms

Definition that a parallelogram

A parallel is a quadrilateral whose opposite sides room parallel.

Properties that a parallelogram

The opposite angle of a parallelogram room equal. The opposite political parties of a parallelogram space equal. The diagonals the a parallel bisect every other.

Tests because that a parallelogram

A quadrilateral is a parallel if:

its the contrary angles are equal, or its the contrary sides space equal, or one pair of opposite sides room equal and parallel, or its diagonals bisect each other.

Rectangles

The indigenous ‘rectangle’ way ‘right angle’, and also this is reflected in its definition.

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Definition that a Rectangle

A rectangle is a square in i m sorry all angle are best angles.

First home of a rectangle − A rectangle is a parallelogram

Each pair of co-interior angles room supplementary, because two best angles add to a directly angle, so the opposite political parties of a rectangle are parallel. This means that a rectangle is a parallelogram, so:

Its the contrary sides room equal and also parallel. That diagonals bisect each other.

Second residential or commercial property of a rectangle − The diagonals are equal

The diagonals that a rectangle have one more important building − they room equal in length. The proof has been collection out in complete as an example, since the overlapping congruent triangles can be confusing.

Theorem

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The diagonals of a rectangle are equal.

Proof

allow ABCD be a rectangle.

we prove that AC = BD.

In the triangle ABC and also DCB:

BC = CB (common)
AB = DC (opposite political parties of a parallelogram)
ABC =DCA = 90° (given)

so abc ≡ DCB (SAS)

hence AC = DB (matching sides of congruent triangles).

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This way that am = BM = cm = DM, wherein M is the intersection of the diagonals. Thus we can draw a solitary circle through centre M v all four vertices. Us can describe this instance by speak that, ‘The vertices the a rectangle room concyclic’.


First test because that a rectangle − A parallelogram v one right angle

If a parallel is well-known to have actually one right angle, then repetitive use that co-interior angles proves that all its angles are appropriate angles.

Theorem

If one angle of a parallel is a right angle, then it is a rectangle.

Because that this theorem, the meaning of a rectangle is periodically taken to it is in ‘a parallelogram through a right angle’.

Construction that a rectangle

We have the right to construct a rectangle with given side lengths by creating a parallelogram through a best angle top top one corner. An initial drop a perpendicular native a point P to a heat . Mark B and then note off BC and BA and also complete the parallel as presented below.

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Second test because that a rectangle − A quadrilateral with equal diagonals the bisect every other

We have shown over that the diagonals the a rectangle space equal and also bisect every other. Conversely, these 2 properties taken with each other constitute a test for a quadrilateral to be a rectangle.

Theorem

A square whose diagonals are equal and also bisect each various other is a rectangle.


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EXERCISE 8

a Why is the quadrilateral a parallelogram?

b use congruence to prove that the number is a rectangle.


As a an effect of this result, the endpoints of any two diameters that a circle type a rectangle, due to the fact that this quadrilateral has equal diagonals the bisect each other.

Thus we deserve to construct a rectangle an extremely simply through drawing any kind of two intersecting lines, then drawing any circle centred in ~ the point of intersection. The quadrilateral formed by involvement the 4 points wherein the circle cuts the lines is a rectangle since it has equal diagonals that bisect each other.

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Rectangles

Definition that a rectangle

A rectangle is a square in i beg your pardon all angles are appropriate angles.

Properties the a rectangle

A rectangle is a parallelogram, therefore its the contrary sides are equal. The diagonals the a rectangle space equal and bisect each other.

Tests because that a rectangle

A parallelogram v one right angle is a rectangle. A square whose diagonals are equal and also bisect each other is a rectangle.

Links forward

The continuing to be special quadrilaterals come be treated by the congruence and also angle-chasing methods of this module space rhombuses, kites, squares and also trapezia. The succession of theorems associated in dealing with all these special quadrilaterals at as soon as becomes quite complicated, therefore their discussion will it is in left until the module Rhombuses, Kites, and Trapezia. Each individual proof, however, is well within Year 8 ability, noted that students have actually the ideal experiences. In particular, it would be beneficial to prove in Year 8 that the diagonals of rhombuses and also kites fulfill at ideal angles − this an outcome is essential in area formulas, that is useful in applications that Pythagoras’ theorem, and also it offers a much more systematic explanation of several necessary constructions.

The following step in the development of geometry is a rigorous therapy of similarity. This will enable various results around ratios of lengths to be established, and likewise make feasible the definition of the trigonometric ratios. Similarity is required for the geometry the circles, where one more class of distinct quadrilaterals arises, namely the cyclic quadrilaterals, who vertices lie on a circle.

Special quadrilaterals and their properties are essential to develop the typical formulas because that areas and volumes that figures. Later, these outcomes will be vital in occurring integration. Theorems around special quadrilaterals will be widely used in name: coordinates geometry.

Rectangles are so common that they walk unnoticed in most applications. One special role worth note is they room the communication of the works with of point out in the cartesian aircraft − to find the coordinates of a point in the plane, we complete the rectangle created by the point and the two axes. Parallelograms arise when we include vectors by completing the parallel − this is the reason why they end up being so essential when complicated numbers are stood for on the Argand diagram.


History and applications

Rectangles have been valuable for as lengthy as there have actually been buildings, since vertical pillars and also horizontal crossbeams room the most obvious means to construct a structure of any size, providing a framework in the shape of a rectangle-shaped prism, every one of whose faces are rectangles. The diagonals that we constantly usage to examine rectangles have actually an analogy in structure − a rectangular frame with a diagonal has actually far much more rigidity than a an easy rectangular frame, and also diagonal struts have always been supplied by building contractors to provide their building more strength.

Parallelograms room not as typical in the physical world (except together shadows of rectangle-shaped objects). Their major role historically has remained in the depiction of physical concepts by vectors. Because that example, as soon as two pressures are combined, a parallelogram can be drawn to assist compute the size and direction that the combined force. When there room three forces, we complete the parallelepiped, i m sorry is the three-dimensional analogue of the parallelogram.


REFERENCES

A background of Mathematics: one Introduction, 3rd Edition, Victor J. Katz, Addison-Wesley, (2008)

History that Mathematics, D. E. Smith, Dover publications new York, (1958)


ANSWERS to EXERCISES

EXERCISE 1

a In the triangles ABM and also CDM :

1.BAM= DCM(alternate angles, abdominal || DC )
2.ABM= CDM(alternate angles, ab || DC )
3. AB = CD (opposite sides of parallelogram ABCD)
ABM = CDM (AAS)

b hence AM = CM and also DM = BM (matching political parties of congruent triangles)

EXERCISE 2

From the diagram,2α + 2β= 360o(angle sum of quadrilateral ABCD)
α + β= 180o
HenceAB || DC(co-interior angles room supplementary)
andAD || BC(co-interior angles are supplementary).

EXERCISE 3

First display that alphabet ≡ CDA utilizing the SSS congruence test.
HenceACB = CAD and CAB = ACD(matching angle of congruent triangles)
soAD || BC and abdominal || DC(alternate angles space equal.)

EXERCISE 4

First prove the ABD ≡ CDB making use of the SAS congruence test.
HenceADB = CBD(matching angles of congruent triangles)
soAD || BC(alternate angles room equal.)

EXERCISE 5

First prove the ABM ≡ CDM using the SAS congruence test.
HenceAB = CD(matching sides of congruent triangles)
AlsoABM = CDM(matching angles of congruent triangles)
soAB || DC(alternate angles are equal):

Hence ABCD is a parallelogram, since one pair of the contrary sides room equal and also parallel.

EXERCISE 6

Join AM. V centre M, draw an arc with radius AM that meets AM produced at C . Then ABCD is a parallelogram due to the fact that its diagonals bisect every other.

EXERCISE 7

The square on every diagonal is the amount of the squares on any type of two surrounding sides. Since opposite sides are equal in length, the squares top top both diagonals space the same.

EXERCISE 8

a We have currently proven that a square whose diagonals bisect each other is a parallelogram.
b Because ABCD is a parallelogram, its opposite sides space equal.
HenceABC ≡ DCB(SSS)
soABC = DCB(matching angle of congruent triangles).
ButABC + DCB = 180o(co-interior angles, ab || DC )
soABC = DCB = 90o .

hence ABCD is rectangle, due to the fact that it is a parallelogram with one appropriate angle.

EXERCISE 9

ADM= α(base angles of isosceles ADM )
andABM= β(base angle of isosceles ABM ),
so2α + 2β= 180o(angle amount of ABD)
α + β= 90o.

Hence A is a best angle, and also similarly, B, C and also D are best angles.

The improving Mathematics education in institutions (TIMES) project 2009-2011 to be funded by the Australian government Department that Education, Employment and also Workplace Relations.

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