Parallelograms and Rectangles

Measurement and Geometry : Module 20Years : 8-9

June 2011

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PDF variation of module


Assumed knowledge

Introductory aircraft geometry involving points and lines, parallel lines and also transversals, angle sums that triangles and quadrilaterals, and also general angle-chasing.The four standard congruence tests and their applications in problems and also proofs.Properties that isosceles and equilateral triangles and also tests because that them.Experience through a logical argument in geometry being written as a sequence of steps, every justified by a reason.Ruler-and-compasses constructions.Informal experience with special quadrilaterals.

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Motivation

There are just three vital categories of special triangles − isosceles triangles, equilateral triangles and right-angled triangles. In contrast, over there are numerous categories of one-of-a-kind quadrilaterals. This module will deal with two of castle − parallelograms and rectangles − leaving rhombuses, kites, squares, trapezia and also cyclic quadrilaterals come the module, Rhombuses, Kites, and also Trapezia.

Apart from cyclic quadrilaterals, these unique quadrilaterals and their properties have been introduced informally over number of years, yet without congruence, a rigorous discussion of castle was no possible. Each congruence proof supplies the diagonals to divide the quadrilateral right into triangles, after ~ which we can apply the techniques of congruent triangles arisen in the module, Congruence.

The current treatment has 4 purposes:

The parallelogram and also rectangle are closely defined.Their significant properties room proven, largely using congruence.Tests because that them are created that have the right to be offered to inspect that a given quadrilateral is a parallel or rectangle − again, congruence is mostly required.Some ruler-and-compasses build of lock are occurred as simple applications of the definitions and tests.

The product in this module is an ideal for Year 8 as further applications that congruence and also constructions. Since of its moment-g.comanized development, it provides fantastic introduction come proof, converse statements, and sequences that theorems. Considerable guidance in such ideas is normally compelled in Year 8, i m sorry is consolidated by further discussion in later on years.

The complementary concepts of a ‘property’ of a figure, and a ‘test’ for a figure, become particularly important in this module. Indeed, clarity about these concepts is one of the many reasons for to teach this product at school. Many of the tests the we satisfy are converses the properties the have already been proven. Because that example, the truth that the base angles of one isosceles triangle space equal is a residential property of isosceles triangles. This property deserve to be re-formulated together an ‘If …, then … ’ statement:

If two sides that a triangle room equal, climate the angles opposite those sides space equal.

Now the equivalent test for a triangle to it is in isosceles is clearly the converse statement:

If two angles the a triangle space equal, climate the sides opposite those angles are equal.

Remember the a statement may be true, but its converse false. The is true that ‘If a number is a lot of of 4, then it is even’, but it is false the ‘If a number is even, climate it is a multiple of 4’.


Quadrilaterals

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In various other modules, we characterized a quadrilateral to be a closed plane figure bounded by four intervals, and also a convex quadrilateral to be a quadrilateral in which each internal angle is less than 180°. We showed two crucial theorems about the angles of a quadrilateral:

The amount of the internal angles of a square is 360°.The sum of the exterior angles of a convex square is 360°.

To prove the first result, we constructed in each situation a diagonal that lies fully inside the quadrilateral. This separated the quadrilateral right into two triangles, every of whose angle amount is 180°.

To prove the 2nd result, we produced one side at every vertex that the convex quadrilateral. The sum of the 4 straight angle is 720° and the sum of the four interior angles is 360°, therefore the amount of the four exterior angles is 360°.


Parallelograms

We start with parallelograms, because we will be making use of the results around parallelograms when pointing out the other figures.

Definition that a parallelogram

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A parallel is a square whose opposite sides space parallel. For this reason the quadrilateral ABCD shown opposite is a parallelogram because abdominal muscle || DC and DA || CB.

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

Constructing a parallelogram making use of the definition

To construct a parallelogram using the definition, we have the right to use the copy-an-angle construction to kind parallel lines. For example, intend that us are offered the intervals abdominal and advertisement in the diagram below. Us extend advertisement and abdominal and copy the angle at A to equivalent angles at B and also D to identify C and complete the parallel ABCD. (See the module, Construction.)

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This is not the easiest means to build a parallelogram.

First home of a parallel − opposing angles room equal

The three properties that a parallelogram emerged below worry first, the internal angles, secondly, the sides, and thirdly the diagonals. The first property is most quickly proven utilizing angle-chasing, yet it can additionally be proven making use of congruence.

Theorem

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The opposite angles of a parallelogram room equal.

Proof

Let ABCD it is in a parallelogram, v A = α and B = β.
Prove that C = α and D = β.
α + β = 180°(co-interior angles, advertisement || BC),
soC = α(co-interior angles, abdominal || DC)
and D = β(co-interior angles, ab || DC).

Second home of a parallel − opposing sides are equal

As one example, this proof has actually been collection out in full, v the congruence test totally developed. Most of the staying proofs however, are presented together exercises, v an abbreviation version offered as one answer.

Theorem

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The opposite sides of a parallelogram room equal.

Proof

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

Third home of a parallel − The diagonals bisect every other

Theorem

The diagonals the a parallelogram bisect each other.


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EXERCISE 1

a Prove that ABM ≡ CDM.

b therefore prove the the diagonals bisect every other.


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

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

First test because that a parallelogram − the contrary angles room equal

Besides the an interpretation itself, there space four valuable tests because that a parallelogram. Our first test is the converse that our very first property, that the opposite angle of a quadrilateral room equal.

Theorem

If the opposite angles of a quadrilateral room equal, then the quadrilateral is a parallelogram.


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EXERCISE 2

Prove this an outcome using the number below.

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Second test because that a parallel − opposite sides space equal

This check is the converse of the home that the opposite political parties of a parallelogram room equal.

Theorem

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


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EXERCISE 3

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Prove this result using congruence in the figure to the right, whereby the diagonal AC has been joined.


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This test offers a basic construction of a parallelogram offered two adjacent sides − ab and advertisement in the number to the right. Attract a circle through centre B and also radius AD, and also another circle v centre D and also radius AB. The circles crossing at 2 points − let C it is in the point of intersection within the non-reflex edge BAD. Climate ABCD is a parallelogram due to the fact that its the contrary sides room equal.

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It likewise gives a method of drawing the line parallel come a provided line with a given point P. Choose any kind of two points A and B top top , and also complete the parallelogram PABQ.

Then PQ ||

Third test for a parallel − One pair of the opposite sides room equal and parallel

This test transforms out come be very useful, since it provides only one pair of the contrary sides.

Theorem

If one pair the opposite sides of a quadrilateral room equal and parallel, then the quadrilateral is a parallelogram.


This test because that a parallelogram gives a quick and easy way to build a parallelogram using a two-sided ruler. Draw a 6 cm interval on every side the the ruler. Joining up the endpoints offers a parallelogram.

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The test is particularly important in the later theory that vectors. Intend that
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and also
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are two directed intervals that space parallel and have the same length − the is, they stand for the exact same vector. Climate the figure ABQP to the right is a parallelogram.

Even a straightforward vector property like the commutativity the the enhancement of vectors relies on this construction. The parallelogram ABQP shows, because that example, that

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+
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=
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=
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+
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Fourth test because that a parallel − The diagonals bisect each other

This check is the converse the the building that the diagonals that a parallelogram bisect every other.

Theorem

If the diagonals of a quadrilateral bisect each other, climate the quadrilateral is a parallelogram:


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This test gives a very straightforward construction that a parallelogram. Draw two intersecting lines, then draw two circles with various radii centred on your intersection. Join the point out where alternate circles reduced the lines. This is a parallelogram since the diagonals bisect each other.

It also enables yet another method of completing an angle bad to a parallelogram, as shown in the following exercise.


EXERCISE 6

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Given 2 intervals ab and ad meeting at a typical vertex A, build the midpoint M that BD. Complete this to a building and construction of the parallel ABCD, justifying your answer.


Parallelograms

Definition of a parallelogram

A parallelogram is a square whose the opposite sides room parallel.

Properties the a parallelogram

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

Tests for a parallelogram

A quadrilateral is a parallelogram if:

its the opposite angles are equal, or its opposite sides are equal, or one pair of the opposite sides are equal and parallel, or the diagonals bisect every other.

Rectangles

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

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Definition that a Rectangle

A rectangle is a quadrilateral in i beg your pardon all angle are best angles.

First residential or commercial property of a rectangle − A rectangle is a parallelogram

Each pair of co-interior angles space supplementary, due to the fact that two appropriate angles add to a right angle, therefore the opposite sides of a rectangle room parallel. This way that a rectangle is a parallelogram, so:

Its the opposite sides space equal and parallel. The diagonals bisect every other.

Second property of a rectangle − The diagonals room equal

The diagonals of a rectangle have an additional important residential or commercial property − they room equal in length. The proof has been collection out in complete as one example, since the overlapping congruent triangles have the right to be confusing.

Theorem

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The diagonals that a rectangle room equal.

Proof

permit ABCD it is in a rectangle.

us prove that AC = BD.

In the triangles ABC and also DCB:

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

so alphabet ≡ DCB (SAS)

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

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This way that am = BM = cm = DM, where M is the intersection the the diagonals. Therefore we can draw a solitary circle through centre M v all four vertices. Us can explain this case by speak that, ‘The vertices of a rectangle space concyclic’.


First test for a rectangle − A parallelogram v one appropriate angle

If a parallelogram is known to have one right angle, then recurring use the co-interior angles proves the all its angle are right angles.

Theorem

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

Because the this theorem, the an interpretation of a rectangle is periodically taken to be ‘a parallelogram with a best angle’.

Construction the a rectangle

We can construct a rectangle with given side lengths by creating a parallelogram through a appropriate angle top top one corner. An initial drop a perpendicular native a allude P to a line . Note B and then note off BC and also BA and also complete the parallel as displayed below.

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Second test for a rectangle − A quadrilateral through equal diagonals the bisect every other

We have shown over that the diagonals of a rectangle space equal and also bisect every other. Vice versa, these 2 properties taken with each other constitute a test for a quadrilateral to it is in a rectangle.

Theorem

A quadrilateral whose diagonals space 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 the the number is a rectangle.


As a consequence of this result, the endpoints of any type of two diameters that a circle form a rectangle, because this quadrilateral has equal diagonals the bisect each other.

Thus we deserve to construct a rectangle really simply by drawing any kind of two intersecting lines, then drawing any type of circle centred in ~ the suggest of intersection. The quadrilateral created by involvement the 4 points where the circle cuts the present is a rectangle since it has actually equal diagonals that bisect every other.

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Rectangles

Definition that a rectangle

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

Properties the a rectangle

A rectangle is a parallelogram, so its the contrary sides space equal. The diagonals of a rectangle space equal and also bisect every other.

Tests because that a rectangle

A parallelogram with one appropriate angle is a rectangle. A quadrilateral whose diagonals room equal and also bisect each other is a rectangle.

Links forward

The staying special quadrilaterals to be cure by the congruence and angle-chasing techniques of this module are rhombuses, kites, squares and also trapezia. The succession of theorems associated in dealing with all these unique quadrilaterals at once becomes fairly complicated, so their discussion will it is in left until the module Rhombuses, Kites, and Trapezia. Every individual proof, however, is well within Year 8 ability, provided that students have actually the appropriate experiences. In particular, it would be useful to prove in Year 8 the the diagonals that rhombuses and also kites fulfill at right angles − this an outcome is essential in area formulas, it is valuable in applications the Pythagoras’ theorem, and also it gives a an ext systematic explanation the several important constructions.

The following step in the development of geometry is a rigorous therapy of similarity. This will permit various results around ratios of lengths to be established, and also make feasible the meaning of the trigonometric ratios. Similarity is required for the geometry the circles, where another class of one-of-a-kind quadrilaterals arises, namely the cyclic quadrilaterals, whose vertices lied on a circle.

Special quadrilaterals and their properties are required to create the traditional formulas for areas and also volumes of figures. Later, these results will be crucial in occurring integration. Theorems around special quadrilaterals will certainly be widely provided in coordinate geometry.

Rectangles are so ubiquitous that they go unnoticed in many applications. One special duty worth noting is they room the communication of the collaborates of clues in the cartesian airplane − to uncover the coordinates of a suggest in the plane, we finish the rectangle created by the suggest and the 2 axes. Parallelograms arise when we include vectors by perfect the parallelogram − this is the factor why they become so necessary when complex numbers are represented on the Argand diagram.


History and also applications

Rectangles have been advantageous for as long as there have actually been buildings, since vertical pillars and also horizontal crossbeams room the most obvious way to build a building of any kind of size, giving a framework in the shape of a rectangular prism, all of whose faces are rectangles. The diagonals that we constantly use to research rectangles have an analogy in structure − a rectangular frame with a diagonal has actually far an ext rigidity 보다 a straightforward rectangular frame, and also diagonal struts have constantly been offered by contractors to give their building more strength.

Parallelograms are not as typical in the physical world (except together shadows of rectangular objects). Their significant role in the history has remained in the depiction of physical principles by vectors. Because that example, when two pressures are combined, a parallelogram deserve to be attracted to aid compute the size and also direction that the combined force. Once there are three forces, we finish the parallelepiped, i beg your pardon is the three-dimensional analogue that the parallelogram.


REFERENCES

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

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


ANSWERS to EXERCISES

EXERCISE 1

a In the triangle ABM and CDM :

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

b thus AM = CM and 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 space supplementary)
andAD || BC(co-interior angles space supplementary).

EXERCISE 3

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

EXERCISE 4

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

EXERCISE 5

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

Hence ABCD is a parallelogram, due to the fact that one pair of the contrary sides room equal and parallel.

EXERCISE 6

Join AM. With centre M, draw an arc with radius AM the meets AM developed at C . Then ABCD is a parallelogram since its diagonals bisect each other.

EXERCISE 7

The square on each diagonal is the amount of the squares on any two adjacent sides. Because opposite sides are equal in length, the squares ~ above both diagonals room the same.

EXERCISE 8

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

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

EXERCISE 9

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

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

The boosting Mathematics education in schools (TIMES) project 2009-2011 was funded through the Australian government Department of Education, Employment and Workplace Relations.

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