Parallelograms and also Rectangles

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

June 2011

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PDF version of module


Assumed knowledge

Introductory aircraft geometry entailing points and lines, parallel lines and also transversals, edge sums the triangles and quadrilaterals, and general angle-chasing.The four standard congruence tests and their application in problems and proofs.Properties of isosceles and equilateral triangles and also tests for them.Experience v a logical argument in geometry being written as a succession of steps, every justified through a reason.Ruler-and-compasses constructions.Informal experience with one-of-a-kind quadrilaterals.

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Motivation

There are just three important categories of unique triangles − isosceles triangles, it is intended triangles and also right-angled triangles. In contrast, there are many categories of one-of-a-kind quadrilaterals. This module will deal with two of lock − parallelograms and rectangles − leave rhombuses, kites, squares, trapezia and cyclic quadrilaterals come the module, Rhombuses, Kites, and also Trapezia.

Apart from cyclic quadrilaterals, these unique quadrilaterals and their properties have actually been presented informally over several years, but without congruence, a rigorous discussion of castle was no possible. Each congruence proof offers the diagonals to divide the quadrilateral into triangles, ~ which us can apply the approaches of congruent triangles developed in the module, Congruence.

The present treatment has four purposes:

The parallelogram and rectangle are very closely defined.Their far-ranging properties room proven, greatly using congruence.Tests for them are developed that deserve to be offered to examine that a given quadrilateral is a parallel or rectangle − again, congruence is mostly required.Some ruler-and-compasses build of lock are arisen as an easy applications of the definitions and also tests.

The material in this module is perfect for Year 8 as more applications the congruence and also constructions. Because of its methodical development, that provides wonderful introduction to proof, converse statements, and sequences the theorems. Considerable guidance in such principles is normally forced in Year 8, i m sorry is consolidated through further conversation in later years.

The complementary concepts of a ‘property’ of a figure, and a ‘test’ because that a figure, become specifically important in this module. Indeed, clarity around these concepts is among the countless reasons for teaching this product at school. Most of the tests that we meet are converses that properties that have already been proven. Because that example, the fact that the base angles of one isosceles triangle room equal is a residential or commercial property of isosceles triangles. This property deserve to be re-formulated together an ‘If …, climate … ’ statement:

If two sides the a triangle room equal, then the angle opposite those sides room equal.

Now the corresponding test for a triangle to it is in isosceles is plainly the converse statement:

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

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


Quadrilaterals

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In various other modules, we defined a square to it is in a closed aircraft figure bounded by 4 intervals, and also a convex quadrilateral to it is in a quadrilateral in i m sorry each inner angle is less than 180°. We showed two crucial theorems about the angle of a quadrilateral:

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

To prove the first result, we built in each case a diagonal the lies completely inside the quadrilateral. This divided the quadrilateral into two triangles, every of who angle sum is 180°.

To prove the 2nd result, we produced one side at each vertex of the convex quadrilateral. The sum of the four straight angles is 720° and also the amount of the four interior angle is 360°, for this reason the sum of the four exterior angles is 360°.


Parallelograms

We start with parallelograms, due to the fact that we will certainly be utilizing the results about parallelograms when mentioning the various other figures.

Definition of a parallelogram

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A parallelogram is a square whose the opposite sides room parallel. Hence the quadrilateral ABCD displayed opposite is a parallelogram because abdominal || DC and DA || CB.

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

Constructing a parallelogram using the definition

To construct a parallelogram utilizing the definition, we can use the copy-an-angle building and construction to kind parallel lines. Because that example, mean that we are provided the intervals abdominal and advertisement in the diagram below. Us extend advertisement and ab and copy the angle at A to equivalent angles in ~ B and also D to determine C and complete the parallelogram ABCD. (See the module, Construction.)

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This is no the easiest way to construct a parallelogram.

First residential or commercial property of a parallelogram − the opposite angles room equal

The 3 properties that a parallelogram occurred below worry first, the interior angles, secondly, the sides, and also thirdly the diagonals. The first property is most easily proven using angle-chasing, yet it can additionally be proven utilizing congruence.

Theorem

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The opposite angles of a parallelogram room equal.

Proof

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

Second home of a parallel − opposing sides are equal

As one example, this proof has been set out in full, with the congruence test completely developed. Many of the remaining proofs however, are presented as exercises, with an abbreviated version provided as one answer.

Theorem

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The opposite political parties of a parallelogram are equal.

Proof

ABCD is a parallelogram.
To prove that abdominal muscle = CD and ad = BC.
Join the diagonal line AC.
In the triangle 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 BC = advertisement (matching political parties of congruent triangles).

Third residential property of a parallel − The diagonals bisect every other

Theorem

The diagonals that a parallelogram bisect every other.


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EXERCISE 1

a Prove that ABM ≡ CDM.

b hence prove the the diagonals bisect each other.


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

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

First test because that a parallel − the contrary angles are equal

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

Theorem

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


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EXERCISE 2

Prove this an outcome using the figure below.

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Second test for a parallelogram − opposite sides room equal

This test is the converse the the building that the opposite political parties of a parallelogram are equal.

Theorem

If the opposite sides of a (convex) quadrilateral room equal, then 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 gives a simple construction the a parallelogram given two adjacent sides − abdominal muscle and ad in the figure to the right. Attract a circle with centre B and also radius AD, and another circle with centre D and also radius AB. The circles crossing at 2 points − let C it is in the allude of intersection in ~ the non-reflex edge BAD. Climate ABCD is a parallelogram because its the contrary sides room equal.

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It likewise gives a method of drawing the heat parallel come a offered line through a given suggest P. Choose any type of two clues A and also B ~ above , and also complete the parallel PABQ.

Then PQ ||

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

This test turns out to be very useful, because it uses only one pair of the opposite sides.

Theorem

If one pair the opposite political parties of a quadrilateral are equal and parallel, then the square is a parallelogram.


This test because that a parallelogram offers a quick and easy means to construct a parallelogram using a two-sided ruler. Attract a 6 centimeter interval on every side that the ruler. Joining up the endpoints offers a parallelogram.

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The check is an especially important in the later on theory that vectors. Expect that
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and also
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room two directed intervals that room parallel and also have the same length − that is, they stand for the same vector. Climate the figure ABQP to the right is a parallelogram.

Even a straightforward vector property favor the commutativity that the enhancement of vectors depends 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 because that a parallelogram − The diagonals bisect each other

This test is the converse that the property that the diagonals the a parallelogram bisect every other.

Theorem

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


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This test gives a very simple construction of a parallelogram. Draw two intersecting lines, then draw two one with different radii centred on your intersection. Sign up with the points where alternating circles reduced the lines. This is a parallelogram since the diagonals bisect each other.

It also enables yet another an approach of completing an angle poor to a parallelogram, as presented in the complying with exercise.


EXERCISE 6

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Given two intervals abdominal muscle and advertisement meeting at a common vertex A, construct the midpoint M of BD. Complete this to a construction of the parallel ABCD, justifying her answer.


Parallelograms

Definition the a parallelogram

A parallel is a square whose the contrary sides room parallel.

Properties the a parallelogram

The opposite angles of a parallelogram are equal. The opposite sides of a parallelogram are equal. The diagonals that a parallel bisect every other.

Tests for a parallelogram

A quadrilateral is a parallel if:

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

Rectangles

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

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Definition that a Rectangle

A rectangle is a quadrilateral in i m sorry all angles are appropriate angles.

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

Each pair the co-interior angles are supplementary, due to the fact that two ideal angles add to a right angle, for this reason the opposite sides of a rectangle space parallel. This means that a rectangle is a parallelogram, so:

Its the contrary sides are equal and also parallel. That is diagonals bisect every other.

Second building of a rectangle − The diagonals room equal

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

Theorem

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The diagonals of a rectangle room equal.

Proof

permit ABCD be a rectangle.

we prove the AC = BD.

In the triangle ABC and DCB:

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

so abc ≡ DCB (SAS)

for this reason AC = DB (matching sides of congruent triangles).

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This means that to be = BM = centimeter = DM, whereby M is the intersection the the diagonals. For this reason we can attract a solitary circle through centre M v all 4 vertices. We can define this case by saying that, ‘The vertices that a rectangle are concyclic’.


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

If a parallelogram is well-known to have actually one ideal angle, then repeated use that co-interior angles proves the all its angle are right angles.

Theorem

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

Because that this theorem, the an interpretation of a rectangle is occasionally taken to it is in ‘a parallelogram through a ideal angle’.

Construction the a rectangle

We deserve to construct a rectangle with given side lengths by building a parallelogram with a ideal angle on one corner. First drop a perpendicular indigenous a point P come a line . Note B and also then note off BC and BA and also complete the parallel as presented below.

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

We have shown above that the diagonals of a rectangle room equal and bisect every other. Vice versa, these 2 properties taken with each other constitute a test because that a quadrilateral to be a rectangle.

Theorem

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


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EXERCISE 8

a Why is the square a parallelogram?

b usage congruence to prove that the figure is a rectangle.


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

Thus we can construct a rectangle very simply by drawing any kind of two intersecting lines, then drawing any type of circle centred in ~ the allude of intersection. The quadrilateral created by involvement the 4 points where the circle cuts the lines is a rectangle since it has equal diagonals the bisect each other.

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Rectangles

Definition of a rectangle

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

Properties of a rectangle

A rectangle is a parallelogram, therefore its opposite sides room equal. The diagonals the a rectangle are equal and bisect every other.

Tests because that a rectangle

A parallelogram v one best angle is a rectangle. A quadrilateral whose diagonals space equal and also bisect each other is a rectangle.

Links forward

The remaining special quadrilaterals come be treated by the congruence and angle-chasing methods of this module are rhombuses, kites, squares and trapezia. The sequence of theorems associated in dealing with all these distinct quadrilaterals at when becomes fairly complicated, therefore their conversation will be left till the module Rhombuses, Kites, and also Trapezia. Every individual proof, however, is well within Year 8 ability, listed that students have actually the best experiences. In particular, it would be advantageous to prove in Year 8 that the diagonals the rhombuses and kites accomplish at ideal angles − this result is required in area formulas, the is beneficial in applications that Pythagoras’ theorem, and also it provides a more systematic explanation of several crucial constructions.

The following step in the advance of geometry is a rigorous therapy of similarity. This will permit various results about ratios that lengths to it is in established, and additionally make possible the definition of the trigonometric ratios. Similarity is compelled for the geometry the circles, where an additional class of one-of-a-kind quadrilaterals arises, specific the cyclic quadrilaterals, whose vertices lie on a circle.

Special quadrilaterals and their nature are needed to establish the standard formulas for areas and volumes that figures. Later, these results will be vital in occurring integration. Theorems about special quadrilaterals will certainly be widely offered in coordinate geometry.

Rectangles are so ubiquitous that they walk unnoticed in many applications. One special function worth noting is they room the basis of the collaborates of clues in the cartesian airplane − to find the coordinates of a suggest in the plane, we finish the rectangle created by the point and the 2 axes. Parallelograms arise when we add vectors by completing the parallel − this is the reason why they come to be so essential when complex numbers are represented on the Argand diagram.


History and applications

Rectangles have actually been useful for as long as there have actually been buildings, due to the fact that vertical pillars and also horizontal crossbeams space the most obvious means to build a structure of any type of size, offering a framework in the form of a rectangular prism, every one of whose deals with are rectangles. The diagonals that we constantly usage to research rectangles have actually an analogy in structure − a rectangular structure with a diagonal has actually far much more rigidity 보다 a an easy rectangular frame, and also diagonal struts have constantly been provided by contractors to offer their building much more strength.

Parallelograms room not as common in the physical human being (except together shadows of rectangle-shaped objects). Their significant role in the history has remained in the depiction of physical principles by vectors. Because that example, as soon as two pressures are combined, a parallelogram have the right to be attracted to aid compute the size and also direction the the an unified force. As soon as there are three forces, we complete the parallelepiped, i m sorry is the three-dimensional analogue the the parallelogram.


REFERENCES

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

History the 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 muscle || DC )
2.ABM= CDM(alternate angles, ab || DC )
3. AB = CD (opposite political parties of parallel 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 square ABCD)
α + β= 180o
HenceAB || DC(co-interior angles are supplementary)
andAD || BC(co-interior angles are supplementary).

EXERCISE 3

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

EXERCISE 4

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

EXERCISE 5

First prove that 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 room equal):

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

EXERCISE 6

Join AM. With centre M, attract an arc through radius AM that meets AM developed at C . Climate ABCD is a parallelogram due to the fact that its diagonals bisect every other.

EXERCISE 7

The square on each diagonal is the sum of the squares on any kind of two surrounding sides. Because opposite sides space equal in length, the squares ~ above both diagonals room the same.

EXERCISE 8

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

therefore ABCD is rectangle, since it is a parallelogram with one best 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 ideal angle, and also similarly, B, C and D are appropriate angles.

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