Parallelograms and Rectangles

Measurement and Geometry : Module 20Years : 8-9

June 2011

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PDF version of module


Assumed knowledge

Introductory aircraft geometry including points and lines, parallel lines and transversals, angle sums the triangles and also quadrilaterals, and general angle-chasing.The 4 standard congruence tests and their applications in problems and also proofs.Properties that isosceles and also equilateral triangles and tests for them.Experience with a logical discussion in geometry being composed as a succession of steps, each justified through a reason.Ruler-and-compasses constructions.Informal experience with special quadrilaterals.

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Motivation

There are just three necessary categories of special triangles − isosceles triangles, it is provided triangles and right-angled triangles. In contrast, there are plenty of categories of one-of-a-kind quadrilaterals. This module will attend to two of castle − parallelograms and also rectangles − leaving rhombuses, kites, squares, trapezia and also cyclic quadrilaterals come the module, Rhombuses, Kites, and Trapezia.

Apart native cyclic quadrilaterals, these unique quadrilaterals and also their properties have actually been presented informally over numerous years, but without congruence, a rigorous conversation of lock was not possible. Each congruence proof provides the diagonals to division the quadrilateral into triangles, after ~ which us can use the techniques of congruent triangles arisen in the module, Congruence.

The current treatment has four purposes:

The parallelogram and also rectangle are carefully defined.Their far-reaching properties are proven, largely using congruence.Tests for them are created that deserve to be used to check that a offered quadrilateral is a parallel or rectangle − again, congruence is largely required.Some ruler-and-compasses build of castle are emerged as basic applications the the definitions and also tests.

The material in this module is an ideal for Year 8 as additional applications the congruence and constructions. Since of its organized development, that provides great introduction come proof, converse statements, and sequences of theorems. Considerable guidance in such principles is normally required in Year 8, i beg your pardon is consolidated through further discussion in later on years.

The complementary principles of a ‘property’ of a figure, and a ‘test’ because that a figure, become an especially important in this module. Indeed, clarity about these ideas is one of the plenty of reasons for teaching this material at school. Many of the tests that we fulfill are converses of properties that have already been proven. Because that example, the reality that the base angles of an isosceles triangle space equal is a building of isosceles triangles. This property have the right to be re-formulated together an ‘If …, climate … ’ statement:

If 2 sides the a triangle are equal, climate the angle opposite those sides space equal.

Now the matching test for a triangle to be isosceles is plainly the converse statement:

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

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


Quadrilaterals

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In other modules, we characterized a square to be a closed airplane figure bounded by four intervals, and also a convex square to be a quadrilateral in i m sorry each interior angle is much less than 180°. We showed two important theorems about the angles of a quadrilateral:

The amount 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 constructed in each case a diagonal the lies totally inside the quadrilateral. This split the quadrilateral right into two triangles, every of whose angle amount is 180°.

To prove the 2nd result, we developed one next at each vertex of the convex quadrilateral. The amount of the four straight angles is 720° and the sum of the four interior angle is 360°, therefore the amount of the four exterior angles is 360°.


Parallelograms

We begin with parallelograms, because we will be using the results about parallelograms when mentioning the other figures.

Definition of a parallelogram

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

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

Constructing a parallelogram utilizing the definition

To construct a parallelogram making use of the definition, we can use the copy-an-angle building and construction to type parallel lines. For example, suppose that we are offered the intervals abdominal and advertisement in the diagram below. Us extend advertisement and abdominal muscle and copy the edge at A to equivalent angles at B and D to recognize C and also complete the parallel ABCD. (See the module, Construction.)

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This is no the easiest way to build a parallelogram.

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

The 3 properties the a parallelogram developed below worry first, the inner angles, secondly, the sides, and also thirdly the diagonals. The very first property is most easily proven utilizing angle-chasing, yet it can likewise be proven making use of congruence.

Theorem

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The opposite angle of a parallelogram are equal.

Proof

Let ABCD be a parallelogram, through A = α and also B = β.
Prove that 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 muscle || DC).

Second building of a parallel − the opposite sides room equal

As one example, this proof has been set out in full, through the congruence test fully developed. Most of the continuing to be proofs however, are presented as exercises, through an abbreviated version provided as an answer.

Theorem

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The opposite sides of a parallelogram are equal.

Proof

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

Third building of a parallel − The diagonals bisect every other

Theorem

The diagonals of a parallel bisect every other.


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EXERCISE 1

a Prove the ABM ≡ CDM.

b for this reason prove the the diagonals bisect each other.


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

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

First test for a parallelogram − opposing angles space equal

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

Theorem

If the opposite angles of a quadrilateral room equal, then 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 because that a parallelogram − the opposite sides space equal

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

Theorem

If the opposite sides 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, where the diagonal AC has been joined.


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This test gives a straightforward construction that a parallelogram offered two adjacent sides − ab and advertisement in the figure to the right. Draw a circle with centre B and also radius AD, and also another circle through centre D and also radius AB. The circles intersect at two points − permit C it is in the suggest of intersection in ~ the non-reflex angle BAD. Climate ABCD is a parallelogram since its the contrary sides are equal.

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It also gives a an approach of illustration the heat parallel come a provided line with a given allude P. Choose any kind of two points A and B top top , and also complete the parallel PABQ.

Then PQ ||

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

This test turns out come be really useful, due to the fact that it uses only one pair of the opposite sides.

Theorem

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


This test because that a parallelogram gives a quick and easy method to construct a parallelogram using a two-sided ruler. Attract a 6 cm interval on every side that the ruler. Joining increase the endpoints gives a parallelogram.

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The test is particularly important in the later theory the vectors. Mean that
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and also
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are two directed intervals that space parallel and also have the same size − that is, they represent the same vector. Then the number ABQP come the best is a parallelogram.

Even a simple vector property prefer the commutativity that the addition of vectors relies on this construction. The parallel ABQP shows, for example, that

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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 the a parallel bisect each other.

Theorem

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


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This test gives a very basic construction of a parallelogram. Attract two intersecting lines, then attract two circles with different radii centred on your intersection. Sign up with the point out 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 presented in the adhering to exercise.


EXERCISE 6

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Given two intervals abdominal muscle and ad meeting at a typical vertex A, build the midpoint M of BD. Finish this to a building and construction of the parallelogram ABCD, justifying your answer.


Parallelograms

Definition the a parallelogram

A parallel is a square whose opposite sides room parallel.

Properties that a parallelogram

The opposite angles of a parallelogram are equal. The opposite political parties of a parallelogram space equal. The diagonals that a parallel bisect every other.

Tests because that a parallelogram

A quadrilateral is a parallelogram if:

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

Rectangles

The word ‘rectangle’ means ‘right angle’, and this is reflect in the definition.

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Definition that a Rectangle

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

First building of a rectangle − A rectangle is a parallelogram

Each pair the co-interior angles room supplementary, due to the fact that two right angles add to a straight angle, for this reason the opposite political parties of a rectangle space parallel. This way that a rectangle is a parallelogram, so:

Its opposite sides are equal and parallel. Its diagonals bisect each other.

Second home of a rectangle − The diagonals room equal

The diagonals the a rectangle have another important home − they space equal in length. The proof has actually been collection out in full as one example, since the overlapping congruent triangles deserve to be confusing.

Theorem

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The diagonals of a rectangle room equal.

Proof

allow ABCD it is in a rectangle.

we prove the AC = BD.

In the triangles ABC and DCB:

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

so alphabet ≡ DCB (SAS)

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

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This means that to be = BM = cm = DM, wherein M is the intersection the the diagonals. Hence we can attract a single circle with centre M v all four vertices. We can explain this instance by saying that, ‘The vertices the a rectangle room concyclic’.


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

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

Theorem

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

Because the this theorem, the definition of a rectangle is occasionally taken to it is in ‘a parallelogram v a appropriate angle’.

Construction the a rectangle

We have the right to construct a rectangle with given side lengths by building a parallelogram v a best angle on one corner. An initial drop a perpendicular from a point P to a line . Mark B and also then note off BC and also BA and also complete the parallel as presented below.

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

We have actually shown over that the diagonals the a rectangle are equal and also bisect each other. Vice versa, these two properties taken with each other constitute a test because that a square to it is in a rectangle.

Theorem

A quadrilateral whose diagonals room equal and also bisect each other is a rectangle.


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EXERCISE 8

a Why is the quadrilateral a parallelogram?

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


As a repercussion of this result, the endpoints of any kind of two diameters the a circle kind a rectangle, due to the fact that this quadrilateral has actually equal diagonals that bisect every other.

Thus we can construct a rectangle really simply by drawing any type of two intersecting lines, climate drawing any type of circle centred in ~ the suggest of intersection. The quadrilateral developed by joining the 4 points whereby the circle cuts the present is a rectangle due to the fact that it has equal diagonals the bisect every other.

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Rectangles

Definition of a rectangle

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

Properties the a rectangle

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

Tests because that a rectangle

A parallelogram with one appropriate angle is a rectangle. A square whose diagonals are equal and bisect each other is a rectangle.

Links forward

The continuing to be special quadrilaterals to be cure by the congruence and angle-chasing approaches of this module space rhombuses, kites, squares and also trapezia. The sequence of theorems associated in dealing with all these one-of-a-kind quadrilaterals at once becomes fairly complicated, therefore their discussion will it is in left until the module Rhombuses, Kites, and also Trapezia. Each individual proof, however, is well within Year 8 ability, listed that students have the ideal experiences. In particular, it would certainly be valuable to prove in Year 8 that the diagonals of rhombuses and kites accomplish at right angles − this an outcome is needed in area formulas, that is helpful in applications of Pythagoras’ theorem, and it provides a more systematic explanation of several important constructions.

The following step in the breakthrough of geometry is a rigorous therapy of similarity. This will enable various results around ratios the lengths to it is in established, and also make feasible the definition of the trigonometric ratios. Similarity is required for the geometry the circles, where an additional class of one-of-a-kind quadrilaterals arises, namely the cyclic quadrilaterals, whose vertices lie on a circle.

Special quadrilaterals and their nature are necessary to establish the traditional formulas for areas and also volumes the figures. Later, these outcomes will be crucial in occurring integration. Theorems about special quadrilaterals will certainly be widely provided in name: coordinates geometry.

Rectangles are so common that they walk unnoticed in many applications. One special duty worth note is they are the basis of the coordinates of points in the cartesian plane − to discover the collaborates of a allude in the plane, we complete the rectangle created by the allude and the two axes. Parallelograms arise once we add vectors by completing the parallel − this is the reason why they come to be so crucial when facility numbers are stood for on the Argand diagram.


History and applications

Rectangles have actually been valuable for as long as there have actually been buildings, because vertical pillars and horizontal crossbeams room the many obvious method to build a building of any kind of size, providing a framework in the form of a rectangular prism, all of whose encounters are rectangles. The diagonals that we constantly usage to research rectangles have an analogy in structure − a rectangular structure with a diagonal has far much more rigidity than a basic rectangular frame, and diagonal struts have constantly been used by building contractors to give their building an ext strength.

Parallelograms space not as typical in the physical world (except together shadows of rectangular objects). Their significant role in the history has remained in the representation of physical ideas by vectors. Because that example, once two forces are combined, a parallelogram can be attracted to assist compute the size and also direction that the merged force. When there space three forces, we finish the parallelepiped, i beg your pardon is the three-dimensional analogue the the parallelogram.


REFERENCES

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

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


ANSWERS come EXERCISES

EXERCISE 1

a In the triangles ABM and CDM :

1.BAM= DCM(alternate angles, abdominal || 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 also DM = BM (matching political parties of congruent triangles)

EXERCISE 2

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

EXERCISE 3

First show that abc ≡ CDA using the SSS congruence test.
HenceACB = CAD and CAB = ACD(matching angles of congruent triangles)
soAD || BC and abdominal muscle || DC(alternate angles are equal.)

EXERCISE 4

First prove the 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 the ABM ≡ CDM utilizing the SAS congruence test.
HenceAB = CD(matching political parties of congruent triangles)
AlsoABM = CDM(matching angles of congruent triangles)
soAB || DC(alternate angles room equal):

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

EXERCISE 6

Join AM. With 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 sum of the squares on any two surrounding sides. Because opposite sides space equal in length, the squares on both diagonals space 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 contrary sides room equal.
HenceABC ≡ DCB(SSS)
soABC = DCB(matching angles of congruent triangles).
ButABC + DCB = 180o(co-interior angles, abdominal || DC )
soABC = DCB = 90o .

thus ABCD is rectangle, due to the fact that it is a parallelogram v 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 appropriate angle, and similarly, B, C and also D are right angles.

The boosting Mathematics education in institutions (TIMES) job 2009-2011 was funded by the Australian government Department of Education, Employment and also Workplace Relations.

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