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 including points and also lines, parallel lines and transversals, angle sums of triangles and also quadrilaterals, and also general angle-chasing.The four standard congruence tests and also their application in problems and also proofs.Properties of isosceles and also equilateral triangles and also tests for them.Experience with a logical discussion in geometry being composed as a sequence of steps, every justified by a reason.Ruler-and-compasses constructions.Informal endure with unique quadrilaterals.

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Motivation

There are just three vital categories of special triangles − isosceles triangles, it is intended triangles and also right-angled triangles. In contrast, over there are countless categories of one-of-a-kind quadrilaterals. This module will address two of lock − parallelograms and rectangles − leave rhombuses, kites, squares, trapezia and also cyclic quadrilaterals to the module, Rhombuses, Kites, and also Trapezia.

Apart from cyclic quadrilaterals, these unique quadrilaterals and also their properties have actually been introduced informally over several years, but without congruence, a rigorous conversation of lock was no possible. Every congruence proof supplies the diagonals to division the quadrilateral into triangles, after which we can apply the techniques of congruent triangles developed in the module, Congruence.

The current treatment has four purposes:

The parallelogram and also rectangle are closely defined.Their far-ranging properties are proven, largely using congruence.Tests because that them are established that deserve to be used to examine that a provided quadrilateral is a parallelogram or rectangle − again, congruence is mainly required.Some ruler-and-compasses build of lock are emerged as simple applications that the definitions and also tests.

The material in this module is perfect for Year 8 as more applications of congruence and constructions. Since of its organized development, it provides wonderful introduction to proof, converse statements, and also sequences of theorems. Considerable guidance in such principles is normally forced in Year 8, which is consolidated by further conversation in later on years.

The complementary principles of a ‘property’ the a figure, and a ‘test’ because that a figure, become an especially important in this module. Indeed, clarity around these concepts is among the countless reasons for to teach this product at school. Most of the tests that we accomplish 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 building of isosceles triangles. This property have the right to be re-formulated together an ‘If …, climate … ’ statement:

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

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

If two angles of a triangle space equal, then the political parties opposite those angles room equal.

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


Quadrilaterals

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In other modules, we identified a quadrilateral to it is in a closed aircraft figure bounded by 4 intervals, and also a convex square to it is in a square in which each inner angle is less than 180°. We confirmed two vital theorems around the angle of a quadrilateral:

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

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

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


Parallelograms

We begin with parallelograms, due to the fact that we will certainly be using the results about parallelograms when stating the other figures.

Definition of a parallelogram

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

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

Constructing a parallelogram utilizing the definition

To build 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 given the intervals ab and advertisement in the diagram below. Us extend ad and abdominal and copy the edge at A to matching angles in ~ B and also D to identify C and complete the parallelogram ABCD. (See the module, Construction.)

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This is no the easiest method to build a parallelogram.

First property of a parallelogram − opposing angles space equal

The three properties of a parallelogram developed below issue first, the inner angles, secondly, the sides, and also thirdly the diagonals. The first property is most quickly proven utilizing angle-chasing, however it can also be proven making use of 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 also B = β.
Prove that C = α and D = β.
α + β = 180°(co-interior angles, ad || BC),
soC = α(co-interior angles, ab || DC)
and D = β(co-interior angles, abdominal muscle || DC).

Second home of a parallelogram − the contrary sides are equal

As one example, this proof has been collection out in full, through the congruence test completely developed. Most of the staying proofs however, space presented together exercises, through an abbreviation version provided as an answer.

Theorem

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The opposite political parties of a parallelogram are equal.

Proof

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

Third residential property of a parallelogram − 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 every 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 v all 4 vertices.

First test for a parallelogram − the contrary angles are equal

Besides the definition itself, there are four helpful tests for a parallelogram. Our very first test is the converse that our very first property, the the opposite angles of a quadrilateral room equal.

Theorem

If the opposite angle of a quadrilateral space equal, then the square is a parallelogram.


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EXERCISE 2

Prove this result using the number below.

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Second test for a parallel − opposite sides are equal

This test is the converse that the building 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, wherein the diagonal line AC has been joined.


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This test gives a basic construction that a parallelogram given two adjacent sides − ab and advertisement in the figure to the right. Draw a circle through centre B and radius AD, and also another circle v centre D and also radius AB. The circles intersect at 2 points − permit C be the suggest of intersection within the non-reflex edge BAD. Climate ABCD is a parallelogram due to the fact that its the opposite sides space equal.

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It additionally gives a an approach of illustration the line parallel come a offered line with a given point P. Choose any two point out A and also B on , and also complete the parallel PABQ.

Then PQ ||

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

This test transforms out to be an extremely useful, since it uses only one pair of the opposite sides.

Theorem

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


This test because that a parallelogram provides a quick and easy method to construct a parallelogram using a two-sided ruler. Draw a 6 centimeter interval on each side that the ruler. Joining increase the endpoints provides a parallelogram.

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The check is an especially important in the later on theory of vectors. Suppose that
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and
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space two command intervals that are parallel and have the same size − that is, they represent the exact same vector. Climate the number ABQP to the best is a parallelogram.

Even a straightforward vector property choose the commutativity the the enhancement of vectors counts 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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Fourth test because that a parallelogram − The diagonals bisect each other

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

Theorem

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


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This test provides a very simple construction that a parallelogram. Attract two intersecting lines, then draw two circles with different radii centred on your intersection. Sign up with the point out where alternate circles reduced the lines. This is a parallelogram since the diagonals bisect each other.

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


EXERCISE 6

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Given two intervals abdominal and ad meeting in ~ a typical vertex A, construct the midpoint M the BD. Complete this to a construction of the parallelogram ABCD, justifying your answer.


Parallelograms

Definition that a parallelogram

A parallel is a quadrilateral whose the opposite sides room parallel.

Properties of a parallelogram

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

Tests for a parallelogram

A quadrilateral is a parallel if:

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

Rectangles

The word ‘rectangle’ method ‘right angle’, and also this is reflect in that is definition.

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Definition of a Rectangle

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

First home of a rectangle − A rectangle is a parallelogram

Each pair that co-interior angles space supplementary, because two right angles include to a straight angle, so the opposite sides of a rectangle room parallel. This method that a rectangle is a parallelogram, so:

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

Second building of a rectangle − The diagonals are 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, because the overlapping congruent triangles can be confusing.

Theorem

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The diagonals that a rectangle space equal.

Proof

permit ABCD be a rectangle.

us prove that AC = BD.

In the triangles 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 way that am = BM = centimeter = DM, whereby M is the intersection of the diagonals. Thus we can draw a solitary circle v centre M through all 4 vertices. Us can explain this situation by speak that, ‘The vertices the a rectangle are concyclic’.


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

If a parallelogram is known to have one appropriate angle, then repetitive use of co-interior angles proves that all its angles are ideal angles.

Theorem

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

Because that this theorem, the meaning of a rectangle is sometimes taken to be ‘a parallelogram through a ideal angle’.

Construction the a rectangle

We can construct a rectangle with provided side lengths by constructing a parallelogram with a appropriate angle on one corner. Very first drop a perpendicular from a allude P to a heat . Mark B and also then mark off BC and also BA and complete the parallel as shown below.

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

We have actually shown above that the diagonals of a rectangle space equal and also bisect each other. Vice versa, these 2 properties taken together constitute a test because that a quadrilateral 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 use congruence to prove the the figure is a rectangle.


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

Thus we deserve to construct a rectangle very simply by drawing any kind of two intersecting lines, climate drawing any type of circle centred at the point of intersection. The quadrilateral created by authorized 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 quadrilateral in i m sorry all angles are right angles.

Properties of a rectangle

A rectangle is a parallelogram, for this reason its the opposite sides room equal. The diagonals of a rectangle are equal and also bisect every other.

Tests because that a rectangle

A parallelogram v one appropriate angle is a rectangle. A square whose diagonals room equal and bisect each various other is a rectangle.

Links forward

The remaining special quadrilaterals come be cure by the congruence and also angle-chasing methods of this module room rhombuses, kites, squares and also trapezia. The succession of theorems involved in dealing with all these special quadrilaterals at once becomes quite complicated, so their discussion will it is in left till the module Rhombuses, Kites, and also Trapezia. Each individual proof, however, is well within Year 8 ability, provided that students have actually the right experiences. In particular, it would certainly be valuable to prove in Year 8 the the diagonals of rhombuses and also kites accomplish at appropriate angles − this an outcome is needed in area formulas, the is useful in applications that Pythagoras’ theorem, and also it provides a an ext systematic explanation of several vital constructions.

The next step in the development of geometry is a rigorous treatment of similarity. This will permit various results about ratios that lengths to it is in established, and also make feasible the an interpretation of the trigonometric ratios. Similarity is required for the geometry that circles, where another class of unique quadrilaterals arises, specific the cyclic quadrilaterals, who vertices lie on a circle.

Special quadrilaterals and their properties are necessary to develop the typical formulas for areas and also volumes of figures. Later, these outcomes will be necessary in developing integration. Theorems around special quadrilaterals will be widely offered in name: coordinates geometry.

Rectangles room so common that they walk unnoticed in most applications. One special function worth note is they space the communication of the coordinates of points in the cartesian aircraft − to find the works with of a allude in the plane, we complete the rectangle developed by the allude and the two axes. Parallelograms arise when we include vectors by completing the parallelogram − this is the reason why they become so essential when complicated numbers are stood for on the Argand diagram.


History and applications

Rectangles have been beneficial for as long as there have been buildings, because vertical pillars and also horizontal crossbeams room the many obvious means to construct a building of any kind of size, giving a structure in the shape of a rectangle-shaped prism, all of whose faces are rectangles. The diagonals that us constantly use to examine rectangles have actually an analogy in building − a rectangular frame with a diagonal has actually far much more rigidity than a basic rectangular frame, and also diagonal struts have always been provided by home builders to provide their building more strength.

Parallelograms are not as usual in the physical people (except together shadows of rectangular objects). Their major role historically has been in the depiction of physical concepts by vectors. Because that example, once two forces are combined, a parallelogram can be drawn to aid compute the size and direction that the merged force. Once there space three forces, we finish the parallelepiped, i beg your pardon is the three-dimensional analogue the the parallelogram.


REFERENCES

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

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


ANSWERS come EXERCISES

EXERCISE 1

a In the triangle ABM and also CDM :

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

b therefore AM = CM and 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 room 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 angles of congruent triangles)
soAD || BC and ab || DC(alternate angles are equal.)

EXERCISE 4

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

EXERCISE 5

First prove that ABM ≡ CDM utilizing 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, because one pair of the contrary sides are equal and also parallel.

EXERCISE 6

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

EXERCISE 7

The square on every diagonal is the sum of the squares on any type of two nearby sides. Because opposite sides are equal in length, the squares on both diagonals room the same.

EXERCISE 8

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

thus ABCD is rectangle, since it is a parallelogram with one appropriate angle.

EXERCISE 9

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

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

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