Parallelograms and Rectangles

Measurement and Geometry : Module 20Years : 8-9

June 2011

PDF version of moduleAssumed knowledge

Introductory airplane geometry involving points and lines, parallel lines and transversals, edge sums of triangles and also quadrilaterals, and also general angle-chasing.The four standard congruence tests and their application in problems and also proofs.Properties that isosceles and equilateral triangles and also tests because that them.Experience with a logical debate in geometry being written as a sequence of steps, every justified through a reason.Ruler-and-compasses constructions.Informal suffer with special quadrilaterals.You are watching: The diagonals of a parallelogram bisect each other.

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Motivation

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

Apart from cyclic quadrilaterals, these unique quadrilaterals and also their properties have been presented informally over number of years, but without congruence, a rigorous discussion of lock was not possible. Each congruence proof offers the diagonals to division the quadrilateral right into triangles, after which us can use the techniques of congruent triangles emerged in the module, Congruence.

The existing treatment has 4 purposes:

The parallelogram and also rectangle are very closely defined.Their far-ranging properties room proven, mainly using congruence.Tests for them are developed that can be supplied to inspect that a offered quadrilateral is a parallelogram or rectangle − again, congruence is largely required.Some ruler-and-compasses build of lock are emerged as basic applications that the definitions and tests.The material in this module is an ideal for Year 8 as more applications of congruence and also constructions. Since of its methodical development, the provides terrific introduction to proof, converse statements, and also sequences of theorems. Substantial guidance in such principles is normally required in Year 8, which is consolidated through further discussion in later on years.

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

If two sides of a triangle space 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 two angles of a triangle are equal, climate the political parties opposite those angles room equal.Remember that a statement might be true, however its converse false. That is true the ‘If a number is a multiple of 4, then it is even’, yet it is false the ‘If a number is even, then it is a many of 4’.

Quadrilaterals

In various other modules, we defined a square to it is in a closed plane figure bounded by 4 intervals, and a convex square to it is in a quadrilateral in which each internal angle is much less than 180°. We verified two necessary theorems about the angle of a quadrilateral:

The amount of the inner angles of a square is 360°.The sum of the exterior angles of a convex square is 360°.To prove the very first result, we created in each instance a diagonal that lies fully inside the quadrilateral. This separated the quadrilateral right into two triangles, each of who angle amount is 180°.

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

Parallelograms

We start with parallelograms, because we will be utilizing the results around parallelograms when mentioning the other figures.

Definition of a parallelogram

A parallelogram is a quadrilateral whose the opposite sides space parallel. Therefore the square ABCD shown opposite is a parallel because ab || DC and also DA || CB.The native ‘parallelogram’ comes from Greek words an interpretation ‘parallel lines’.

Constructing a parallelogram using the definition

To build a parallelogram using the definition, we can use the copy-an-angle building and construction to form parallel lines. For example, mean that we are offered the intervals abdominal and advertisement in the chart below. We extend advertisement and abdominal muscle and copy the edge at A to matching angles in ~ B and also D to recognize C and also complete the parallelogram ABCD. (See the module, Construction.)

This is no the easiest means to build a parallelogram.

First building of a parallelogram − the contrary angles space equal

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

Theorem

The opposite angles of a parallelogram space equal.Proof

Let ABCD be a parallelogram, through A = α and also B = β. | ||||||

Prove the C = α and D = β. | ||||||

α + β | = 180° | (co-interior angles, ad || BC), | ||||

so | C | = α | (co-interior angles, abdominal muscle || DC) | |||

and | D | = β | (co-interior angles, ab || DC). |

Second property of a parallelogram − the opposite sides room equal

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

Theorem

The opposite political parties of a parallelogram are equal.Proof

ABCD is a parallelogram. | ||||

To prove that ab = CD and advertisement = BC. | ||||

Join the diagonal line AC. | ||||

In the triangle ABC and also CDA: | ||||

BAC | = DCA | (alternate angles, abdominal muscle || 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 parallel − The diagonals bisect every other

Theorem

The diagonals that a parallel bisect each other.

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EXERCISE 1

a Prove that ABM ≡ CDM.

b therefore prove that the diagonals bisect each other.

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

Notice that, in general, a parallelogram does not have actually a circumcircle v all four vertices.

First test for a parallel − opposing angles are equal

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

Theorem

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

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EXERCISE 2

Prove this an outcome using the figure below.

Second test because that a parallelogram − the contrary sides are equal

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

Theorem

If the opposite sides of a (convex) quadrilateral are equal, climate the square is a parallelogram.

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EXERCISE 3

Prove this result using congruence in the number to the right, whereby the diagonal line AC has actually been joined.This test gives a simple construction that a parallelogram given two adjacent sides − abdominal and ad in the number to the right. Attract a circle v centre B and also radius AD, and another circle with centre D and also radius AB. The circles crossing at 2 points − let C be the allude of intersection in ~ the non-reflex angle BAD. Then ABCD is a parallelogram due to the fact that its the contrary sides space equal.It likewise gives a an approach of illustration the heat parallel to a provided line with a given suggest P. Choose any kind of two point out A and also B top top , and also complete the parallelogram PABQ.

Then PQ ||

Third test because that a parallel − One pair of opposite sides are equal and also parallel

This test transforms out to be an extremely 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 room equal and parallel, climate the square is a parallelogram.

This test for a parallelogram gives a quick and also easy means to build a parallelogram making use of a two-sided ruler. Attract a 6 cm interval on every side of the ruler. Joining increase the endpoints offers a parallelogram.

The test is particularly important in the later on theory that vectors. Expect that and also are two command intervals that room parallel and have the same length − that is, they stand for the same vector. Climate the number ABQP come the right is a parallelogram.Even a an easy vector property choose the commutativity the the enhancement of vectors counts on this construction. The parallel ABQP shows, because that example, that

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

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

Theorem

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

This test gives a very simple construction the a parallelogram. Draw two intersecting lines, then attract two one with various radii centred on your intersection. Join the clues where alternative circles reduced the lines. This is a parallelogram since the diagonals bisect every other.

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

EXERCISE 6

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

Definition the a parallelogram

A parallel is a quadrilateral whose the contrary sides space parallel.

Properties the a parallelogram

The opposite angles of a parallelogram room equal. The opposite political parties of a parallelogram are equal. The diagonals of a parallelogram bisect each other.Tests because that a parallelogram

A quadrilateral is a parallel if:

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

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

Definition of a RectangleA rectangle is a square in i m sorry all angle are right angles.

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

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

Its the contrary sides space equal and parallel. That is diagonals bisect each other.Second home of a rectangle − The diagonals room equal

The diagonals of a rectangle have another important residential property − they space equal in length. The proof has been set out in full as an example, since the overlapping congruent triangles have the right to be confusing.

Theorem

The diagonals that a rectangle space equal.Proof

allow ABCD be a rectangle.

we prove that AC = BD.

In the triangles ABC and also DCB:

BC | = CB | (common) | ||

AB | = DC | (opposite political parties of a parallelogram) | ||

ABC | =DCA = 90° | (given) |

so alphabet ≡ DCB (SAS)

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

This method that to be = BM = cm = DM, where M is the intersection that the diagonals. Hence we can draw a solitary circle through centre M with all four vertices. We can describe this situation by speak that, ‘The vertices the a rectangle are concyclic’.First test because that a rectangle − A parallelogram v one right angle

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

Theorem

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

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

Construction of a rectangle

We can construct a rectangle with provided side lengths by constructing a parallelogram v a ideal angle on one corner. An initial drop a perpendicular from a suggest P come a heat . Mark B and then note off BC and also BA and also complete the parallel as shown below.

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

We have shown above that the diagonals that a rectangle are equal and also bisect each other. Conversely, these two properties taken together constitute a test for a quadrilateral to it is in a rectangle.

Theorem

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

EXERCISE 8

a Why is the square a parallelogram?

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

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

Thus we have the right to construct a rectangle really simply by drawing any kind of two intersecting lines, then drawing any kind of circle centred in ~ the point of intersection. The quadrilateral developed by joining the four points where the circle cut the present is a rectangle due to the fact that it has actually equal diagonals the bisect every other.

Rectangles

Definition the a rectangle

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

Properties that a rectangle

A rectangle is a parallelogram, for this reason its opposite sides room equal. The diagonals the a rectangle room equal and bisect every other.Tests because that a rectangle

A parallelogram v one right angle is a rectangle. A quadrilateral whose diagonals room equal and also bisect each various other is a rectangle.Links forward

The staying special quadrilaterals come be treated by the congruence and 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 when becomes rather complicated, so 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, noted that students have the right experiences. In particular, it would certainly be useful to prove in Year 8 the the diagonals the rhombuses and kites satisfy at ideal angles − this an outcome is needed in area formulas, the is beneficial in applications that Pythagoras’ theorem, and also it gives a an ext systematic explanation the several necessary constructions.

The following step in the development of geometry is a rigorous therapy of similarity. This will allow various results around ratios of lengths to be established, and also make feasible the meaning of the trigonometric ratios. Similarity is forced for the geometry the circles, where an additional class of special quadrilaterals arises, specific the cyclic quadrilaterals, who vertices lie on a circle.

Special quadrilaterals and also their nature are needed to create the standard formulas for areas and also volumes that figures. Later, these results will be crucial in developing integration. Theorems around special quadrilaterals will certainly be widely offered in name: coordinates geometry.

Rectangles are so common that they walk unnoticed in most applications. One special duty worth note is they room the communication of the collaborates of points in the cartesian airplane − to uncover the coordinates of a allude in the plane, we complete the rectangle created by the suggest and the two axes. Parallelograms arise as soon as we include vectors by perfect the parallelogram − this is the reason why they come to be so crucial when complicated numbers are represented on the Argand diagram.

History and also applications

Rectangles have actually been advantageous for as long as there have actually been buildings, because vertical pillars and also horizontal crossbeams are the many obvious method to build a building of any type of size, giving a framework in the form of a rectangle-shaped prism, every one of whose faces are rectangles. The diagonals that we constantly use to research rectangles have actually an analogy in building − a rectangular framework with a diagonal has actually far much more rigidity than a basic rectangular frame, and also diagonal struts have constantly been offered by contractors to offer their building an ext strength.

Parallelograms are not as typical in the physical people (except as shadows of rectangle-shaped objects). Their major role in history has remained in the representation of physical concepts by vectors. For example, once two forces are combined, a parallelogram have the right to be attracted to aid compute the size and direction that the an unified force. Once there room three forces, we finish the parallelepiped, i m sorry is the three-dimensional analogue of the parallelogram.

REFERENCES

A background of Mathematics: one Introduction, third 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 triangle ABM and CDM :

1. | BAM | = DCM | (alternate angles, ab || DC ) | |||

2. | ABM | = CDM | (alternate angles, abdominal muscle || DC ) | |||

3. | AB | = CD | (opposite political parties of parallel ABCD) | |||

ABM = CDM (AAS) |

b therefore AM = CM and also DM = BM (matching sides of congruent triangles)

EXERCISE 2

From the diagram, | 2α + 2β | = 360o | (angle amount of quadrilateral ABCD) | ||

α + β | = 180o |

Hence | AB || DC | (co-interior angles space supplementary) | ||

and | AD || BC | (co-interior angles are supplementary). |

EXERCISE 3

First show that alphabet ≡ CDA using the SSS congruence test. | ||||

Hence | ACB = CAD and also CAB = ACD | (matching angle of congruent triangles) | ||

so | AD || BC and abdominal muscle || DC | (alternate angles room equal.) |

EXERCISE 4

First prove the ABD ≡ CDB using the SAS congruence test. | ||||

Hence | ADB = CBD | (matching angle of congruent triangles) | ||

so | AD || BC | (alternate angles room equal.) |

EXERCISE 5

First prove that ABM ≡ CDM making use of the SAS congruence test. | ||||

Hence | AB = CD | (matching sides of congruent triangles) | ||

Also | ABM = CDM | (matching angle of congruent triangles) | ||

so | AB || DC | (alternate angles are equal): |

Hence ABCD is a parallelogram, because one pair of the opposite sides space equal and also parallel.

EXERCISE 6

Join AM. With centre M, draw an arc through radius AM the meets AM produced at C . Climate ABCD is a parallelogram because its diagonals bisect each other.

EXERCISE 7

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

EXERCISE 8

a | We have currently proven the a square whose diagonals bisect each various other is a parallelogram. |

b | Because ABCD is a parallelogram, its the contrary sides space equal. | ||||

Hence | ABC ≡ DCB | (SSS) | |||

so | ABC = DCB | (matching angles of congruent triangles). | |||

But | ABC + DCB = 180o | (co-interior angles, abdominal muscle || DC ) | |||

so | ABC = DCB = 90o . |

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

EXERCISE 9

ADM | = α | (base angles of isosceles ADM ) | |||

and | ABM | = β | (base angles of isosceles ABM ), | ||

so | 2α + 2β | = 180o | (angle amount of ABD) | ||

α + β | = 90o. |

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

The enhancing Mathematics education in schools (TIMES) task 2009-2011 was funded by the Australian government Department the Education, Employment and Workplace Relations.

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