Parallelograms and also Rectangles

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

June 2011

PDF Version of moduleAssumed knowledge

Introductory aircraft geomeattempt entailing points and also lines, parallel lines and transversals, angle sums of triangles and quadrilaterals, and basic angle-chasing.The 4 typical congruence tests and their application in troubles and also proofs.Properties of isosceles and also equilateral triangles and tests for them.Experience with a logical debate in geometry being created as a sequence of measures, each justified by a factor.Ruler-and-compasses constructions.Informal endure with unique quadrilaterals.You are watching: Do the diagonals of a parallelogram bisect the angles

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Motivation

There are just 3 vital categories of special triangles − isosceles triangles, equilateral triangles and also right-angled triangles. In comparison, tbelow are many type of categories of one-of-a-kind quadrilaterals. This module will certainly resolve two of them − parallelograms and also rectangles − leaving rhombprovides, kites, squares, trapezia and also cyclic quadrilaterals to the module, Rhombuses, Kites, and Trapezia.

Acomponent from cyclic quadrilaterals, these one-of-a-kind quadrilaterals and their properties have been introduced informally over several years, however without congruence, a rigorous conversation of them was not feasible. Each congruence proof offers the diagonals to divide the quadrilateral right into triangles, after which we can use the methods of congruent triangles emerged in the module, Congruence.

The current therapy has actually 4 purposes:

The parallelogram and also rectangle are very closely defined.Their considerable properties are prcooktop, mostly using congruence.Tests for them are establimelted that can be offered to inspect that a provided quadrilateral is a parallelogram or rectangle − aobtain, congruence is greatly required.Some ruler-and-compasses constructions of them are arisen as basic applications of the meanings and also tests.The material in this module is suitable for Year 8 as even more applications of congruence and constructions. Due to the fact that of its methodical breakthrough, it gives an excellent advent to proof, converse statements, and sequences of theorems. Considerable guidance in such ideas is usually forced in Year 8, which is consolidated by even more discussion in later years.

The complementary concepts of a ‘property’ of a number, and also a ‘test’ for a figure, end up being specifically important in this module. Without a doubt, clarity about these ideas is just one of the many kind of factors for teaching this material at school. Many of the tests that we fulfill are converses of properties that have actually already been prrange. For instance, the reality that the base angles of an isosceles triangle are equal is a residential property of isosceles triangles. This residential or commercial property can be re-formulated as an ‘If …, then … ’ statement:

If two sides of a triangle are equal, then the angles opposite those sides are equal.Now the corresponding test for a triangle to be isosceles is plainly the converse statement:

If two angles of a triangle are equal, then the sides oppowebsite those angles are equal.Remember that a statement may be true, yet its converse false. It is true that ‘If a number is a multiple of 4, then it is even’, yet it is false that ‘If a number is even, then it is a multiple of 4’.

Quadrilaterals

In various other modules, we identified a quadrilateral to be a closed aircraft figure bounded by four intervals, and also a convex quadrilateral to be a quadrilateral in which each interior angle is much less than 180°. We confirmed two necessary theorems about the angles of a quadrilateral:

The amount of the inner angles of a quadrilateral is 360°.The amount of the exterior angles of a convex quadrilateral is 360°.To prove the initially result, we constructed in each case a diagonal that lies entirely inside the quadrilateral. This split the quadrilateral into two triangles, each of whose angle sum is 180°.

To prove the second outcome, we produced one side at each vertex of the convex quadrilateral. The sum of the 4 straight angles is 720° and the sum of the 4 inner angles is 360°, so the sum of the four exterior angles is 360°.

Parallelograms

We begin via parallelograms, because we will certainly be using the results about parallelograms once stating the various other figures.

Definition of a parallelogram

A parallelogram is a quadrilateral whose oppowebsite sides are parallel. Thus the quadrilateral ABCD presented opposite is a parallelogram because AB || DC and DA || CB.Words ‘parallelogram’ originates from Greek words meaning ‘parallel lines’.

Constructing a parallelogram making use of the definition

To construct a parallelogram utilizing the meaning, we deserve to use the copy-an-angle building to develop parallel lines. For example, expect that we are provided the intervals AB and ADVERTISEMENT in the diagram below. We extend ADVERTISEMENT and also AB and copy the angle at A to equivalent angles at B and also D to determine C and complete the parallelogram ABCD. (See the module, Construction.)

This is not the simplest way to construct a parallelogram.

First property of a parallelogram − The opposite angles are equal

The 3 properties of a parallelogram developed listed below worry first, the inner angles, secondly, the sides, and thirdly the diagonals. The initially residential property is the majority of quickly proven making use of angle-chasing, yet it deserve to likewise be prrange utilizing congruence.

Theorem

The opposite angles of a parallelogram are equal.Proof

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

Prove that C = α and also D = β. | ||||||

α + β | = 180° | (co-internal angles, AD || BC), | ||||

so | C | = α | (co-internal angles, AB || DC) | |||

and | D | = β | (co-internal angles, AB || DC). |

2nd residential or commercial property of a parallelogram − The oppowebsite sides are equal

As an example, this proof has actually been set out in full, via the congruence test totally developed. Most of the continuing to be proofs but, are presented as exercises, through an abbreviated variation provided as an answer.

Theorem

The oppowebsite sides of a parallelogram are equal.Proof

ABCD is a parallelogram. | ||||

To prove that AB = CD and also ADVERTISEMENT = BC. | ||||

Join the diagonal AC. | ||||

In the triangles ABC and CDA: | ||||

BAC | = DCA | (alternative angles, AB || DC) | ||

BCA | = DAC | (different angles, ADVERTISEMENT || BC) | ||

AC | = CA | (common) | ||

so ABC ≡ CDA (AAS) | ||||

Hence AB = CD and BC = ADVERTISEMENT (matching sides of congruent triangles). |

Third property of a parallelogram − The diagonals bisect each other

Theorem

The diagonals of a parallelogram bisect each various other.

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EXERCISE 1

a Prove that ABM ≡ CDM.

b Hence prove that the diagonals bisect each other.

As a consequence of this residential property, the interarea of the diagonals is the centre of 2 concentric circles, one through each pair of oppowebsite vertices.

Notice that, in basic, a parallelogram does not have a circumcircle via all 4 vertices.

First test for a parallelogram − The opposite angles are equal

Besides the definition itself, tbelow are four useful tests for a parallelogram. Our first test is the converse of our first property, that the oppowebsite angles of a quadrilateral are equal.

Theorem

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

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EXERCISE 2

Prove this outcome utilizing the figure below.

2nd test for a parallelogram − Oppowebsite sides are equal

This test is the converse of the building that the oppowebsite sides of a parallelogram are equal.

Theorem

If the oppowebsite sides of a (convex) quadrilateral are equal, then the quadrilateral is a parallelogram.

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EXERCISE 3

Prove this outcome utilizing congruence in the figure to the right, wright here the diagonal AC has actually been joined.This test offers a simple building of a parallelogram provided two surrounding sides − AB and also AD in the figure to the appropriate. Draw a circle with centre B and also radius ADVERTISEMENT, and an additional circle with centre D and radius AB. The circles intersect at 2 points − let C be the suggest of intersection within the non-reflex angle BAD. Then ABCD is a parallelogram because its opposite sides are equal.It likewise gives a method of illustration the line parallel to a offered line with a given allude P. Choose any type of 2 points A and also B on , and finish the parallelogram PABQ.

Then PQ ||

Third test for a parallelogram − One pair of oppowebsite sides are equal and also parallel

This test transforms out to be exceptionally useful, because it uses just one pair of opposite sides.

Theorem

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

This test for a parallelogram gives a quick and also basic method to construct a parallelogram using a two-sided ruler. Draw a 6 cm interval on each side of the ruler. Joining up the endpoints offers a parallelogram.

The test is specifically crucial in the later concept of vectors. Suppose that and are two directed intervals that are parallel and also have actually the exact same length − that is, they recurrent the exact same vector. Then the figure ABQP to the best is a parallelogram.Even an easy vector property like the commutativity of the addition of vectors depends on this construction. The parallelogram ABQP reflects, for instance, that

+ = = +4th test for a parallelogram − The diagonals bisect each other

This test is the converse of the home that the diagonals of a parallelogram bisect each other.

Theorem

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

This test offers a really basic building and construction of a parallelogram. Draw 2 intersecting lines, then attract 2 circles with different radii centred on their intersection. Join the points wbelow alternative circles cut the lines. This is a parallelogram because the diagonals bisect each other.

It likewise enables yet another method of completing an angle BADVERTISEMENT to a parallelogram, as shown in the adhering to exercise.

EXERCISE 6

Given two intervals AB and also AD meeting at a common vertex A, construct the midpoint M of BD. Complete this to a construction of the parallelogram ABCD, justifying your answer.Parallelograms

Definition of a parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel.

Properties of a parallelogram

The oppowebsite angles of a parallelogram are equal. The opposite sides of a parallelogram are equal. The diagonals of a parallelogram bisect each other.Tests for a parallelogram

A quadrilateral is a parallelogram if:

its oppowebsite angles are equal, or its opposite sides are equal, or one pair of opposite sides are equal and parallel, or its diagonals bisect each various other.Rectangles

Words ‘rectangle’ implies ‘ideal angle’, and this is reflected in its interpretation.

Definition of a RectangleA rectangle is a quadrilateral in which all angles are right angles.

First Property of a rectangle − A rectangle is a parallelogram

Each pair of co-interior angles are supplementary, bereason two best angles include to a straight angle, so the oppowebsite sides of a rectangle are parallel. This suggests that a rectangle is a parallelogram, so:

Its opposite sides are equal and also parallel. Its diagonals bisect each other.Second property of a rectangle − The diagonals are equal

The diagonals of a rectangle have actually an additional vital building − they are equal in size. The proof has actually been collection out in full as an example, bereason the overlapping congruent triangles can be confutilizing.

Theorem

The diagonals of a rectangle are equal.Proof

Let ABCD be a rectangle.

We 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 ABC ≡ DCB (SAS)

Hence AC = DB (equivalent sides of congruent triangles).

This indicates that AM = BM = CM = DM, where M is the interarea of the diagonals. Hence we have the right to draw a single circle through centre M with all four vertices. We can define this situation by saying that, ‘The vertices of a rectangle are concyclic’.First test for a rectangle − A parallelogram via one ideal angle

If a parallelogram is recognized to have actually one ideal angle, then recurring use of co-interior angles proves that all its angles are ideal angles.

Theorem

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

Thus theorem, the definition of a rectangle is sometimes taken to be ‘a parallelogram through a right angle’.

Construction of a rectangle

We deserve to construct a rectangle with given side lengths by creating a parallelogram through a best angle on one edge. First drop a perpendicular from a allude P to a line . Mark B and also then mark off BC and BA and also complete the parallelogram as shown listed below.

2nd test for a rectangle − A quadrilateral with equal diagonals that bisect each other

We have actually presented over that the diagonals of a rectangle are equal and bisect each various other. Conversely, these 2 properties taken together constitute a test for a quadrilateral to be a rectangle.

Theorem

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

EXERCISE 8

a Why is the quadrilateral a parallelogram?

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

As a consequence of this result, the endpoints of any type of 2 diameters of a circle create a rectangle, because this quadrilateral has actually equal diagonals that bisect each other.

Thus we have the right to construct a rectangle exceptionally simply by drawing any kind of 2 intersecting lines, then drawing any circle centred at the allude of interarea. The quadrilateral formed by joining the 4 points wbelow the circle cuts the lines is a rectangle bereason it has actually equal diagonals that bisect each various other.

Rectangles

Definition of a rectangle

A rectangle is a quadrilateral in which all angles are best angles.

Properties of a rectangle

A rectangle is a parallelogram, so its oppowebsite sides are equal. The diagonals of a rectangle are equal and also bisect each various other.Tests for a rectangle

A parallelogram via one best angle is a rectangle. A quadrilateral whose diagonals are equal and bisect each other is a rectangle.Links forward

The continuing to be unique quadrilaterals to be treated by the congruence and angle-chasing techniques of this module are rhombprovides, kites, squares and also trapezia. The sequence of theorems connected in dealing with all these unique quadrilaterals at when becomes quite facility, so their conversation will be left until the module Rhombuses, Kites, and Trapezia. Each individual proof, but, is well within Year 8 capacity, offered that students have the ideal experiences. In specific, it would be useful to prove in Year 8 that the diagonals of rhombsupplies and also kites accomplish at best angles − this result is needed in area formulas, it is helpful in applications of Pythagoras’ theorem, and it provides a more lifwynnfoundation.organized explanation of numerous important constructions.

The next step in the development of geomeattempt is a rigorous treatment of similarity. This will enable various results around ratios of lengths to be established, and likewise make possible the meaning of the trigonometric ratios. Similarity is required for the geomeattempt of circles, wbelow one more course of distinct quadrilaterals arises, namely the cyclic quadrilaterals, whose vertices lie on a circle.

Special quadrilaterals and also their properties are necessary to create the typical formulas for locations and also volumes of figures. Later, these outcomes will be necessary in developing integration. Theorems about special quadrilaterals will be commonly supplied in coordinate geomeattempt.

Rectangles are so common that they go unnoticed in the majority of applications. One distinct duty worth noting is they are the basis of the coordinates of points in the cartesian aircraft − to uncover the coordinates of a point in the aircraft, we finish the rectangle formed by the suggest and also the 2 axes. Parallelograms aclimb as soon as we add vectors by completing the parallelogram − this is the factor why they become so necessary when facility numbers are represented on the Argand also diagram.

History and applications

Rectangles have been advantageous for as lengthy as there have been structures, bereason vertical pillars and horizontal crossbeams are the many evident way to construct a structure of any type of size, providing a structure in the form of a rectangular prism, all of whose faces are rectangles. The diagonals that we constantly use to study rectangles have actually an analogy in building − a rectangular frame with a diagonal has actually far more rigidity than a straightforward rectangular frame, and also diagonal struts have constantly been used by contractors to give their building even more toughness.

Parallelograms are not as common in the physical people (except as shadows of rectangular objects). Their major function historically has actually remained in the depiction of physical ideas by vectors. For instance, when 2 pressures are linked, a parallelogram can be attracted to help compute the dimension and direction of the combined force. When tbelow are 3 pressures, we complete the parallelepiped, which is the three-dimensional analogue of the parallelogram.

REFERENCES

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

History of Mathematics, D. E. Smith, Dover publications New York, (1958)

ANSWERS TO EXERCISES

EXERCISE 1

a In the triangles ABM and CDM :

1. | BAM | = DCM | (alternative angles, AB || DC ) | |||

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

3. | AB | = CD | (opposite sides of parallelogram ABCD) | |||

ABM = CDM (AAS) |

b Hence AM = CM and DM = BM (corresponding 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 are supplementary) | ||

and | ADVERTISEMENT || BC | (co-internal angles are supplementary). |

EXERCISE 3

First show that ABC ≡ CDA making use of the SSS congruence test. | ||||

Hence | ACB = CAD and also CAB = ACD | (corresponding angles of congruent triangles) | ||

so | ADVERTISEMENT || BC and AB || DC | (alternate angles are equal.) |

EXERCISE 4

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

Hence | ADB = CBD | (equivalent angles of congruent triangles) | ||

so | ADVERTISEMENT || BC | (alternative angles are equal.) |

EXERCISE 5

First prove that ABM ≡ CDM utilizing the SAS congruence test. | ||||

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

Also | ABM = CDM | (corresponding angles of congruent triangles) | ||

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

Hence ABCD is a parallelogram, because one pair of oppowebsite sides are equal and parallel.

EXERCISE 6

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

EXERCISE 7

The square on each diagonal is the sum of the squares on any two surrounding sides. Because oppowebsite sides are equal in size, the squares on both diagonals are the very same.

EXERCISE 8

a | We have actually already proven that a quadrilateral whose diagonals bisect each various other is a parallelogram. |

b | Due to the fact that ABCD is a parallelogram, its oppowebsite sides are equal. | ||||

Hence | ABC ≡ DCB | (SSS) | |||

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

But | ABC + DCB = 180o | (co-inner angles, AB || DC ) | |||

so | ABC = DCB = 90o . |

Hence ABCD is rectangle, because it is a parallelogram via one ideal angle.

EXERCISE 9

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

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

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

α + β | = 90o. |

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

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and also Worklocation Relations.

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The views expressed here are those of the writer and execute not necessarily reexisting the views of the Australian Government Department of Education, Employment and also Worklocation Relations.

© The University of Melbourne on behalf of the Internationwide Centre of Excellence for Education in Mathematics (ICE-EM), the education and learning division of the Australian Mathematical Sciences Institute (lifwynnfoundation.lifwynnfoundation.org), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. https://creativecommons.lifwynnfoundation.org/licenses/by-nc-nd/3.0/