Parallelograms and also Rectangles

Measurement and Geometry : Module 20Years : 8-9

June 2011

PDF variation of moduleAssumed knowledge

Introductory aircraft geometry including points and lines, parallel lines and transversals, edge sums that triangles and quadrilaterals, and also general angle-chasing.The four standard congruence tests and also their application in problems and proofs.Properties of isosceles and also equilateral triangles and tests for them.Experience through a logical discussion in geometry being created as a sequence of steps, every justified by a reason.Ruler-and-compasses constructions.Informal endure with distinct quadrilaterals.You are watching: Prove that the diagonals of a parallelogram bisect each other

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Motivation

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

Apart from cyclic quadrilaterals, these one-of-a-kind quadrilaterals and their properties have been presented informally over several years, but without congruence, a rigorous discussion of lock was not possible. Every congruence proof uses the diagonals to division the quadrilateral right into triangles, ~ which we can use the approaches of congruent triangles emerged in the module, Congruence.

The present treatment has 4 purposes:

The parallelogram and also rectangle are carefully defined.Their far-reaching properties are proven, mainly using congruence.Tests for them are established that can be used to inspect that a offered quadrilateral is a parallel or rectangle − again, congruence is mostly required.Some ruler-and-compasses build of them are occurred as basic applications of the definitions and also tests.The material in this module is an ideal for Year 8 as further applications of congruence and constructions. Due to the fact that of its organized development, that provides terrific introduction to proof, converse statements, and sequences of theorems. Significant guidance in such concepts is normally required in Year 8, i beg your pardon is consolidated through further discussion in later years.

The complementary principles of a ‘property’ the a figure, and a ‘test’ because that a figure, become specifically important in this module. Indeed, clarity about these concepts is among the many reasons for teaching this product at school. Many of the tests the we meet are converses the properties the have already been proven. For example, the truth that the base angles of one isosceles triangle are equal is a building of isosceles triangles. This property have the right to be re-formulated as an ‘If …, then … ’ statement:

If two sides of a triangle are equal, climate the angle opposite those sides room equal.Now the matching test because that a triangle to be isosceles is plainly the converse statement:

If 2 angles the a triangle space equal, then the political parties opposite those angles are equal.Remember that a statement might be true, yet its converse false. It is true the ‘If a number is a lot of of 4, climate it is even’, yet it is false that ‘If a number is even, then it is a multiple of 4’.

Quadrilaterals

In other modules, we defined a quadrilateral to it is in a closed airplane figure bounded by 4 intervals, and a convex quadrilateral to it is in a quadrilateral in i beg your pardon each inner angle is less than 180°. We confirmed two essential theorems about the angles of a quadrilateral:

The sum of the interior angles that a square is 360°.The sum of the exterior angles of a convex square is 360°.To prove the an initial result, we created in each situation a diagonal that lies completely inside the quadrilateral. This separated the quadrilateral into two triangles, every of whose angle sum is 180°.

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

Parallelograms

We start with parallelograms, due to the fact that we will be utilizing the results around parallelograms when pointing out the various other figures.

Definition that a parallelogram

A parallel is a square whose opposite sides space parallel. Thus the square ABCD displayed opposite is a parallel because ab || DC and DA || CB.The native ‘parallelogram’ originates from Greek words an interpretation ‘parallel lines’.

Constructing a parallelogram using the definition

To build a parallelogram making use of the definition, we deserve to use the copy-an-angle construction to type parallel lines. For example, mean that us are provided the intervals ab and advertisement in the chart below. We extend ad and abdominal and copy the edge at A to corresponding angles in ~ B and D to recognize C and complete the parallelogram ABCD. (See the module, Construction.)

This is no the easiest method to build a parallelogram.

First property of a parallel − opposing angles space equal

The three properties the a parallelogram occurred below issue 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 also be proven making use of congruence.

Theorem

The opposite angle of a parallelogram space equal.Proof

Let ABCD it is in a parallelogram, with A = α and B = β. | ||||||

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

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

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

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

Second residential or commercial property of a parallel − opposing sides room equal

As one example, this proof has actually been set out in full, v the congruence test fully developed. Many of the staying proofs however, are presented as exercises, with an abbreviation version offered as an answer.

Theorem

The opposite sides of a parallelogram space 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, ab || DC) | ||

BCA | = DAC | (alternate angles, advertisement || BC) | ||

AC | = CA | (common) | ||

so alphabet ≡ CDA (AAS) | ||||

Hence abdominal = CD and BC = ad (matching political parties of congruent triangles). |

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

Theorem

The diagonals the a parallel bisect every other.

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EXERCISE 1

a Prove that ABM ≡ CDM.

b thus prove the the diagonals bisect every other.

As a consequence of this property, the intersection that the diagonals is the centre of 2 concentric circles, one v 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 − the opposite angles are equal

Besides the an interpretation itself, there room four advantageous tests for a parallelogram. Our very first test is the converse of our an initial property, the the opposite angle 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.

Second test because that a parallel − opposite sides are equal

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

Theorem

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

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EXERCISE 3

Prove this an outcome using congruence in the number to the right, whereby the diagonal AC has been joined.This test provides a an easy construction that a parallelogram offered two nearby sides − ab and ad in the number to the right. Draw a circle through centre B and radius AD, and another circle with centre D and radius AB. The circles intersect at 2 points − permit C be the point of intersection in ~ the non-reflex angle BAD. Climate ABCD is a parallelogram since its the opposite sides space equal.It additionally gives a technique of illustration the line parallel to a offered line with a given suggest P. Choose any kind of two clues A and also B ~ above , and also complete the parallel PABQ.

Then PQ ||

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

This test turns out to be very useful, since it supplies only one pair of opposite sides.

Theorem

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

This test because that a parallelogram provides a quick and also easy method to build a parallelogram making use of a two-sided ruler. Draw a 6 cm interval on each side the the ruler. Joining increase the endpoints offers a parallelogram.

The check is particularly important in the later theory of vectors. Suppose that and also are two directed intervals that room parallel and have the same size − that is, they represent the very same vector. Then the number ABQP to the ideal is a parallelogram.Even a an easy vector property prefer the commutativity that the enhancement of vectors counts on this construction. The parallel ABQP shows, because that example, that

+ = = +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 every other.

Theorem

If the diagonals of a square bisect every other, then the square is a parallelogram:

This test offers a very simple construction of a parallelogram. Draw two intersecting lines, then draw two circles with various radii centred on their intersection. Sign up with the points where alternate circles reduced the lines. This is a parallelogram because the diagonals bisect every other.

It also allows yet another method of perfect an angle poor to a parallelogram, as shown in the following exercise.

EXERCISE 6

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

Definition of a parallelogram

A parallel is a square whose the opposite sides space parallel.

Properties the a parallelogram

The opposite angle of a parallelogram room equal. The opposite sides of a parallelogram space equal. The diagonals of a parallelogram bisect every other.Tests because that a parallelogram

A square is a parallel if:

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

The native ‘rectangle’ method ‘right angle’, and this is reflected in its definition.

Definition that a RectangleA rectangle is a square in which all angles are right angles.

First home of a rectangle − A rectangle is a parallelogram

Each pair the co-interior angles are supplementary, due to the fact that two best angles include to a directly angle, therefore the opposite sides of a rectangle space parallel. This way that a rectangle is a parallelogram, so:

Its the contrary sides room equal and also parallel. That is diagonals bisect every other.Second residential or commercial property of a rectangle − The diagonals are equal

The diagonals of a rectangle have one more important residential property − they are equal in length. The proof has been collection out in full as an example, due to the fact that the overlapping congruent triangles deserve to be confusing.

Theorem

The diagonals that a rectangle space equal.Proof

permit ABCD it is in a rectangle.

we prove that 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)

hence AC = DB (matching political parties of congruent triangles).

This method that to be = BM = centimeter = DM, whereby M is the intersection of the diagonals. Thus we can draw a single circle v centre M through all four vertices. Us can describe this instance by speak that, ‘The vertices of a rectangle space concyclic’.First test because that a rectangle − A parallelogram through one best angle

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

Theorem

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

Because the this theorem, the definition of a rectangle is periodically taken to be ‘a parallelogram v a best angle’.

Construction that a rectangle

We have the right to construct a rectangle with provided side lengths by building a parallelogram with a appropriate angle ~ above one corner. First drop a perpendicular native a point P to a line . Mark B and then note off BC and also BA and complete the parallelogram as displayed below.

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

We have actually shown over that the diagonals that a rectangle are equal and bisect every other. Conversely, these two properties taken with each other constitute a test for a quadrilateral to be a rectangle.

Theorem

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

EXERCISE 8

a Why is the square a parallelogram?

b use congruence come prove that the number is a rectangle.

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

Thus we have the right to construct a rectangle very simply through drawing any type of two intersecting lines, climate drawing any type of circle centred at the allude of intersection. The quadrilateral formed by joining the four points whereby the circle cuts the present is a rectangle because it has equal diagonals that bisect every other.

Rectangles

Definition of a rectangle

A rectangle is a square in i m sorry all angles are best angles.

Properties of a rectangle

A rectangle is a parallelogram, for this reason its opposite sides space equal. The diagonals that a rectangle are equal and also bisect each other.Tests because that a rectangle

A parallelogram v one best angle is a rectangle. A square whose diagonals are equal and bisect each various other is a rectangle.Links forward

The continuing to be special quadrilaterals to be treated by the congruence and also angle-chasing methods of this module space rhombuses, kites, squares and trapezia. The succession of theorems connected in dealing with all these unique quadrilaterals at as soon as becomes fairly complicated, therefore their discussion will be left till the module Rhombuses, Kites, and Trapezia. Each individual proof, however, is fine within Year 8 ability, noted that students have actually the ideal experiences. In particular, it would certainly be valuable to prove in Year 8 the the diagonals that rhombuses and kites accomplish at appropriate angles − this an outcome is necessary in area formulas, the is advantageous in applications the Pythagoras’ theorem, and also it provides a much more systematic explanation that several necessary constructions.

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

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

Rectangles space so common that they walk unnoticed in most applications. One special function worth noting is they room the communication of the works with of points in the cartesian airplane − to find the works with of a suggest in the plane, we complete the rectangle created by the point and the two axes. Parallelograms arise when we include vectors by perfect the parallelogram − this is the factor why they come to be so necessary when facility numbers are represented on the Argand diagram.

History and also applications

Rectangles have been valuable for as long as there have actually been buildings, because vertical pillars and also horizontal crossbeams space the most obvious means to construct a building of any type of size, providing a structure in the form of a rectangle-shaped prism, all of whose faces are rectangles. The diagonals that us constantly usage to study rectangles have an analogy in structure − a rectangular framework with a diagonal has actually far an ext rigidity than a simple rectangular frame, and diagonal struts have constantly been used by building contractors to offer their building more strength.

Parallelograms are not as usual in the physical human being (except as shadows of rectangle-shaped objects). Their major role historically has remained in the depiction of physical ideas by vectors. For example, when two pressures are combined, a parallelogram have the right to be drawn to aid compute the size and direction of the an unified force. Once there room three forces, we complete the parallelepiped, i beg your pardon is the three-dimensional analogue the the parallelogram.

REFERENCES

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

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

ANSWERS to EXERCISES

EXERCISE 1

a In the triangles ABM and CDM :

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

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

3. | AB | = CD | (opposite sides 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 square ABCD) | ||

α + β | = 180o |

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

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

EXERCISE 3

First display that abc ≡ CDA making use of the SSS congruence test. | ||||

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

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

EXERCISE 4

First prove that ABD ≡ CDB making use of the SAS congruence test. | ||||

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

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

EXERCISE 5

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

Hence | AB = CD | (matching political parties of congruent triangles) | ||

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

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

Hence ABCD is a parallelogram, since one pair of the contrary sides are equal and also parallel.

EXERCISE 6

Join AM. Through centre M, attract an arc v radius AM that meets AM created at C . Then ABCD is a parallelogram because its diagonals bisect each other.

EXERCISE 7

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

EXERCISE 8

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

b | Because ABCD is a parallelogram, its opposite sides room equal. | ||||

Hence | ABC ≡ DCB | (SSS) | |||

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

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

so | ABC = DCB = 90o . |

thus ABCD is rectangle, since it is a parallelogram v one right angle.

EXERCISE 9

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

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

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

α + β | = 90o. |

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

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