Parallelograms and Rectangles

Measurement and Geometry : Module 20Years : 8-9

June 2011

PDF variation of moduleAssumed knowledge

Introductory airplane geometry entailing points and also lines, parallel lines and transversals, edge sums the triangles and quadrilaterals, and also general angle-chasing.The 4 standard congruence tests and also their applications in problems and also proofs.Properties of isosceles and also equilateral triangles and tests for them.Experience v a logical discussion in geometry being composed as a succession of steps, each justified through a reason.Ruler-and-compasses constructions.Informal endure with one-of-a-kind quadrilaterals.You are watching: Diagonals of a parallelogram bisect each other

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Motivation

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

Apart indigenous cyclic quadrilaterals, these special quadrilaterals and their properties have been introduced informally over numerous years, yet without congruence, a rigorous conversation of castle was not possible. Every congruence proof provides the diagonals to divide the quadrilateral into triangles, ~ which we can apply the methods of congruent triangles occurred in the module, Congruence.

The current treatment has 4 purposes:

The parallelogram and also rectangle are carefully defined.Their far-ranging properties space proven, mainly using congruence.Tests because that them are developed that deserve to be used to check that a provided quadrilateral is a parallelogram or rectangle − again, congruence is mostly required.Some ruler-and-compasses build of castle are emerged as straightforward applications that the definitions and tests.The product in this module is an ideal for Year 8 as further applications the congruence and constructions. Due to the fact that of its systematic development, that provides great introduction come proof, converse statements, and sequences the theorems. Substantial guidance in such principles is normally required in Year 8, which is consolidated through further discussion in later years.

The complementary ideas of a ‘property’ the a figure, and a ‘test’ because that a figure, become particularly important in this module. Indeed, clarity about these principles is among the numerous reasons for teaching this material at school. Many of the tests that we meet are converses that properties that have currently been proven. Because that example, the truth that the base angle of an isosceles triangle room equal is a residential or commercial property of isosceles triangles. This property have the right to be re-formulated as an ‘If …, then … ’ statement:

If 2 sides the a triangle are equal, then the angles opposite those sides are equal.Now the equivalent test for a triangle to it is in isosceles is plainly the converse statement:

If 2 angles the a triangle space equal, climate the sides opposite those angles space equal.Remember that a statement may be true, however its converse false. That is true that ‘If a number is a multiple of 4, climate it is even’, however it is false the ‘If a number is even, climate it is a multiple of 4’.

Quadrilaterals

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

The amount of the inner angles of a square is 360°.The amount of the exterior angle of a convex square is 360°.To prove the very first result, we built in each situation a diagonal that lies completely inside the quadrilateral. This split the quadrilateral right into two triangles, each of whose angle sum is 180°.

To prove the second result, we produced one next at each vertex of the convex quadrilateral. The sum of the 4 straight angles is 720° and also the sum of the 4 interior angles is 360°, for this reason the sum of the 4 exterior angle is 360°.

Parallelograms

We begin with parallelograms, since we will certainly be utilizing the results about parallelograms when discussing the various other figures.

Definition the a parallelogram

A parallel is a square whose opposite sides room parallel. Hence the square ABCD presented opposite is a parallel because abdominal || DC and also DA || CB.The word ‘parallelogram’ comes from Greek words meaning ‘parallel lines’.

Constructing a parallelogram utilizing the definition

To build a parallelogram utilizing the definition, we deserve to use the copy-an-angle construction to form parallel lines. Because that example, mean that us are offered the intervals ab and advertisement in the diagram below. Us extend ad and ab and copy the edge at A to corresponding angles at B and D to identify C and complete the parallel ABCD. (See the module, Construction.)

This is no the easiest way to construct a parallelogram.

First residential property of a parallelogram − opposing angles are equal

The three properties of a parallelogram arisen below worry first, the interior angles, secondly, the sides, and also thirdly the diagonals. The very first property is most conveniently proven making use of angle-chasing, yet it can likewise be proven using congruence.

Theorem

The opposite angle of a parallelogram space equal.Proof

Let ABCD be a parallelogram, through 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 || DC) | |||

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

Second residential property of a parallelogram − opposing 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, room presented together exercises, v an abbreviated version provided as one answer.

Theorem

The opposite political parties of a parallelogram are equal.Proof

ABCD is a parallelogram. | ||||

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

Join the diagonal line AC. | ||||

In the triangles 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 ab = CD and also BC = advertisement (matching sides of congruent triangles). |

Third property of a parallel − The diagonals bisect every other

Theorem

The diagonals the a parallelogram bisect every other.

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EXERCISE 1

a Prove the ABM ≡ CDM.

b therefore prove the the diagonals bisect every other.

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

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

First test for a parallelogram − opposing angles are equal

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

Theorem

If the opposite angles of a quadrilateral room equal, climate the quadrilateral 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 space equal

This test is the converse of the home that the opposite sides of a parallelogram room equal.

Theorem

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

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EXERCISE 3

Prove this an outcome using congruence in the number to the right, where the diagonal AC has actually been joined.This test offers a simple construction the a parallelogram offered two nearby sides − abdominal muscle and advertisement in the figure to the right. Draw a circle through centre B and radius AD, and another circle through centre D and radius AB. The circles crossing at two points − allow C be the allude of intersection in ~ the non-reflex angle BAD. Then ABCD is a parallelogram because its opposite sides room equal.It also gives a method of illustration the line parallel come a given line with a given point P. Choose any type of two clues A and also B top top , and complete the parallel PABQ.

Then PQ ||

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

This test turns out to be very useful, because it offers only one pair of the opposite sides.

Theorem

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

This test because that a parallelogram provides a quick and easy means to build a parallelogram making use of a two-sided ruler. Draw a 6 centimeter interval on every side that the ruler. Joining up the endpoints provides a parallelogram.

The test is specifically important in the later on theory that vectors. Mean that and space two directed intervals that room parallel and also have the same length − that is, they stand for the exact same vector. Climate the figure ABQP come the best is a parallelogram.Even a basic vector property favor the commutativity that the addition of vectors depends on this construction. The parallelogram ABQP shows, because that example, that

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

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

Theorem

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

This test provides a very simple construction the a parallelogram. Draw two intersecting lines, then attract two one with different radii centred on your intersection. Join the clues where alternating circles reduced the lines. This is a parallelogram due to the fact that the diagonals bisect each other.

It also allows yet another technique of perfect an angle poor to a parallelogram, as presented in the complying with exercise.

EXERCISE 6

Given 2 intervals ab and ad meeting at a typical vertex A, construct the midpoint M of BD. Complete this to a construction of the parallel ABCD, justifying her answer.Parallelograms

Definition of a parallelogram

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

Properties the a parallelogram

The opposite angles of a parallelogram space equal. The opposite political parties of a parallelogram room equal. The diagonals that 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 are equal, or one pair of the opposite sides room equal and also parallel, or that is diagonals bisect every other.Rectangles

The native ‘rectangle’ way ‘right angle’, and this is reflect in that is definition.

Definition that a RectangleA rectangle is a quadrilateral in i beg your pardon all angle are ideal angles.

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

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

Its opposite sides room equal and also parallel. The diagonals bisect every other.Second building of a rectangle − The diagonals space equal

The diagonals that a rectangle have one more important residential or commercial property − they room equal in length. The proof has actually been collection out in full as one example, because the overlapping congruent triangles deserve to be confusing.

Theorem

The diagonals the a rectangle space equal.Proof

permit ABCD be a rectangle.

us prove the AC = BD.

In the triangle ABC and also DCB:

BC | = CB | (common) | ||

AB | = DC | (opposite sides of a parallelogram) | ||

ABC | =DCA = 90° | (given) |

so alphabet ≡ DCB (SAS)

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

This way that am = BM = cm = DM, whereby M is the intersection the the diagonals. Therefore we can draw a solitary circle through centre M v all 4 vertices. We can explain this instance by saying that, ‘The vertices that a rectangle are concyclic’.First test for a rectangle − A parallelogram v one ideal angle

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

Theorem

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

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

Construction the a rectangle

We can construct a rectangle with offered side lengths by constructing a parallelogram v a appropriate angle ~ above one corner. First drop a perpendicular from a allude P come a line . Mark B and also then note off BC and also BA and complete the parallel as shown below.

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

We have actually shown above that the diagonals that a rectangle room equal and bisect every other. Vice versa, these two properties taken together constitute a test because that a square to be a rectangle.

Theorem

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

EXERCISE 8

a Why is the quadrilateral a parallelogram?

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

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

Thus we have the right to construct a rectangle an extremely simply by drawing any kind of two intersecting lines, climate drawing any type of circle centred at the allude of intersection. The quadrilateral created by joining the 4 points whereby the circle cut the present is a rectangle because it has equal diagonals that bisect each other.

Rectangles

Definition of a rectangle

A rectangle is a quadrilateral in which 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 space equal and bisect every other.Tests for a rectangle

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

The staying special quadrilaterals come be cure by the congruence and angle-chasing techniques of this module are rhombuses, kites, squares and trapezia. The succession of theorems associated in treating all these special quadrilaterals at as soon as becomes rather complicated, so their conversation will be left till the module Rhombuses, Kites, and also Trapezia. Each individual proof, however, is fine within Year 8 ability, noted that students have the right experiences. In particular, it would be useful to prove in Year 8 the the diagonals the rhombuses and kites fulfill at right angles − this result is essential in area formulas, the is valuable in applications of Pythagoras’ theorem, and also it offers a an ext systematic explanation that several necessary constructions.

The next step in the breakthrough of geometry is a rigorous treatment of similarity. This will permit various results around ratios that lengths to be established, and likewise make feasible the an interpretation of the trigonometric ratios. Similarity is compelled for the geometry that circles, where another class of special quadrilaterals arises, namely the cyclic quadrilaterals, who vertices lied on a circle.

Special quadrilaterals and also their properties are needed to establish the standard formulas because that areas and volumes that figures. Later, these outcomes will be essential in arising integration. Theorems around special quadrilaterals will certainly be widely used in coordinate geometry.

Rectangles room so ubiquitous that they walk unnoticed in most applications. One special role worth noting is they room the basis of the collaborates of points in the cartesian plane − to discover the coordinates of a point in the plane, we complete the rectangle created by the allude and the 2 axes. Parallelograms arise when we add vectors by completing the parallel − this is the reason why they come to be so essential when complex numbers are represented on the Argand diagram.

History and applications

Rectangles have been advantageous for as long as there have been buildings, because vertical pillars and also horizontal crossbeams room the most obvious method to build a structure of any size, providing a structure in the form of a rectangular prism, all of whose encounters are rectangles. The diagonals that us constantly usage to examine rectangles have an analogy in building − a rectangular structure with a diagonal has far much more rigidity than a an easy rectangular frame, and diagonal struts have constantly been offered by contractors to provide their building an ext strength.

Parallelograms are not as usual in the physical human being (except together shadows of rectangular objects). Their significant role historically has been in the depiction of physical concepts by vectors. For example, once two forces are combined, a parallelogram can be attracted to aid compute the size and direction that the an unified force. As soon as there space three forces, we complete the parallelepiped, i m sorry is the three-dimensional analogue the the parallelogram.

REFERENCES

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

History that 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 sides of parallel ABCD) | |||

ABM = CDM (AAS) |

b hence 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 are supplementary) | ||

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

EXERCISE 3

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

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

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

EXERCISE 4

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

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

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

EXERCISE 5

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

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

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

so | AB || DC | (alternate angles space 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. Through centre M, attract an arc with radius AM that meets AM developed at C . Climate ABCD is a parallelogram since 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. Due to the fact that opposite sides room 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 the contrary sides space equal. | ||||

Hence | ABC ≡ DCB | (SSS) | |||

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

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

so | ABC = DCB = 90o . |

for this reason ABCD is rectangle, since it is a parallelogram with one best angle.

EXERCISE 9

ADM | = α | (base angle 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 ideal angle, and similarly, B, C and D are appropriate angles.

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