Parallelograms and also Rectangles

Measurement and Geometry : Module 20Years : 8-9

June 2011

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PDF variation of module


Assumed knowledge

Introductory airplane geometry entailing points and lines, parallel lines and transversals, angle sums of triangles and also quadrilaterals, and 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 tests because that them.Experience with a logical discussion in geometry being created as a succession of steps, each justified by a reason.Ruler-and-compasses constructions.Informal endure with unique quadrilaterals.

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Motivation

There are only three vital categories of special triangles − isosceles triangles, equilateral triangles and also right-angled triangles. In contrast, there are numerous categories of one-of-a-kind quadrilaterals. This module will deal with two of lock − parallelograms and rectangles − leaving rhombuses, kites, squares, trapezia and also cyclic quadrilaterals come the module, Rhombuses, Kites, and also Trapezia.

Apart indigenous cyclic quadrilaterals, these distinct quadrilaterals and their properties have been presented informally over several years, but without congruence, a rigorous discussion of lock was no possible. Every congruence proof offers the diagonals to divide the quadrilateral right into triangles, after which us can apply the methods of congruent triangles arisen in the module, Congruence.

The existing treatment has four purposes:

The parallelogram and also rectangle are carefully defined.Their far-reaching properties are proven, largely using congruence.Tests for them are created that can be provided to examine that a offered quadrilateral is a parallel or rectangle − again, congruence is mainly required.Some ruler-and-compasses build of them are developed as basic applications the the definitions and tests.

The product in this module is an ideal for Year 8 as more applications the congruence and constructions. Due to the fact that of its methodical development, it provides terrific introduction come proof, converse statements, and sequences the theorems. Substantial guidance in such concepts is normally required in Year 8, which is consolidated by further discussion in later on years.

The complementary principles of a ‘property’ that a figure, and also a ‘test’ because that a figure, become an especially important in this module. Indeed, clarity about these principles is one of the plenty of reasons for to teach this product at school. Many of the tests that we accomplish are converses that properties that have already been proven. For example, the reality that the base angles of one isosceles triangle space equal is a home of isosceles triangles. This property have the right to be re-formulated together an ‘If …, climate … ’ statement:

If 2 sides that a triangle are equal, climate the angle opposite those sides room equal.

Now the equivalent test because that a triangle to it is in isosceles is clearly the converse statement:

If 2 angles of a triangle room equal, climate the sides opposite those angles are equal.

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


Quadrilaterals

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In various other modules, we identified a square to be a closed airplane figure bounded by 4 intervals, and a convex square to it is in a quadrilateral in i m sorry each internal angle is much less than 180°. We proved two crucial theorems around the angle of a quadrilateral:

The sum of the interior angles of a quadrilateral 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 right into two triangles, each of who angle sum is 180°.

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


Parallelograms

We start with parallelograms, since we will be making use of the results around parallelograms when stating the various other figures.

Definition that a parallelogram

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A parallelogram is a square whose the contrary sides room parallel. Thus the quadrilateral ABCD shown opposite is a parallelogram because ab || DC and also DA || CB.

The word ‘parallelogram’ comes from Greek words an interpretation ‘parallel lines’.

Constructing a parallelogram making use of the definition

To build a parallelogram using the definition, we can use the copy-an-angle building to type parallel lines. Because that example, expect that us are provided the intervals abdominal muscle and ad in the diagram below. Us extend ad and abdominal muscle and copy the angle at A to matching angles at B and also D to identify C and complete the parallelogram ABCD. (See the module, Construction.)

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This is not the easiest way to construct a parallelogram.

First residential or commercial property of a parallel − the contrary angles room equal

The three properties the a parallelogram occurred below worry first, the inner angles, secondly, the sides, and thirdly the diagonals. The very first property is most quickly proven using angle-chasing, yet it can additionally be proven using congruence.

Theorem

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The opposite angle of a parallelogram room equal.

Proof

Let ABCD be a parallelogram, v 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, abdominal muscle || DC)
and D = β(co-interior angles, ab || DC).

Second residential or commercial property of a parallelogram − the opposite sides room equal

As one example, this proof has actually been collection out in full, v the congruence test completely developed. Many of the remaining proofs however, are presented together exercises, v an abbreviation version provided as one answer.

Theorem

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The opposite political parties of a parallelogram are equal.

Proof

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

Third residential or commercial property 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 hence prove that the diagonals bisect every other.


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

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

First test because that a parallel − the opposite angles space equal

Besides the an interpretation itself, there are four valuable tests because that a parallelogram. Our first test is the converse that our an initial property, that the opposite angles of a quadrilateral room equal.

Theorem

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


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EXERCISE 2

Prove this result using the number below.

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

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

Theorem

If the opposite political parties of a (convex) quadrilateral room 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 number to the right, wherein the diagonal line AC has been joined.


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This test provides a basic construction that a parallelogram offered two nearby sides − abdominal and ad in the figure to the right. Draw a circle with centre B and radius AD, and another circle through centre D and radius AB. The circles crossing at two points − let C it is in the point of intersection within the non-reflex angle BAD. Then ABCD is a parallelogram due to the fact that its the opposite sides room equal.

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

Then PQ ||

Third test for a parallelogram − One pair of opposite sides room equal and also parallel

This test transforms out come be very useful, because it offers only one pair of the contrary sides.

Theorem

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


This test because that a parallelogram gives a quick and easy way to build a parallelogram making use of a two-sided ruler. Draw a 6 centimeter interval on each side that the ruler. Joining increase the endpoints gives a parallelogram.

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The test is an especially important in the later theory of vectors. Expect that
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and also
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are two command intervals that space parallel and have the same size − the is, they represent the very same vector. Climate the figure ABQP to the ideal is a parallelogram.

Even a an easy vector property choose the commutativity that the addition of vectors depends on this construction. The parallelogram ABQP shows, for example, that

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+
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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 every other

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

Theorem

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


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This test provides a very an easy construction of a parallelogram. Attract two intersecting lines, then draw two circles with different radii centred on their intersection. Join the point out where alternative circles cut the lines. This is a parallelogram due to the fact that the diagonals bisect each other.

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


EXERCISE 6

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Given 2 intervals ab and ad meeting at a usual vertex A, build the midpoint M of BD. Complete this come a building and construction of the parallel ABCD, justifying your answer.


Parallelograms

Definition the a parallelogram

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

Properties that a parallelogram

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

Rectangles

The native ‘rectangle’ means ‘right angle’, and this is reflect in its definition.

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Definition of a Rectangle

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

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

Each pair the co-interior angles room supplementary, due to the fact that two appropriate angles include to a directly angle, so the opposite sides of a rectangle room parallel. This means that a rectangle is a parallelogram, so:

Its the opposite sides space equal and parallel. That diagonals bisect every other.

Second building of a rectangle − The diagonals are equal

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

Theorem

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The diagonals the a rectangle are equal.

Proof

permit ABCD be a rectangle.

we prove the 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)

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

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This means that to be = BM = centimeter = DM, where M is the intersection of the diagonals. Hence we can attract a solitary circle through centre M through all 4 vertices. We can describe this situation by saying that, ‘The vertices the a rectangle room concyclic’.


First test for a rectangle − A parallelogram with one ideal angle

If a parallelogram is well-known to have one right angle, then repetitive use that co-interior angle proves the all its angle are best angles.

Theorem

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

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

Construction that a rectangle

We can construct a rectangle with offered side lengths by building a parallelogram with a right angle ~ above one corner. Very first drop a perpendicular indigenous a allude P come a line . Note B and also then mark off BC and BA and complete the parallelogram as shown below.

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

We have actually shown over that the diagonals of a rectangle space equal and also bisect every other. Whereas these 2 properties taken together constitute a test for a square to be a rectangle.

Theorem

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


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EXERCISE 8

a Why is the square a parallelogram?

b usage congruence come prove the the figure is a rectangle.


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

Thus we have the right to construct a rectangle really simply through drawing any kind of two intersecting lines, climate drawing any type of circle centred at the allude of intersection. The quadrilateral created by involvement the 4 points where the circle cuts the lines is a rectangle since it has equal diagonals that bisect each other.

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Rectangles

Definition that a rectangle

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

Properties the a rectangle

A rectangle is a parallelogram, therefore its opposite sides are equal. The diagonals of a rectangle are equal and also bisect each other.

Tests because that a rectangle

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

Links forward

The staying special quadrilaterals come be treated by the congruence and also angle-chasing approaches of this module space rhombuses, kites, squares and also trapezia. The sequence of theorems involved in treating all these special quadrilaterals at when becomes quite complicated, therefore their discussion will it is in left till the module Rhombuses, Kites, and Trapezia. Each individual proof, however, is well within Year 8 ability, noted that students have the appropriate experiences. In particular, it would be valuable to prove in Year 8 that the diagonals the rhombuses and kites satisfy at appropriate angles − this an outcome is necessary in area formulas, it is beneficial in applications the Pythagoras’ theorem, and it gives a much more systematic explanation that several vital constructions.

The next step in the advancement of geometry is a rigorous treatment of similarity. This will allow various results around ratios that lengths to be established, and additionally make possible the an interpretation of the trigonometric ratios. Similarity is forced for the geometry of circles, where an additional class of one-of-a-kind quadrilaterals arises, specific the cyclic quadrilaterals, who vertices lie on a circle.

Special quadrilaterals and also their nature are necessary to establish the traditional formulas because that areas and also volumes that figures. Later, these results will be necessary in occurring integration. Theorems around special quadrilaterals will be widely offered in coordinate geometry.

Rectangles room so common that they go unnoticed in most applications. One special duty worth noting is they room the communication of the collaborates of clues in the cartesian plane − to uncover the coordinates of a suggest in the plane, we finish the rectangle formed by the allude and the 2 axes. Parallelograms arise as soon as we add vectors by completing the parallel − this is the factor why they end up being so important when complicated numbers are represented on the Argand diagram.


History and also applications

Rectangles have actually been helpful for as lengthy as there have actually been buildings, due to the fact that vertical pillars and also horizontal crossbeams are the most obvious means to construct a building of any type of size, providing a framework in the shape of a rectangular prism, all of whose deals with are rectangles. The diagonals that us constantly usage to research rectangles have an analogy in structure − a rectangular framework with a diagonal has actually far an ext rigidity than a straightforward rectangular frame, and diagonal struts have always been offered by building contractors to provide their building much more strength.

Parallelograms room not as typical in the physical civilization (except together shadows of rectangle-shaped objects). Their major role historically has been in the depiction of physical concepts by vectors. Because that example, when two forces are combined, a parallelogram can be attracted to assist compute the size and also direction that the merged force. When there space three forces, we complete the parallelepiped, i m sorry is the three-dimensional analogue that the parallelogram.


REFERENCES

A history of Mathematics: one 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 triangle ABM and also CDM :

1.BAM= DCM(alternate angles, ab || DC )
2.ABM= CDM(alternate angles, abdominal muscle || DC )
3. AB = CD (opposite political parties of parallelogram 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 square ABCD)
α + β= 180o
HenceAB || DC(co-interior angles are supplementary)
andAD || BC(co-interior angles room supplementary).

EXERCISE 3

First display that alphabet ≡ CDA making use of the SSS congruence test.
HenceACB = CAD and also CAB = ACD(matching angles of congruent triangles)
soAD || BC and abdominal || DC(alternate angles space equal.)

EXERCISE 4

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

EXERCISE 5

First prove the 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 are equal):

Hence ABCD is a parallelogram, due to the fact that one pair of opposite sides are equal and parallel.

EXERCISE 6

Join AM. V centre M, attract an arc with radius AM the 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 each diagonal is the amount of the squares on any two surrounding sides. Because opposite sides are equal in length, the squares on both diagonals are the same.

EXERCISE 8

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

hence ABCD is rectangle, because it is a parallelogram with one appropriate angle.

EXERCISE 9

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

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

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