Theorems on Quadrilateral

8/21/2019 Theorems on Quadrilateral

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Theorems on Quadrilateral

Theorem 7.1If a quadrilateral is a parallelogram, then its oppositesides are congruent.

Theorem 7.12 A parallelogram is a rectangle if and only if itsdiagonals are congruent.

Theorem 7.3If a quadrilateral is a parallelogram, then itsconsecutive angles are supplementary

Theorem 7.10 A parallelogram is a rhomus if and only if itsdiagonals are perpendicular.

Theorem 7.! "onverse of 7.1

Theorem 7.# "onverse of 7.$

Theorem 7.7 "onverse of 7.3

Theorem 7.% "onverse of 7.2

Theorem 7.&If one pair of opposite sides of a quadrilateral arecongruent and parallel, then the quadrilateral is aparallelogram.

Theorem 7.$ If a quadrilateral is a parallelogram, then itsdiagonals isect each other.

Theorem 7.11 A quadrilateral is a rhomus if and only if eachdiagonal isects a pair of opposite angles.

Theorem 7.2If a quadrilateral is a parallelogram, then its oppositeangles are congruent.

Theorem 7.13 A parallelogram is a square if and only if itsdiagonals are oth perpendicular and congruent.


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Key Points About Quadrilaterals

1. 'um of the angles of a quadrilateral is 3%0(.

2. A diagonal of a parallelogram divides it into t)o congruent triangles.

3. In a parallelogram,

*i+ opposite sides are equal

*ii+ opposite angles are equal

*iii+ diagonals isect each other

$. A quadrilateral is a parallelogram, if

*i+ opposite sides are equal or

*ii+ opposite angles are equal or

*iii+ diagonals isect each other or

*iv+ a pair of opposite sides is equal and parallel

!. iagonals of a rectangle isect each other and are equal and vice-versa.

%. iagonals of a rhomus isect each other at right angles and vice-versa.

7. iagonals of a square isect each other at right angles and are equal, and vice-

versa.

#. The line-segment oining the mid-points of any t)o sides of a triangle is parallel to the

third side and is half of it.

&. A line through the mid-point of a side of a triangle parallel to another side isects the

third side.

10. The quadrilateral formed y oining the mid-points of the sides of a quadrilateral, in

order, is a parallelogram.


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Quadrilateral and Its Types

To vie) the complete lesson clic/ the video elo)

We know that a quadrilateral has four sides, four angles, four vertices and twodiagonals. Quadrilateralscan be classied into dierent types based on their sides and angles.In case of rectangle and square, all the angles are right angles, the opposite sides are parallel,and the diagonals bisect each other. In a rectangle, the opposite sides are equal, whereas ina square, all the sides are equal. Hence, a square is a rectangle with adjacent sides equal. parallelogram is a quadrilateral in which the opposite sides are parallel and equal in length.In aparallelogram, the opposite angles are equal and the diagonals bisect each other. Ina parallelogram, the angles are not right angles. When we co!pare aparallelogram witha rectangle, we see that it is dierent fro! the rectangle in ter!s of the !easure of its angles. rhombus is a quadrilateral in which all the sides are equal in length, the opposite sides areparallel, the opposite angles are equal and the diagonals bisect each other at right angles.

square is a rhombus in which, each angle !easures"##. Squares, rectangles and rhombuses are all e$a!ples of parallelograms.

trapezium is a quadrilateral in which one pair of opposite sides is parallel. trape%iu! withthe non&parallel sides equal and the base angles equal is known as an isosceles trapezium.


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kite is a quadrilateral in which two pairs of adjacent sides are equal in length and one pair ofopposite angles, 'the ones that are between the sides of unequal length,( are equal in !easureand the diagonals intersect at right angles.

Properties of a Parallelogram

To vie) the complete lesson clic/ the video elo)

•


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quadrilateral is a closed fgure which has our sides, our angles and ourvertices. )here are dierent types o quadrilaterals suchasParallelogram, Rectangle, Square, Rhombus,Trapezium and ite.

parallelogram is a quadrilateral in which

• opposite sides are parallel and equal,

• opposite angles are equal,

• the diagonals bisect each other,

• each diagonal divides it into t!o congruent triangles,

• ad"acent angles are supplementary.

square is a parallelogram in which

• all the sides are equal,

• each angle measures #$$,

• diagonals are equal and bisect at right angles. rectangle is a parallelogram in which

• diagonals are equal and bisect each other,

• each angle !easures "##.

rhombus is a parallelogram in which

• all our sides are equal,

• diagonals bisect each other at right angles.


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)heore!* diagonal o a parallelogram divides it into t!o congruent triangles.

)heore!* If each pair o opposite sides is equal in a quadrilateral, then it isa parallelogram.

)heore!* If the diagonals o a quadrilateral bisect each other, then it is a parallelogram.


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Theorem 1- In a parallelogram, the opposite sides are of equal length.

Given: AB || C, A || BC

To prove: AB ! C, A ! BC

ra" in the diagonal AC


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Theorem# If the opposite sides in a quadrilateral are the same length, then the figure is

a parallelogram.

Given: AB ! C, A ! BC

To prove: AB || C, A || BC

ra" in AC


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Theorem$- A quadrilateral is a parallelogram if and onl% if the diagonals &ise't ea'h

other.

(in'e this is an )if and onl% if) proof, there are t"o things to prove.

1. Given: ABC is a parallelogram

To prove: A* ! *C, B* ! *

and the 'onverse:

#. Given: A* ! *C, B* ! *

To prove: ABC is a parallelogram

There is another "a% to prove this.


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The definition of a parallelogram is that the opposite sides are parallel. In the se'ond

"a% "e esta&lish that the opposite sides are parallel, so "e 'an use the definition to

'on'lude that the figure is a prarllelogram. It is simpler to sho" that the opposite sides

are equal in length "hi'h "e did in the first "a%. If "e do it that "a% the reason that

the figure is a parallelogram is that "e proved that if the opposite sides are the samelength then the figure is a parallelogram in Theorem #.

Theorem +: If one pair of opposite sides in a four sided figure are &oth opposite and

parallel, then the figure is a parallelogram

roof: Given: AB ! C

AB || C

To prove: ABC is a parallelogram

http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/T2.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/T2.html


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Theorem : A re'tangle is a parallelogram.

e are given that ABC is a re'tangle. That means that all of the angles are the same

si/e. (in'e a quadrilateral 'an &e &ro0en up into t"o triangles, there are a total of

$2o in the angles of a quadrilateral. If all of the four angles are the same si/e, the% all

have to &e 32 o. (o "e 'an state our )Given) as

Given: All four angles are 32o

.

To prove: ABC is a parallelogram.

Theorem : A parallelogram is a re'tangle if and onl% if the diagonals are the same

length.

http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.html


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(in'e this is an )if and onl% if) proof, there are t"o things to prove.

1. Given: ABC is a re'tangle.

To prove: AC ! B

and the 'onverse

#. Given: ABC is a parallelogram, AC ! B

To prove: ABC is a re'tangle.

Theorem 4: A rhom&us is a parallelogram.

Given: ABCD is a rhom&us

To prove: ABCD is a parallelogram.

http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.html


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Theorem 5: A quadrilateral is a rhom&us if and onl% if the diagonals are perpendi'ular

&ise'tors of ea'h other.

This is an )if and onl% if) proof, so there are t"o things "e have to prove:

6ne "a% to prove this is to use 'ongruent triangles.



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7o"ever, note that this essentiall% runs through the proof of one of the isos'eles

triangle theorems "hi'h "e have alread% proved. 8ote the ((( reason after "e

esta&lished the refle9ive side. e do not have to do it again. e 'an simpl% refer to it.

or the 'onverse,

there is the 'ongruent tirangle proof.

But, again "e are running through a proof of one of the isos'eles triangle

theorems. e 'ould simpl% refer to it instead of proving it again.

6ne 'ould also use the fa't that a point is equidistant from t"o given points if and

onl% if it is on the perpendi'ular &ise'tor of the line segment &et"een them.

http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I3.html#p1http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I3.html#p1http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I4.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I4.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I7.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I3.html#p1http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I3.html#p1http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I4.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I4.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Isosceles/I7.html


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Theorem 9: A quadrilateral is a rhom&us if and onl% if the diagonals &ise't all the

verte9 angles.

1. Given: ABCD is a rhom&us

To prove: ; DAC ! ; BAC ; ABD ! ; CBD; BCA ! ; DCA

; CDB ! ; ADB

#. Given: ; DAC ! ; BAC ; ABD ! ; CBD; BCA ! ; DCA; CB ! ; ADB

To prove: ABCD is a rhom&us

http://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/Definitions.html#rhombushttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/theorems/Parallelograms/default.htmlhttp://www.sonoma.edu/users/w/wilsonst/courses/math_150/Definitions.html#rhombus


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Theorem 12 : If one of the diagonals in a parallelogram &ise'ts one of the verte9

angles, then the parallelogram is a rhom&us.

Given: ABC is a parallelogram.

To prove: ABC is a rhom&us

Question #2



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Statements Reasons

1. 1. Given

2. 2. 6n the 'oordinate plane, themidpoint of a line segment is found &% ta0ing the average of the

endpoints < is equal to half the sumof the endpoints=.

3. 3. (u&stitution. The midpoint of ea'hsegment equals the same set of

'oordinates.

4. 4. The &ise'tor of a segment interse'tsthe segment at its midpoint.

5. 5. If the diagonals of a quadrilateral &ise't ea'h other, the quadrilateral is

a parallelogram.

Question #3

Statements Reasons


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1. 1. Given

2. 2. The opposite sides of a parallelogram are 'ongruent.

3. 3.Transitive propert%: (egments 'ongruent to the same

segment are 'ongruent to ea'h other.

4. 4. The opposite sides of a parallelogram are parallel.

5. 5. >ines parallel to the same line are parallel to ea'h other.

6. 6.A quadrilateral "ith one set of sides &oth parallel and

'ongruent is a parallelogram.

Question #4

Statements Reasons

1. 1. Given

2. 2.

erpendi'ular lines meet to form right

angles.

3. 3. All right angles are 'ongruent.

4. 4.The opposite sides of a parallelogramare parallel.

5. 5.If # parallel lines are 'ut &% a

transversal, the alternate interior

angles are 'ongruent.

6. 6.The opposite sides of a parallelogram

are 'ongruent.

7. 7.

AA(: If t"o angles and the non-

in'luded side of one triangle are

'ongruent to the 'orresponding partsof a se'ond triangle, the triangles are

'ongruent

Question #4


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Statements Reasons

1. 1. Given


3. 3.If # parallel lines are 'ut &% a transversal, the alternate

interior angles are 'ongruent.

4. 4. Congruent angles are angles of equal measure.

5. 5.The measure of the e9terior angle of a triangle is greater

than the measure of either nonad?a'ent interior angle.

6. 6. (u&stitution: A quantit% ma% &e su&stituted for its equal

Question #6

Statements Reasons

1. 1. Given

2. 2. erpendi'ular lines meet to form right angles.


4. 4. In a plane, lines perpendi'ular to the same line are parallel.

5. 5. A quadrilateral "ith t"o sets of opposite parallel sides is a


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parallelogram.

6. 6. A parallelogram "ith one right angle is a re'tangle

Question #7

slope: slope of slope of (in'e the slopes are thesame,

slope of slope of (in'e the slopes are not

equal,

distan'e:

AD = CB

ABCD is an isos'eles trape/oid &e'ause it has onl% one set of parallel sides and its legs are

'ongruent

Question #8


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Statements Reasons

1. 1. Given


3. 3. The opposite sides of a parallelogram are 'ongruent.

4. 4. Congruent segments are segments of equal measure.

5. 5. Bet"eeness of points.


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2. 2. A re'tangle is a parallelogram "ith four right angles.

3. 3. The diagonals of a re'tangle are 'ongruent.

4. 4. The diagonals of a parallelogram &ise't ea'h other.

5. 5. Bise'tor of a segment interse'ts the segment at itsmidpoint.

6. 6. @idpoint of a segment divides a segment into t"o'ongruent segments.

7. 7. Congruent segments are segments of equal measure.

8. 8. Bet"eeness of points


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Statements Reasons

1. 1. Given

2. 2. If t"o angles form a linear pair the% are supplementar%.

3. 3. (upplements of the same angle, or 'ongruent angles are'ongruent.

4. 4. (u&stitution

5. 5. If # lines are 'ut &% a transversal and the interior angles onthe same side of the transversal are supplementar%, the

lines are parallel.

6. 6. A parallelogram has # sets of opposite sides parallel


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Prove% )he diagonals of a

parallelogra! divide theparallelogra! into two congruenttriangles

Question #11

Statements Reasons

1. 1. Given

2. 2. The opposite sides of a parallelogramare 'ongruent.

3. 3. The opposite angles of a parallelogramare 'ongruent.

4. 4. (A(: If t"o sides and the in'ludedangle of one triangle are 'ongruent to

the 'orresponding parts of anothertriangle, the triangles are 'ongruent

Understanding Quadrilaterals

Exercise 1

Question: 1 The angles of !uadrilateral are in the ratio " : # : $ : 1" %ind all theangles of the !uadrilateral

Ans&er: As you /no) angle sum of a quadrilateral 3%0(


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ence, angles are 3%(, %0(, 10#(, 1!%(

Question: ' If the diagonals of a parallelogram are e!ual( then sho& that it is arectangle

Ans&er: In the follo)ing parallelogram oth diagonals are equal

As all are right angles so the parallelogram is a rectangle.

Question: " )ho& that if the diagonals of a !uadrilateral bisect each other at rightangles( then it is a rhombus


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Ans&er: In the given quadrilateral A" diagonals A" and isect each other at

right angle. 4e have to prove that A""A

'o, AA

'imilarly A""A can e proved )hich means that A" is a rhomus.

Question: * )ho& that the diagonals of a s!uare are e!ual and bisect each other at right angles

Ans&er: In the figure given aove let us assume that

5A5 *'ides opposite equal angles are equal+

'imilarly A555" can e proved

This gives the proof of diagonals of square eing equal.


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Question: # )ho& that if the diagonals of a !uadrilateral are e!ual and bisecteach other at right angles( then it is a s!uare

Ans&er: 6sing the same figure,

If 5A5

*Angles opposite to equal sides are equal+

'o, all angles of the quadrilateral are right angles ma/ing it a square.

Question: + ,iagonal A- of a parallelogram A.-, bisects angle A )ho& that

/i0 it bisects angle - also(

/ii0 A.-, is a rhombus

Ans&er: A" is a parallelogram )here diagonal A" isects angle A


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As diagonals are intersecting at right angles so it is a rhomus

Question: In parallelogram A.-,( t&o points P and Q are ta2en on diagonal .,such that ,P 3 .Q )ho& that:


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4ith equal opposite angles and equal opposite sides it is proved that A"8 is aparallelogram

Question: 4 A.-, is a parallelogram and AP and -Q are perpendiculars from5ertices A and - on diagonal ., )ho& that


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Question: $ In 6 A.- and 6 ,E%( A. 3 ,E( A. 77 ,E( .- 3 E% and .- 77 E%8ertices A( . and - are 9oined to 5ertices ,( E and % respecti5ely )ho& that

/i0 !uadrilateral A.E, is a parallelogram

/ii0 !uadrilateral .E%- is a parallelogram

/iii0 A, 77 -% and A, 3 -%

/i50 !uadrilateral A-%, is a parallelogram

/50 A- 3 ,%

In quadrilateral A9


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A 9

A::9

'o, A9 is a parallelogram *opposite sides are equal and parallel+

'o, 9::A ------------ *1+

'imilarly quadrilateral A"; can e proven to e a parallelogram

'o, 9::"; ------------ *2+

;rom equations *1+ < *2+

It is proved that

A::";

'o, A";

'imilarly A"; and A"::; can e proved

1 A.-, is a trape;ium in &hich A. 77 -, and A, 3 .- )ho& that


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Understanding Quadrilaterals

Exercise '

Question: 1 A.-, is a !uadrilateral in &hich P( Q( < and ) are mid=points of the

sides A.( .-( -, and ,A A- is a diagonal )ho& that :

Ans&er: =et us e>tend the line '? to T so that "T is parallel to A'

As '? is touching the mid points of A and " so as per mid point theorem '?::A"

'imilarly A" :: 8 can e proven )hich )ill prove that 8?' is a parallelogram.


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Question:' A.-, is a rhombus and P( Q( < and ) are the mid=points of the sidesA.( .-( -, and ,A respecti5ely )ho& that the !uadrilateral PQ


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Ans&er: In @ A

A parallel line to the ase originating from mid point of second side )ill intersect at themidpoint of the third side.

A :: "

A :: 9;

'o, 9; :: "

'o, In @ A

9 :: A

9 is the mid point of A

'o, is the mid point of

Bo), in @ "

; :: "

is the mid point of

'o, ; )ill e mid point of " * Cid point theorem+

* In a parallelogram A.-,( E and % are the mid=points of sides A. and -,respecti5ely )ho& that the line segments A% and E- trisect the diagonal .,


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ence, A9 ";

In quadrilateral A9";

9" :: A; < 9" A;

A9 ";

'o, A9 :: ";

'o, A9"; is a parallelogram.

In @ 8"

9 :: 8" *proved earlier y proving A9 :: ";+

9 is the mid point of "

'o, is the mid point of 8

'o, 8

In @ A


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;8 :: A

; is the mid point of A

'o, 8 8

'o, 8 8 proved

# )ho& that the line segments 9oining the mid=points of the opposite sides of a!uadrilateral bisect each other

Ans&er: A" is a quadrilateral in )hich , 8, ?, < ' are mid points of A, ", " < A

In @ A"

'? is touching mid points of " and A

'o, '? :: A"

'imilarly follo)ing can e proved

8 :: A"

8? ::

' ::


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'o, 8?' is a parallelogram.

? and 8' are diagonals of the parallelogram 8?', so they )ill isect each other.

+ A.- is a triangle right angled at - A line through the mid=point > of

hypotenuse A. and parallel to .- intersects A- at , )ho& that

Ans&er: C :: "

C is the mid point of A

'o, is the mid point of A" *Cid point theorem+


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Theorems on Quadrilateral

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