Math, asked by princeverma90, 1 month ago

State and prove all the properties of parallelogram {(CLASS - 9th)}
The project must include
1. Table of content (INDEX)
2. observation and conclusions
3. bibliography
4. project is of 100 MARKS

*this question points is also 100* *(POINTS)*​

Answers

Answered by arth696
1

Answer:

OPPOSITE ANGLES OF PARALLELOGRAM ARE CONGRUENT - DEFINITION

concept

Diagonal of Parallelogram: Parallelogram is a Quadrilateral whose both pairs of opposite sides are parallel and equal. In a parallelogram, the Diagonals Bisect one another. One pair of opposite sides is Parallel and Equal in length.

Given: □PQRSis a Parallelogram.

To prove: ∠SPQ≅∠QRS

and ∠PSR≅∠RQP

Construction: Draw a diagonal SQ

Proof: □PQRS is a parallelogram.

∴sidePS∣∣sideQR and seg SQ is a transveral

∴∠PSQ≅∠RQS ...(Alternate angles) ...(1)

Also, sidePQ∣∣sideSR and seg SQ is a transversal.

∠PQS≅∠RSQ ...(Alternate angles)...(2)

In △PQS and △RSQ

∠PSQ≅∠RQS ...from (1)

sideSQ≅sideQS ...(common side)

∠PQS≅∠RSQ ...from (2)

∴△PQS≅△RSQ ...(ASA test)

∴∠SPQ≅∠QRS ...(c.a.c.t.)

Similarly, we can prove by drawing diagonal PR.

∠PSR≅∠RQP

Hence, the opposite angles of a parallelogram are congruent.

TEST OF PARALLELOGRAM 2 - DEFINITION

concept

Given: □PQRS is a quadrilateral in which

∠SPQ≅∠QRS

∠PQR≅∠RSP

To prove: □PQRS is a parallelogram.

Proof: Let ∠SPQ=∠QRS=x

0

Opposite angle of

and ∠PQR=∠RSP=y

0

a quadrilateral.

∠SPQ+∠PQR+∠QRS+∠RSP=360

0

. (Angle sum property of a quadrilateral)

∴x+y+x+y=360

0

∴2x+2y=360

0

∴x+y=180

0

...(dividing by 2)

∠SPQ+∠RSP=180

0

∴sidePQ∣∣sideSR ...(interior angles test)..(1)

Similarly, we can prove that

sidePS∣∣sideQR ...(2)

∴□PQRS is a parallelgram ... from (1) and (2)

Hence, if opposite angles of a quadrilateral are congruent, then it is a parallelogram.

DIAGONAL OF PARALLELOGRAM BISECT EACH OTHER - DEFINITION

concept

Given: □ABCD is a parallelogram in which the diagonals AC and BD intersect in M.

To prove: segAM≅segCM

and segBM≅segDM

Proof: since □ABCD is a paeallelogram.

sideAb∣∣sideCD and segAC is a transersal.

∴∠BAC≅∠DCA ...(Alternate angles)

i.e. ∠BAM≅∠DCM ...(A-M-C)...(1)

Also, sideAB∣∣sideDC and segDB is a transversal.

∴∠ABD≅∠CDB ...(Alternate angels)

i.e. ∠ABM≅∠CDM ...(B-M-D)..(2)

Now, In △ABM and △CDM

∠BAM≅∠DCM ...(from 1)

sideAB≅sideDC ...(opposite side)

∠ABM≅∠CDM ...(from 2)

∴△ABM≅△CDM ...(ASA test)

∴segAM≅segCM ...(c.s.c.t.)

and segBM≅segDM

Hence, diagonals of parallelogram bisect each other.

TEST OF PARALLELOGRAM 3 - DEFINITION

concept

Given: □PQRS is a quadrilateral in which diagonals PR and QS intersect in M.

segPM≅segRM and

segQM≅segSM

To Prove: □PQRS is a parallelogram

Proof: In △PMQ and △RMS

segPMQ≅segRM ...(given)

∠PMQ≅∠RMS ...(vertically opposite angles)

segQM≅segSM ...(given)

∴△PMQ≅△RMS ...(SAS test)

∴∠PQM≅∠RSM ...(c.a.c.t.)

i.e. sidePQ∣∣sideSR ...(alternate angle test)..(1)

similarly, we can prove that

sidePS∣∣sideQR ...(2)

□PQRS is a parallelgram ...from (1) and (2)

Hence, if the diagonals of a quadrilateral bisect each ther then it is a parallelogram.

TEST OF PARALLELOGRAM 4 - DEFINITION

concept

Given: □LMNK is a given quadriateral in which

sideLM∣∣sideNK and

sideLM≅sideNK

To prove: □LMNK is a parallelogram

Construction: Draw diagonal MK

Prrof: since □LMNK is a quadrilateral in which

sideLM∣∣sideNK

and segMK is a transversal.

∠LMK≅∠NKM ...(Alternate angles)..(1)

Now, In △KLM and △MNK

segLM≅segNK ...(given)

∠LMK≅∠NKM ...(from 1)

segKM≅segMK ...(common side)

∴△KLM≅△MNK ...(SAS test)

∴∠LKM≅∠NMK ...(c.a.c.t.)

∴sideLK∣∣sideMN ...(alternate angles test)..(2)

and sideLM∣∣sideNK ...(given)..(3)

∴ from (2) and (3) we have □LKMN is a parallelogram

Hence, if a pair of opposite side of a quadrilateral is parallel and congruent then the quadrilateral is a parallelogram.

Hope it helps. : )

Answered by xXItzSujithaXx34
6

Diagonal of Parallelogram: Parallelogram is a Quadrilateral whose both pairs of opposite sides are parallel and equal. In a parallelogram, the Diagonals Bisect one another. One pair of opposite sides is Parallel and Equal in length.

Given: □PQRSis a Parallelogram.

To prove: ∠SPQ≅∠QRS

and ∠PSR≅∠RQP

Construction: Draw a diagonal SQ

Proof: □PQRS is a parallelogram.

∴sidePS∣∣sideQR and seg SQ is a transveral

∴∠PSQ≅∠RQS ...(Alternate angles) ...(1)

Also, sidePQ∣∣sideSR and seg SQ is a transversal.

∠PQS≅∠RSQ ...(Alternate angles)...(2)

In △PQS and △RSQ

∠PSQ≅∠RQS ...from (1)

sideSQ≅sideQS ...(common side)

∠PQS≅∠RSQ ...from (2)

∴△PQS≅△RSQ ...(ASA test)

∴∠SPQ≅∠QRS ...(c.a.c.t.)

Similarly, we can prove by drawing diagonal PR.

∠PSR≅∠RQP

Hence, the opposite angles of a parallelogram are congruent.

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