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
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. : )
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.