Question

in parallelogram ABCD two points P and Q are taken on diagonal BD such that DP=BQ.show that APCQ is a parallelogram.
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Answer 1

Answer:

Here, ABCD is a parallelogram.

AB∥DC and BD is a transversal.

∴ ∠ABQ=∠CDP [ Alternate angles ] ---- ( 1 )

In △AQB and △CPD,

⇒ AB=CD [ Opposite sides of parallelogram are equal ]

⇒ ∠ABQ=∠CDP [ From ( 1 ) ]

⇒ BQ=DP [ Given ]

∴ △AQB≅△CPD [ By SAS congruence ]

⇒ AQ=CP [ CPCT ]

Answer 2

Step-by-step explanation:

In ∆APD and ∆CQB

• DP=BQ {given}
• AD=BC {ABCD is a parallelogram}
• DBC=BDA {Alternative opposite}

Hence, ∆APD≈∆CQB

Similarly,

AP=QC {By c.p.c.t}...... (1)

Now, In ∆AQB and ∆CPD

• BQ=DP {given}
• AB=DC {Opposite sides of parallelogram ABCD}
• ABQ=CDP {Alternative opposite}

Hence, ∆AQB≈∆CPD

So,

AQ=CP {By c.p.c.t}...... (2)

Now, In Quadrilateral APCQ

• AP=QC {from 1st eq. }
• AQ=CP {from 2nd eq. }

Hence, APCQ is a parallelogram

{two opposite sides are equal}

.......proved

Hope it helps you ☺I'm also in 9th ✌

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