Advertisements
Advertisements
प्रश्न
In the given figure, AP is the bisector of ∠A and CQ is the bisector of ∠C of parallelogram ABCD. 
Prove that APCQ is a parallelogram.
Advertisements
उत्तर
Construction: Join AC
Proof:
∠BAP = `1/2`∠A ...( AP is the bisector of ∠A )
∠DCQ = `1/2`∠C ...( CQ is the bisector of ∠C )
⇒ ∠BAP = ∠DCQ ....(i)....[ ∠A = ∠R ( Opposite angles of a parallelogram.) ]
Now,
∠BAC = ∠DCA ....(ii)....[ Alternate angles since AB || DC ]
Subtracting (ii) from (i), We get
∠BAP - ∠BAC = ∠DCQ - ∠DCA
⇒ ∠CAP = ∠ACQ
⇒ AP || QC .....( Alternate angles are equal )
Similarly, PC || AQ.
Hence, APCQ is a parallelogram.
APPEARS IN
संबंधित प्रश्न
In a parallelogram `square`ABCD, If ∠A = (3x + 12)°, ∠B = (2x - 32)° then find the value of x and then find the measures of ∠C and ∠D.
In case of a parallelogram
prove that:
(i) The bisectors of any two adjacent angles intersect at 90o.
(ii) The bisectors of the opposite angles are parallel to each other.
In the following figure, ABCD is a parallelogram. 
Prove that:
(i) AP bisects angle A.
(ii) BP bisects angle B
(iii) ∠DAP + ∠BCP = ∠APB
In the following figures, find the remaining angles of the parallelogram
In the following figures, find the remaining angles of the parallelogram
In the following figures, find the remaining angles of the parallelogram
In the following figures, find the remaining angles of the parallelogram
In the given figure, ABCD is a parallelogram, find the values of x and y.
PQR is a triangle formed by the adjacent sides PQ and QR and diagonal PR of a parallelogram PQRS. If in ΔPQR, ∠P : ∠Q : ∠R = 3 : 8 : 4, Calculate the measures of all the angles of parallelogram PQRS.
If PQRS is a parallelogram, then ∠P – ∠R is equal to ______.
