0 AP and BP bisect LA and LB in a parallelogram ABID. Also CR and DR bisect LL and Lo intersecting BP and AP ant o and a respectively. Find LAOD, (BOC, LAPR and LORO Is O P Q R rectangle? P
Answers
Answer:
Step-by-step explanation:
ABCD is a parallelogram, in which ∠A=60
o
.
⇒ ∠A+∠B=180
o
[ Sum of adjacent angles of parallelogram are supplementary ]
⇒ 60
o
+∠B=180
o
.
∴ ∠B=120
o
.
⇒ ∠D=∠B=120
o
[ Opposite angles of parallelogram are equal ]
⇒ ∠A=∠C=60
o
[ Opposite angles of parallelogram are equal ]
AP bisects ∠A
∴ ∠DAP=∠PAB=30
o
.
BP bisects ∠B
∴ ∠CBP=∠PBA=60
o
In △PAB,
⇒ ∠PAB+∠PBA+∠APB=180
o
.
⇒ 30
o
+60
o
+∠APB=180
o
⇒ ∠APB=90
o
.
In △PBC,
⇒ ∠PBC+∠PCB+∠BPC=180
o
.
⇒ 60
o
+60
o
+∠BPC=180
o
.
⇒ ∠BPC=60
o
In △ADP,
⇒ ∠PAD+∠ADP+∠APD=180
o
.
⇒ 30
o
+120
o
+∠APD=180
o
.
⇒ ∠APD=30
o
.
In △PBC,
⇒ ∠BPC=∠CBP=60
o
. [ linear angles are supplementary ]
⇒ BC=PC [ Sides opposite to equal angles of a triangle are equal ] ----- ( 1 )
In △ADP,
⇒ ∠APD=∠DAP=30
o
.
⇒ AD=DP [ Sides opposite to equal angles of triangle are equal ]
But AD=BC [ Opposite sides of parallelogram are equal ]
⇒ So, BC=DP ----- ( 2 )
From ( 1 ) and ( 2 ), we get
⇒ DP=PC
⇒ P is the mid-point of CD