Match the following;
Column−IColumn−II(A) In the given figure,(p) 60∘ ABCD is a cyclic Quadrilateral. O is the centre of the circle. If ∠BOD=160∘ find the Measure of ∠BPD.
(B) In given figure, ABCD is a cyclic(q) 65∘ quadrilateral whose side AB is a diameter of the circle through A, B, C, D. If ∠ADC=130∘,find ∠BAC.
(C) In the given figure BD = DC and(r) 40∘ ∠CBD=30∘ Find m(∠BAC)
(D) In the given figure, O is the centre of the(s) 100∘ arc ABC subtends an angle of130∘ at the centre. If AB is extended to Pm find ∠PBC.
Choose the correct option:
A-s; B-r; C-p; D-q
(a) (A)→s;(B)→r;(C)→p,(D)→q
(A) Consider the arc BCD of the circle. This arc makes angle ∠BOD=160∘ at the centre of the circle and ∠BAD at a point A on the circumference.
∴∠BAD=12∠BOD=80∘
Now, ABPD is a cyclic quadrilateral.
⇒∠BAD+∠BPD=180∘⇒80∘+∠BPD=180∘⇒∠BPD=100∘
(B) Since ABCD is a cyclic quadrilateral.
∴∠ADC+∠ABC=180∘⇒130∘+∠ABC=180∘⇒∠ABC=50∘
Since ∠ACB is the angle in a semicircle.
∴∠ACB=90∘
Now, in ΔABC, we have
∠BAC+∠ACB+∠ABC=180∘⇒∠BAC=90∘+50∘=180∘⇒∠BAC=40∘
(C) BD=DC
∠BCD=∠CBD=30∘
So, ∠BDC=120∘
As ABCD is cyclic
∴∠BAC=60∘
(D) ∠PBC=65∘ (If one side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite opposite angle)