ABCD is a cyclic quadrilateral in which AB and CD when produced meet in E and EA=ED prove that
(I)AD||BC (ll)ED=EC
Answers
(I)AD||BC (ll)ED=EC
Proved Below.
Step-by-step explanation:
Given:
ABCD is a cyclic quadrilateral. AB and CD produced meet at E and EA=ED
(I) Now in the triangle EAD,
EA=ED
Let ∠EAD=x
Therefore ∠EAD=∠EDA=x (1) [angles opposite to equal sides are equal]
∠BCD+∠DAB=180 deg [in a cyclic quadrilateral opposiite angles are equal]
∠BCD+x=180 [since ∠DAB=∠DAE=x]
∠BCD=180-x (2)
Similarly
∠ABC=180-x (3)
∠DAB+∠ABC=x+180-x
∠DAB+∠ABC=180 deg (4)
Similarly
∠BCD+∠CDA=180 deg (5)
by (4) and (5)
Since the adjacent interior angles are supplementary, therefore AD||BC.
(ll) Since,
∠EBC=∠EAD=x (6) [corresponding angles are equal]
similarly ∠ECB=∠EDA=x.....(7)
Now in triangle EBC,
∠EBC=∠ECB=x
therefore EB=EC [sides opposite to equal angles are equal]
