ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA=ED. Prove that:
(i) AD||BC (ii) EB=EC
Answers
Given: ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED.
To Prove : (i) AD || BC (ii) EB = EC
Solution :
(i) Since, EA = ED
Then,
∠EAD = ∠EDA ……………...(1)
[ Angles opposite to equal sides of a triangle are equal]
Since, ABCD is a cyclic quadrilateral, and Sum of Opposite pair of angles in a cyclic quadrilateral is 180° :
Then,
∠ABC + ∠ADC = 180° ………………(2)
But,
∠ABC + ∠EBC = 180° ……………(3)
[Linear pair]
From eq 2 & 3,
∠ADC = ∠EBC …………………………...(4)
On Comparing eq (1) and (4), we get
∠EAD = ∠EBC ……………..(5)
Since, corresponding angles are equal, Then,
BC ‖ AD
Hence proved BC ‖ AD
(ii) From eq (5), we have
∠EAD = ∠EBC
Similarly,
∠EDA = ∠ECB …………...(6)
On Comparing equations (1), (5) and (6), we get
∠EBC = ∠ECB
EB = EC
[ Angles opposite to equal sides of a triangle are equal]
Hence proved EB = EC
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Answer:
Step-by-step explanation:
(i) Since, EA = ED
Then,
∠EAD = ∠EDA ……………...(1)
[ Angles opposite to equal sides of a triangle are equal]
Since, ABCD is a cyclic quadrilateral, and Sum of Opposite pair of angles in a cyclic quadrilateral is 180° :
Then,
∠ABC + ∠ADC = 180° ………………(2)
But,
∠ABC + ∠EBC = 180° ……………(3)
[Linear pair]
From eq 2 & 3,
∠ADC = ∠EBC …………………………...(4)
On Comparing eq (1) and (4), we get
∠EAD = ∠EBC ……………..(5)
Since, corresponding angles are equal, Then,
BC ‖ AD
Hence proved BC ‖ AD
(ii) From eq (5), we have
∠EAD = ∠EBC
Similarly,
∠EDA = ∠ECB …………...(6)
On Comparing equations (1), (5) and (6), we get
∠EBC = ∠ECB
EB = EC