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Circles — Maths STD 9 — Question

CBSE BoardEnglish MediumSTD 9MathsCircles4 Marks
Question
ABCD ia a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that:
  1. AD || BC.
  2. EB = EC.
✓

Answer



Given ABCD is a cyclic quadrilateral in which EA = ED To prove:
  1. AD || BC.
  2. EB = EC.
Proof:
  1. Since EA = ED
    Then, $\angle\text{EAD}=\angle\text{EDA}\dots(1)$ [Oppo. angles to equal sides]
    Since, ABCD is a cyclic quadrilateral
    Then, $\angle\text{ABC}+\angle\text{ADC}=180^\circ$
    But $\angle\text{ABC}+\angle\text{EBC}=180^\circ$ [Linear pair of angles]
    Then, $\angle\text{ADC}=\angle\text{EBC}\dots(2)$
    Compare equations (1) and (2)
    $\angle\text{EAD}=\angle\text{EBC}\dots(3)$
    Since, corresponding angles are equal
    Then, BC || AD
  1. From equation (3)
    $\angle\text{EAD}=\angle\text{EBC}\dots(3)$
    Similarly $\angle\text{EDA}=\angle\text{ECB}\dots(4)$
    Compare equations (1)(3) and (4)
    $\angle\text{EBC}=\angle\text{ECB}$
    $\Rightarrow \text{EB}=\text{EC}$ [Opposite angles to equal sides]

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