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प्रश्न
ABCD is a cyclic quadrilateral AB and DC are produced to meet in E. Prove that Δ EBC ∼ Δ EDA.
उत्तर
In Δ EBC and Δ EDA, we have
∠ EBC = ∠ EDA ...[ Exterior or angle in a cyclic quadrilateral is equal to opposite interior angle]
∠ ECB = ∠ EAD ...[ Exterior or angle in a cyclic quadrilateral is equal to opposite interior angle]
and ∠ E = ∠ E
So, by AAA exterior of similarly, we get
Δ EBC ∼ Δ EDA
Hence proved.
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संबंधित प्रश्न
ABCD is a quadrilateral inscribed in a circle, having ∠ = 60°; O is the center of the circle.
Show that: ∠OBD + ∠ODB =∠CBD +∠CDB.
In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If ∠DCQ = 40° and ∠ABD = 60°, find;
- ∠DBC
- ∠BCP
- ∠ADB
The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.
ABCD is a cyclic quadrilateral. Sides AB and DC produced meet at point E; whereas sides BC and AD produced meet at point F. If ∠DCF : ∠F : ∠E = 3 : 5 : 4, find the angles of the cyclic quadrilateral ABCD.
Two circles intersect in points P and Q. A secant passing through P intersects the circles in A and B respectively. Tangents to the circles at A and B intersect at T. Prove that A, Q, B and T lie on a circle.
Prove that any four vertices of a regular pentagon are concylic (lie on the same circle).
In following fig., O is the centre of the circle, prove that ∠x =∠ y + ∠ z.
If ABCD is a cyclic quadrilateral in which AD || BC. Prove that ∠B = ∠C.
In the adjoining figure, AB is the diameter of the circle with centre O. If ∠BCD = 120°, calculate:
(i) ∠BAD (ii) ∠DBA
An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle, to prove the theorem complete the activity.
Given: ABCD is cyclic,
`square` is the exterior angle of ABCD
To prove: ∠DCE ≅ ∠BAD
Proof: `square` + ∠BCD = `square` .....[Angles in linear pair] (I)
ABCD is a cyclic.
`square` + ∠BAD = `square` ......[Theorem of cyclic quadrilateral] (II)
By (I) and (II)
∠DCE + ∠BCD = `square` + ∠BAD
∠DCE ≅ ∠BAD
