Advertisements
Advertisements
प्रश्न
ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that AD || BC .
Advertisements
उत्तर
If ABCD is a cyclic quadrilateral in which AB and CD when produced meet in E such that EA = ED, then we have to prove the following, AD || BC
It is given that EA = ED, so
Since, ABCD is cyclic quadrilateral
`x + angleABC = 180 ⇒ angleDAB = 180 - x`
And ; ` x + angleBCD = 180 ⇒ angle BCD = 180- x `
Now,
`angle DAB + angle ABC = x + 180 - x = 180`
Therefore, the adjacent angles `angleDAB ` and `angleABC` are supplementary
Hence, AD || BC
APPEARS IN
संबंधित प्रश्न
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.
Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.
AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.
In the given figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD.
In the given figure, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E such that ∠AED = 95° and ∠OBA = 30°. Find ∠OAC.
PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If ∠QPR = 67° and ∠SPR = 72°, then ∠QRS =
ABCD is a cyclic quadrilateral such that ∠A = 90°, ∠B = 70°, ∠C = 95° and ∠D = 105°.
If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.
