QQuestion
Prove that in a cyclic quadrilateral opposite angle are supplementary.
✓Answer
Given: ABCD is a cyclic quadrilateral
To prove: ∠ABC + ∠ADC = 2 right angles and ∠BAD + ∠BCD = 2 right angles
Construction: Two diagonals AC and BD are drawn.
Proof: ∠ADB = ∠ACB [angles in the same segment of the circle]
Again, ∠BAC = ∠BDC [angles in the same segment of the circle]
Again, ∠ADC = ∠ADB + ∠BDC
= ∠ACB + ∠BAC
∴ ∠ADC + ∠ABC = ∠ACB + ∠BAC + ∠ABC
∴ ∠ADC + ∠ABC = 2 right angles [∴ sum of three angles of a triangle is 2 right angles]
Similarly we can prove that, ∠BAD + ∠BCD = 2 right angles
Related Questions
More WBBSE Moderate level questions
Prove that a cyclic trapezium is an isosceles trapezium.
ABCD is a cyclic quadrilateral. Bisectors of ∠DAB and ∠BCD intersect the circle at X and Y respectively. If O be the centre of the circle, find ∠XOY.
AB and CD are two chords of a circle. If we extend BA and DC, they intersect each other at point P. Prove that ∠PCB = ∠PAD.
Prove that in a cyclic quadrilateral opposite angles are supplementary.
Prove that the opposite angles of a cyclic quadrilateral are supplementary.
ABCD is a cyclic quadrilateral. If ∠ADB = x° and ∠ABD = y° then ∠BCD is (x + y)°.
Write the magnitude of each angle of a cyclic parallelogram
BOC is a diameter of a circle whose centre is O. ABCD is a cyclic quadrilateral. If ∠ADC =110°, find ∠ACB
What is the relation between the opposite angles of a cyclic quadrilateral?
ABCD is a cyclic quadrilateral. Chord DE is the external bisector of ∠BDC. Prove that AE (or produced AE) is the external bisector of ∠BAC.
If the opposite angles of a quadrilateral be supplementary, then the vertices of the quadrilateral will be ___.
Distance between two pillars is 150 m, height of one pillar is three times the other pillar. From the mid-point of the line joining their foot the angles of elevation of the top of the two pillars are complementary. Find the height of the smaller pillar