Advertisements
Advertisements
Question
If ABCD is a cyclic quadrilateral in which AD || BC (In the given figure). Prove that ∠B = ∠C.
Advertisements
Solution
It is given that, ABCD is cyclic quadrilateral in which AD || BC
We have to prove `angleB = angle C`
Since, ABCD is a cyclic quadrilateral
So,
`angle B + angle D = 180°` and `angle A + angle C = 180°` ..… (1)
`⇒ angleB + angle A ` = 180° and `angle C + angle D = 180°` (Sum of pair of consecutive interior angles is 180°) …… (2)
From equation (1) and (2) we have
`angleB + angleD + angleB + angle A ` = 360° …… (3)
`angleA + angleC + angleC + angle D ` = 360° …… (4)
`2angleB + angleD + angleA = 2 angleC + angleA + angleD`
`2angleB = 2angleC`
`angleB = angleC`
Hence Proved
APPEARS IN
RELATED QUESTIONS
In figure PA and PB are tangents from an external point P to the circle with centre O. LN touches the circle at M. Prove that PL + LM = PN + MN
In Fig. 5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.
In the given figure, ABCD is a cyclic quadrilateral. If ∠BCD = 100° and ∠ABD = 70°, find ∠ADB.
In the given figure, if ∠BAC = 60° and ∠BCA = 20°, find ∠ADC.
Draw a circle of any radius. Show one diameter, one radius and one chord on that circle.
A chord is at a distance of 15 cm from the centre of the circle of radius 25 cm. The length of the chord is
In the figure, segment PQ is the diameter of the circle with center O. The tangent to the tangent circle drawn from point C on it, intersects the tangents drawn from points P and Q at points A and B respectively, prove that ∠AOB = 90°
Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.
The circumcentre of the triangle ABC is O. Prove that ∠OBC + ∠BAC = 90º.
The circumcentre of a triangle is the point which is ______.
