Q4 of 12 Page 167

In the figure, the diagonal AC or cyclic quadrilateral ABCD bisects ∠C. Prove that the diagonal BD is parallel to the tangent at point A of the circle passing through A, B, C and D.


∠ ACD = ∠ ACB …………. (1) [In the question it is given that AC bisects the angle C which means that the two angles ∠ 1 and ∠2 has to be equal]


Here we can see that ∠ PAD = ∠ ABD [Angle in the alternate segment]


Similarly, ∠ QAB = ∠ ADB


Also AB is common arc and ADB and ACB are the angle in the same segment, so


∠ ABD = ∠ ACD


But by the equation (1) we can say that


∠ 1 = ∠ 2


So ∠ PAD = ∠ ADB [Alternate Interior angle]


So PQ ∥ BD


More from this chapter

All 12 →
8

In the figure, O is the centre of the circle. Tangents PA and PB drawn from an external point P touch the circle at A and B respectively. Prove that PAOB is a cyclic quadrilateral.

 1

Answer the following questions from the figure.


(i) Alternate segment of ∠BAQ.


(ii) Alternate segment of ∠DAP.


(iii) If C is jointed to B, then ∠ACB is equal to which angles?


(iv) ∠ABD and ∠ADB are equal to which angles?

 2

In the figure, if ∠BAC = 80° then find the value of ∠BCP.

 3

In the figure, PQ and XY are parallel tangents. If ∠QRT = 30°, then find the value of ∠TSY.