CYCLIC QUADRILATERAL PROBLEMS WITH SLUTIONS

A cyclic quadrilateral is any four-sided geometric figure whose vertices all lie on a circle.

The opposite angles of a cyclic quadrilateral are supplementary.

a+b  =  180 and c+d  =  180

One side subtends equal angles at the other two vertices.

If α = β, then ABCD is a cyclic quadrilateral.

Find the missing angles in the following problems given below.

Example 1 :

Find x given : 

Solution :

In a cyclic quadrilateral, sum of opposite angles are supplementary.

x+15+x-21  =  180

2x-6  =  180

2x  =  186

x  =  93

Example 2 :

Solution :

73 + x  =  180

x  =  180-73

x  =  107

Example 3 :

Solution :

x  =  2x - 70

70  =  2x-x

x  =  70

Example 4 :

Solution :

Let "y" be <DCB.

<DCB + <BCE  =  180

y + 180 - x  =  180

y-x  =  0  ----(1)

(opposite angles of cyclic quadrilateral)

x+y  =  180   ------(2)

(1)  =  (2)

2y  =  180

y  =  90

So, x is 90.

Example 5 :

Solution :

<DAB  =  109

<DAB + <DCB  =  180

109 + <DCB  =  180

<DCB  =  180 - 109

<DCB  =  71

In triangle DBC,

<DBC + <DCB + <BDC  =  180

<DBC + 71 + 47  =  180

<DBC  =  180-118

<DBC  =  62

x  =  62

Example 6 :

Solution :

Sum of opposite angles are supplementary.

<DAB + <BCD  =  180 

(a+30) + <BCD  =  180  -----(1)

<BCD + <BCE  =  180

<BCD + 110  =  180

<BCD  =  70

By applying the value of <BCD in (1), we get

(a+30) + 70  =  180

a+100  =  180

a  =  80

So, the value of a is 80.

Example 7 :

Solution :

Since the two sides are parallel, a and 60 are co-interior angles.

So, a is 63.

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