# if the sum of a pair of opposite angles of a quadrilateral is 180,then show that the quadrilateral is cyclic

Theorem. Sum of the opposite angles of a cyclic quadrilateral is 180°.
Given : A cyclic quadrilateral ABCD
To prove : ∠BAD + ∠BCD = ∠ABC + ∠ADC = 180°
Construction : Draw AC and DB
and ∠BAC = ∠BDC
[Angles in the same segment]
Adding ∠ABC on both the sides, we get
∠ACB + ∠BAC + ∠ABC = ∠ADC + ∠ABC
But ∠ACB + ∠BAC + ∠ABC = 180° [Sum of the angles of a triangle]
∴ ∠ADC + ∠ABC = 180°
∴ ∠BAD + ∠BCD = 360° – (∠ADC + ∠ABC) = 180°.
Hence proved.
Converse of this theorem is also true.
If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.
• -7

wrong proof

• -1

ya it is

• -7

GIVEN :       ABCD is a quadrilateral

TO PROOF:   ANGLE A+ANGLE C=180

PROOF:   arc BCD  subtitute ANGE BOD at the center                                                                                                                                                                  and ANGLE BAD at the point A on the remaining part of the circle

THERE FOR       ANGLE BOD =2* ANGLE BAD ~~~~~~~~~~~(1)      {THE ANGLE SUPTENTED BY AN ARC AT THE CENTER IS

DOUBLE THE ANGE SUBTENDED BY IT AT THE                                                                                                                                                                 REMAINING PART OF THE CIRCLE}                     similarlly major arc  BAD subtentes reflex ANGLE BOD at the center

and ANGLE BCD at the point C  onthe remaining part of the circle

THERE FOR     reflex ANGLE =2* ANGLE BCD ~~~~~~~~~~~~~~(2)

ANGLE BOD +reflex ANGLE BOD=2*ANGLE BAD +2*ANGLE BCD

360/2 =ANGLE BAD + ANGLE BCD

THERE FOR                                                    ANGLE A + ANGLE C=180 ~~~~~~~~~~~(A)

IN quadrilateral ABCD, ANGLE A + ANGLE B + ANGLE C + ANGLE D =360    {ANGLE SUM PROPERTY OF THE QUADRILATERAL}

180+ANGLE B + ANGLE D = 360

ANGLE B + ANGLE D = 360-180

ANGLE B + ANGLE D= 180

THUS IT IS PROVED THAT THE SUM OF EIGHTHER PAIR OF OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL IS 180

)22

and

• -12

If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

##### Proof

Consider a quadrilateral  , such that  and  .

The aim is to prove that points A, B, P and Q lie on the circumference of a circle.

By contradiction. Assume that point P does not lie on a circle drawn through points A, B and Q. Let the circle cut  (or  extended) at point R. Draw  .

(8)

the assumption that the circle does not pass through P, must be false, and A, B, P and Q lie on the circumference of a circle and  is a cyclic quadrilateral.

• -5
Where 8s the picture
• -1
if sum of a pair of opposite angles of a quadrilateral is 180 then it is cyclic
• 2
Gghhhh
• -1
Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic

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