21 Cyclic Quadrilaterals: 21.1 Angles in A Circle
21 Cyclic Quadrilaterals: 21.1 Angles in A Circle
21 Cyclic Quadrilaterals: 21.1 Angles in A Circle
After studying this lesson you will acquire knowledge about the following : Proof and application of the theorem and its converse which states that the opposite angles of a cyclic quadrilateralare supplementary. Proof and application of the theorem and its converse which states that the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
O A D B
AO B = 2 ACB
A O B (Reflex) = 2 A D B
(2) Theorem : In a circle, the angles in the same segment are equal
T S R P Q
O X Y
P T Q = P SQ = P R Q 274
In the above diagrams, the points A,B,C and D and the points P,Q,R,S and T and the points X,Yand Z are points on the same circle. Thus the pointslying on the circumference of a circle are called concyclic. The converse of the second theorem mentioned above, is a useful theorem for concyclic points. Theorem : If a stright line subtends equal angles at two points lying on the same side of the straight line, then the two points and the two end points of the straight line are concyclic. D C
In the figure given above, the points C and D are on the same side of the straightline AB. According to the theorem
B C
D P Q
L N M
In the above diagram ABCD, PQRS and KLMN are cyclic quadrilaterals. 275
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Theorem The opposite angles of a cyclic quadrilateralare supplementary. Now let us prove this theorem
A
O D B C
Data T.P.T
and A B C + A D C = 180 0 Construction:- Mark O as the centre of the circle and draw OB and OD Proof :- B O D = 2 B A D (Angle subtended at the centre is twice the angle subtended at the circumference) BO D (reflex) = 2 BC D (Angles subtended at the centre is twice the angle subtended at the circumference) BO D + BO D (reflex) = 2 BA D + 2 BC D But BO D + BO D (reflex) = 3600 (Angles at a point)
2 B A D + 2 BC D = 360 0 B A D + B C D = 180 0
But
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This theorem can be proved also by using the knowledge of isosceles triangles Since the radii of a circle are equal the triangles AOB, BOC, COD and AOD are A isosceles triangles
ad
a
B
A BC + A D C 1800
Similarly
BA D BC D 1800 The converse of this theorem too can be used as a theorem. ie If a pair of opposite angles of aquadrilateral is supplementary, then the quadrilateral is a cyclic quadrilateral.
x 1800 -750
x 750 2a
2a 1200 277
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(ii)
A m
400
m 600 1000
600
n 800
Example 2 The vertices of the parallelogram ABCD are concyclic. Prove thatABCD is a rectangle.
> >
>
> >
Data :- The vertices of the parallelogram ABCD are concyclic. T.P.T :- ABCD is a rectangle Proof :- Since ABCD is a cyclic quadrilateral
>
C
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Exercise 21.1 (1) In these digrams O denotes the centre of each circle. Find the angles marked by symbols.
700 x
1000
1300 3x y
R
x 700
1100
700 O
m
R
c
y 380
Q
410
320
S O
840
x 500
P Q R
(2) In the diagram, the vertices of the pentagon ABCDE lie an a circle with centre O. AD is a diameter of the circle. If
E
D A E = 30 0 and B A C = 250 , AB = BC and DC = DE find the values of the angles of the pentagon.
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3) In the cyclic quadrilateral ABCD, AB // DC. Prove that A BC = BA D. (4) In the cyclic quadrilateral KLMN, K L M = K N M Prove that KM is a diameter of the circle. (5) In triangle PQR, the perpendiculars drawn from Q to PR and R to PQ intersect at O. (See figure) P Prove that, (i) PSOT is a cyclic quadrilateral (ii) QRSTis a cyclic quadrilateral
Q T O R S
(6) In cyclic quadrilateral MALE, C is a point on EL such thatAC = AL. Prove that A M E = A C E .
A
M L E C
(7) In quadrilateral PQRS, the diagonals PR and QS intersect at O. If PQ // SR and SO = OR. Prove that P, Q, R and S are concyclic.
P Q
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(8) In triangle ABC, the bisectors of the interior angles B and C meet at D and the bisectors of the exterior angles at B and C meet at E. Prove that BDCE is a cyclic quadrilateral.
A D B C
21.3 The relationship between the exterior angles and the interior angles of a cyclic quadrilateral.
Activity :- In this diagram, we willfind the angles denoted by symbols.
a = 1800 - 700 = . . .
fb
e
a 70
0
b = 1800 - 1000 = . . . c = 1800 - 700 = . . . d = 1800 - 1000 = . . . e = 1800 - 1100 = . . . f= ... - ... = ...
100 d
We will write the values of interior angles and exterior angles of this cyclic quadrilateral separately. Interior angles Exterior angles
According to the values you have got, see whether there is a relationship between the values of the interior angles and exterior angles. When considering an exterior angle of a quadrilateral, the angle opposite the interior angle which is adjacent to the exterior angle is called the interior opposite angle of the exterior angle mentioned above. 281
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We will complete this table Value of the exterior angle Value of the interior opposite angle
1100 1100 800 500 320
The tableshows that the value of the exterior angle is equalto the value of the interior opposite angle. This relationship is considered as a theorem in geometry. Theorem : In a cyclic quadrilateral, the exterior angle formed by producing a side, is equal to the interior opposite angle. Proof of this theorem
A
D B C E
BC D + D AC BC D + BA D DCE BA D
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This theorem can also be proved by usingthe theorems on angles in a triangle and angles in a circle. a = y (Angles in the same segment) A b = z (Angles in the same segment)
a b
D
a b y z But yz x
x
C
a b x
E
x a b
D C E BA D
The converse of this theorem too can be used as a theorem. Hence if the exterior angle formed by producing a side of a quadrilateralis equalto the interior opposite angle, then the quadrilateral is a cyclic quadrilateral. Example 3 Find the values of x, y and z in the figure.
x
x 750
y z
75 0 42 0
z 630
Q A
Example 4 In the figure, the two circles intersect at A and B. The point Alies on the side QR of triangle PQR. Prove that PSBTis a cyclic quadrilateral.
S B T
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:The two circles are interset at Aand B. The pointA lies on the side QR of triangle PQR :PSBT is a cyclic quadrilateral :Join AB :In the cyclic quadrilateralABSQ, R A B BSQ (Exterior angle = interior opposite angle) In the cyclic quadrilateralABTR,
a z
840
yx
1050
O y
780
b a
1000
30 0
x
85 0
y O 330
450
m
250
a O x 700 350 O
200
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(2) In the given diagram the two circles intersect eact other at L and Q KLM and PQR are two straight lines.Name an angle (i) equal to angle K L Q (ii) supplementary to K L Q
(3) BC is a diameter of a circle with centre O. Side CDof the cyclic quadrilateral ABCD is produced to E. Using the given diagram (i) Name a right angle, giving reasons. (ii) Write the relationship between A D E and A B C givingreasons
0 0 (iii) If A D E 65 and C A D 20
A B O
E D
(4) In the cyclic quadrilateral PQRS, PQ and SR produced meet at T. PT= ST. (i) Prove that QRT is an isosceles triangle. (ii) Prove that PS // QR.
P Q T R S
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(5) In the figure, the two circles intersect at B and E. ABC and DEF are two straight lines. Prove that AD // CF.
B A C
D E F
(6) XYZis a triangle inscribed in a circle. Y the perpendicular drawn from Y Ais to XZ. The perpendicular drawn from X X to YZ meets YA at C and YZ at D and the circle at B. (i) Prove that ACDZis a cyclic quadri lateral. (ii) Prove that Y C B Y B C
Y D B
A C Z
(7) Side AD of the cyclic quadrilateralABCD is produced to E. The bisector PD of angle CDE when produced meets the circle again at Q. Prove that the line QBbisects the angle ABC.
A
Q D C F P
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(8) KLMN is a cyclic quadrilateral. KLand NM when produced meet atA. KN and LM when produced meet at B. If ABNL is a cyclic quadrilateral, prove thatAB is a diameter of the circle passingthrough the points A, B, N and L.
K
L M
B A
287