Exercise 15.2
Question 1
If O is the centre of the circle, find the value of x in each of the following figures (using the given information):
Sol :
From the figure
(i) ABCD is a cyclic quadrilateral
Now arc BD subtends ∠BOD at the centre and ∠BAD at the remaining part of the circle.
∴∠BOD=2∠BAD=2x
⇒2x=150∘⇒x=75∘
(ii) ∠BCD+∠DCE=180∘
(Linear pair)
Figure to be added
⇒∠BCD+80∘=180∘
⇒∠BCD=180∘−80∘=100∘
∵Arc BAD subtends reflex ∠BOD at the centre and ∠BCD at the remaining part of the circle
∴ Reflex ∠BOD=2∠BCD
⇒x0=2×100∘=200∘
(iii) In ΔACB
Figure to be added
∠CAB+∠ABC+∠ACB=180∘
(Angles of a triangle)
But ∠ACB=90∘ (Angle in a semicircle)
∴25∘+90∘+∠ABC=180∘
⇒115∘+∠ABC=180∘
⇒∠ABC=180∘−115∘=65∘
∵ABCD is a cyclic quadrilateral
∴∠ABC+∠ADC=180∘
(Opposite angles of a cyclic quadrilateral)
⇒65°+x°=180°
⇒x°=180°-65°=115°
Question 2
(a) In the figure (i) given below, O is the centre of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC.
(b) In the figure (i) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.
Sol :
(a) Given, ∠AOC = 150° and AD = CD
We know that an angle subtends by an arc of a circle
at the centre is twice the angle subtended by the same arc
at any point on the remaining part of the circle.
(i) ∴∠AOC=2×∠ABC
∠ABC=∠AOC2=150∘2=75∘
(ii) From figure,ABCD is a cyclic quadrilateral
∴∠ABC+∠ADC=180∘
∵ Sum of opposite angles in a cyclic quadrilateral is 180∘
⇒75∘+∠ADC=180∘
∠ADC=180∘−75∘
⇒∠ADC=105∘
(b) (i)∵AC is the diameter of the circle.
∴∠ABC=90∘ (Angle in a semi circle)
⇒∠BAD+75∘=180∘
(∵∠BCD=75∘)
⇒∠BAD=180∘−75∘=105∘
But ∠EAF=∠BAD
(Vertically opposite angles)
∴∠EAF=105∘
Question 3
(a) In the figure, (i) given below, if ∠DBC = 58° and BD is a diameter of the circle, calculate:
(i) ∠BDC (ii) ∠BEC (iii) ∠BAC
(b) In the figure (if) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find:
(i) ∠CAD (ii) ∠CBD (iii) ∠ADC (2008)
Sol :
(a) ∠DBC = 58°
BD is diameter
∠DCB = 90° (Angle in semi circle)
(i) In ∆BDC
∠BDC + ∠DCB + ∠CBD = 180°
∠BDC = 180°- 90° – 58° = 32°
(ii) ∠BEC=180∘−32∘=148∘
(opp. angle of cyclic quadrilateral)
(iii) ∠BAC=∠BDC=32∘
(Angles in same segment)
(b) In the figure, AB‖DC
∠BCE=80∘ and ∠BAC=25∘
ABCD is a cyclic quadrilateral and DC is produced to E
(i) ∴Ext. ∠BCE= Interior ∠A
⇒80∘=∠BAC+∠CAD
⇒80∘=25∘+∠CAD
∴∠CAD=80∘−25∘=55∘
(ii) But ∠CAD=∠CBD
∴∠CBD=55°
(iii) ∠BAC=∠BDC
(Angles in the same segment)
∴∠BDC=25° (∵∠BAC=25°)
Now AB||DC and BD is the transversal
∴∠BDC=∠ABD (Alternate angles)
∠ABD=25°
∴∠ABC=∠ABD+∠CBD
=25°+55°=80°
But ∠ABC+∠ADC=180°
(Opposite angles of a cyclic quadrilateral)
⇒80°+∠ADC=180°
⇒∠ADC=180°-80°=100°
Question 4
(a) In the figure given below, ABCD is a cyclic quadrilateral. If ∠ADC = 80° and ∠ACD = 52°, find the values of ∠ABC and ∠CBD.
(b) In the figure given below, O is the centre of the circle. ∠AOE =150°, ∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.
(a) In the given figure, ABCD is a cyclic quadrilateral
∠ADC = 80° and ∠ACD = 52°
To find the measure of ∠ABC and ∠CBD
∵ABCD is a cyclic quadrilateral
∴∠ABC+∠ADC=180∘
(sum of opposite angles =180∘)
⇒∠ABC+80∘=180∘
⇒∠ABC=180∘−80∘=100∘
In ΔADC,
∠DAC=180°-∠ADC+∠ACD
=180°-(80°+52°)
=180°-132°=48°
But ∠CBD=∠DAC
(Angles in the same segment)
∴∠CBD=48∘
(b) In the given figure, O is the centre of the circle.
∠AOE=150∘,∠DAO=51∘
To find ∠BEC and ∠EBC
ABED is a cyclic quadrilateral
∴ Ext. ∠BEC=∠DAB=51∘
∵∠AOE=150∘
∴ Ref. ∠AOE=360∘−150∘=210∘
Now arc ABE subtends ∠AOE at the centre
and ∠ADE at the remaining part of the circle.
∴∠ADE=12 Ref. ∠AOE=12×210∘=105∘
But Ext. ∠EBC=∠ADE=105∘
Hence ∠BEC=51∘ and ∠EBC=105∘
Question 5
(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC.
(b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find:
(i)∠BAD (ii) DBCD.
Sol :
(a) ADFE is a cyclic quadrilateral
Ext. ∠FEB = ∠ADF
⇒ ∠ADF = 80°
ABCD is a parallelogram
∠B = ∠D = ∠ADF = 80°
or ∠ABC = 80°
(b)In trapezium ABCD, AD || BC
(i) ∠B + ∠A = 180°
⇒ 70° + ∠A = 180°
⇒ ∠A = 180° – 70° = 110°
∠BAD = 110°
(ii) ABCD is a cyclic quadrilateral
∠A + ∠C = 180°
⇒ 110° + ∠C = 180°
⇒ ∠C = 180° – 110° = 70°
∠BCD = 70°
Question 6
(a) In the figure given below, O is the centre of the circle. If ∠BAD = 30°, find the values of p, q and r.
(b) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate
(i) ∠QBC (ii) ∠BCP
(a) (i) ABCD is a cyclic quadrilateral

∴∠A+∠C=180∘
⇒30∘+p=180∘
⇒p=180∘−30∘=150∘
(ii) Arc BD subtends ∠BOD at the centre and ∠BAD at the remaining part of the circle
∴∠BOD=2∠BAD
⇒q=2×30∘=60∘
∠BAD and ∠BED are in the same segment of the circle
∴∠BAD=∠BED⇒30∘=r
⇒r=30∘
(b) Join PQ
AQPD is a cyclic quadrilateral.

∴∠A+∠QPD=180∘
⇒80∘+∠QPD=180∘
⇒∠QPD=180∘−80∘=100∘
and ∠D+∠AQP=180∘
⇒84∘+∠AQP=180∘
⇒∠AQP=180∘−84∘=96∘
Now PQBC is a cyclic quadrilateral
∴ Ext. ∠QPD=∠QBC
⇒∠QBC=100∘
and ext. ∠AQP=∠BCP
⇒∠BCP=96∘
Question 7
(a) In the figure given below, PQ is a diameter. Chord SR is parallel to PQ . Given ∠PQR = 58°, calculate (i) ∠RPQ (ii) ∠STP
(T is a point on the minor arc SP)
(b) In the figure given below, if ∠ACE = 43° and ∠CAF = 62°, find the values of a, b and c (2007)
(a) In ∆PQR,
∠PRQ = 90° (Angle in a semi circle) and ∠PQR = 58°
∠RPQ = 90° – ∠PQR = 90° – 58° = 32°
SR || PQ (given)
∴∠SRP=∠RPQ=32∘ (alternate angles)
Now PRST is a cyclic quadrilateral,
∴∠STP+∠SRP=180∘
⇒∠STP+32∘=180∘
⇒∠STP=180∘−32∘=148∘
(b) In the given figure,
∠ACE=43∘ and ∠CAF=62∘
Now, in △AEC
∠ACE+∠CAE+∠AEC=180∘
⇒43∘+62∘+∠AEC=180∘
⇒105∘+∠AEC=180∘
⇒∠AEC=180∘−105∘=75∘
But ∠ABD+∠AED=180∘
(Sum of opposite angles of a cyclic quadrilateral)
and ∠AED=∠AEC
⇒a+75∘=180∘
⇒a=180∘−75∘=105∘
(Angles in the alternate segment)
∴∠c=62∘
Now in Δ BAF
∠a+62∘+∠b=180∘
⇒105∘+62∘+b=180∘
⇒167∘+b=180∘
⇒b=180∘−167∘=13∘
Hence a=105°
, b=13°
, c=62°
(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.
(b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.
Sol :
(a) Construction: Join BC, and AC then
ABCD is a cyclic quadrilateral.
In quadrilateral ABCD
∠ABC+∠ADC=180∘
(∵ Sum of opposite ∠s of a cyclic quadrilateral)
∴∠ABC+120∘=180∘⇒∠ABC=60∘
Now in △ABC
∠ABC+∠ACB+∠CAB=180∘
(∵InΔABC,∠ACB=90∘)
(angle in a semicircle)
∴∠CAB+60∘+90∘=180∘
⇒∠CAB=30∘
(b) ∠DCF=∠BCE=x
(Vertically opposite angle)
Now In ΔDCF
Ext. ∠2=x+z ....(i)
and in ΔCBE
Ext. ∠1=x+y ...(ii)
Adding (i) and (ii)
x+y+x+z=∠1+∠2
⇒2x+y+z=180∘
(\because ABCD is a cyclic quad.) ...(iii)
But x : y : z = 3 : 4 : 5
∴xy=34
⇒y=43x
and xz=35
⇒z=53x
∴ Substituting the value of y and z in (iii)
2x+43x+53x=180∘
⇒6x+4x+5x=180∘×3
⇒15x=180∘×3
⇒x=180×315=36∘
∴y=43x=36∘×43=48∘
and z=53x=36∘×53=60∘
Hence , x=36°, y=48° and z=60°
(a) In the figure (i) given below, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If ∠ADE = 70° and ∠OBA = 45°, calculate
(i) ∠OCA (ii) ∠BAC
(b) In figure (ii) given below, ABF is a straight line and BE || DC. If ∠DAB = 92° and ∠EBF = 20°, find :
(i) ∠BCD (ii) ∠ADC.
Sol :
(a) ABCD is a cyclic quadrilateral
Figure to be added
Given, ABCD is a quadrilateral inscribed in a circle with centre O. Also, CD is a chord produced to E
∠ADE=70∘ and ∠OBA=45∘
∠ADE+∠ADC=180∘ (Linear pair)
⇒∠ADC=180∘−70∘=110∘
Again, ∠D+∠B=180∘
(Opposite angles of a cyclic quadrilateral are supplementary)
⇒∠OBA+∠OBC=70∘
⇒45∘+∠OBC=70∘
⇒∠OBC=70∘−45∘
⇒∠OBC=25∘
Also, ∠AOC=140∘
(Angle subtended by an arc of a circle at the centre is double than the angle drawn subtended by it at any point on the remaining part of a circle)
Now in ΔOAC
∠OAC+∠OCA+∠AOC=180∘
(∵ Sum of all ∠s of a tiangle)
⇒∠OAC+∠OCA+140∘=180∘
⇒∠OAC+∠OAC+140∘=180∘
OA= OC, radii of same circle implies that angle opposite to equal sides are equal)
⇒2∠OAC=40∘⇒∠OAC=20∘
Similarly in ΔOAB
⇒∠OAB=∠OBA=45∘
(∵ OA=OB, radii of same circle )
Now ∠BAC=∠OAB+∠OAC
∠BAC=45∘+20∘
∠BAC=65∘
(b) ABCD is a cyclic quadrilateral
∴∠BCD+∠DAB=180∘
⇒∠BCD+92∘=180∘
⇒∠BCD=180∘−92∘=88∘.
Again Ext. ∠CBF=∠ADC
⇒∠ADC=∠CBF=∠CBE+∠EBF
=∠BCD+∠EBF
(∵DC‖BE and alternate angles are equal)
=88∘+20∘=108∘
(a) In the figure (ii) given below, PQRS is a cyclic quadrilateral in which PQ = QR and RS is produced to T. If ∠QPR = 52°, calculate ∠PST.
(b) In the figure (ii) given below, O is the centre of the circle. If ∠OAD = 50°, find the values of x and y.
Sol :
(a) PQRS is a cyclic quadrilateral in which
PQ = QR
∴ in ΔPQR
∠QPR=∠QRP=52∘
∴∠PQR=180∘−(∠QPR+∠QRP)
=180∘−(52∘+52∘)=180∘−104∘=76∘
Now in cyclic quad. PQRS,
Ext. ∠PST=∠PQR=76∘
(b) ABCD is a cyclic quadrilateral
∴∠OAD+∠BCD=180∘
⇒50∘+x=180∘
⇒x=180∘−50∘=130∘
In ΔOAD,OA=OD
(Radii of the same circle)
∠ODA=∠OAD=50∘
Now Ext. ∠BOD=∠ODA+∠OAD
⇒y=50∘+50∘=100∘
(a) In the figure (i) given below, O is the centre of the circle. If ∠COD = 40° and ∠CBE = 100°, then find :
(i) ∠ADC
(ii) ∠DAC
(iii) ∠ODA
(iv) ∠OCA.
(b) In the figure (ii) given below, O is the centre of the circle. If ∠BAD = 75° and BC = CD, find :
(i) ∠BOD
(ii) ∠BCD
(iii) ∠BOC
(iv) ∠OBD (2009)
(a) (i) ∴ ABCD is a cyclic quadrilateral.
∴ Ext. ∠CBE = ∠ADC
⇒ ∠ADC = 100°
(ii) Arc CD subtends ∠COD at the centre
and ∠CAD at the remaining part of the circle
∴ ∠COD = 2 ∠CAD
⇒∠CAD=12∠COD=12×40∘=20∘
(iii) In ΔCOD, OC=OD
(radii of the same circle)
∴∠CDO=∠DCO
But ∠CDO+∠DCO+∠COD=180∘
⇒∠CDO+∠CDO+∠COD=180∘
⇒2∠CDO+40∘=180∘
⇒2∠CDO=180∘−40∘=140∘
∴∠CDO=140∘2=70∘
Now ∠ODA=∠ADC−∠CDO
=100∘−70∘=30∘
(iv) ∠OCA=∠OCD−∠ACD
=70∘−{180∘−∠ADC−∠CAD}
=70∘−{180∘−100∘−20∘}
=70∘−{180∘−120∘}=70∘−60∘=10∘
(b) Given ∠BAD=75∘, chord BC= chord CD
Construction : Join OC and BD
(i) ∠BOD=2×∠BAD
∠BOD=2×75∘=150∘
(ii) In quadrilateral ABCD,
∠BCD+∠BAD=180∘
(Sum of opposite ∠s in quadrilateral)
(iii) ∠BOC=12∠BOD
∠BOC=12×150∘=75∘
(iv) In ΔOBD
∠OBD+∠ODC+∠BOD=180∘
(Sum of all ∠s of a triangle is 180∘ )
⇒∠OBD+∠ODC+150∘=180∘
⇒∠OBD+∠OBD+150∘=180∘
(∴BC=CD)
⇒2∠OBD=30∘
⇒∠OBD=15∘
In the given figure, O is the centre and AOE is the diameter of the semicircle ABCDE. If AB = BC and ∠AEC = 50°, find :
(i) ∠CBE
(ii) ∠CDE
(iii) ∠AOB.
Prove that OB is parallel to EC.
Sol :
In the given figure,
O is the centre of the semi-circle ABCDE
and AOE is the diameter. AB = BC, ∠AEC = 50°
To prove : OB||EC and find
(i) ∠CBE,
(ii) ∠CDE and ∠AOB
Proof : AECB is a cyclic quadrilateral
∴∠AEC+∠ABC=180°
⇒50∘+∠ABE+∠EBC=180∘
⇒50∘+90∘+∠EBC=180∘
⇒140∘+∠EBC=180∘
(∵∠ABE is in a semi-circle)
⇒∠EBC=180∘−140∘=40∘
But BEDC is a cyclic quadrilateral
∴∠EBC+∠CDE=180∘
40∘+∠CDE=180∘
⇒∠CDE=180∘−40∘=140∘
∵AB=BC
∴∠AEB=∠BEC=12∠AEC
=12×50∘=25∘
In Δ OBE
OB=OE (radii of the same circle)
∴∠OBE=∠OEB=25∘
and Ext. ∠AOB=∠OBE+∠OEB
=25∘+25∘=50∘
∴∠AOB=50∘
∵∠AOB=∠OEC (each 50°)
But these are corresponding angles
∴OB‖EC
(a) In the figure (i) given below, ED and BC are two parallel chords of the circle and ABE, ACD are two st. lines. Prove that AED is an isosceles triangle.
(b) In the figure (ii) given below, SP is the bisector of ∠RPT and PQRS is a cyclic quadrilateral. Prove that SQ = RS.
(a) Given: Chord BC || ED,
ABE and ACD are straight lines.
To Prove: ∆AED is an isosceles triangle.
Proof: BCDE is a cyclic quadrilateral.
Ext. ∠ABC = ∠D …(i)
But BC || ED (given)
∴∠ABC=∠E
(corresponding angle) ...(ii)
From (i) and (ii)
∠D=∠E
∴AE=AD
(sides opposite to equal angles)
Hence, ΔAED is an isosceles triangle Q.E.D
(b) Given : In a circle , PQRS is a cyclic quadrilateral . QP is produced to T sucht that PS is the bisector of ∠RPT
To prove : SQ=RS
Proof : ∵PQRS is a cyclic quadrilateral
∴Ext.∠TPS=∠SRQ...(i)
and ∠RPS=∠RQS..(ii)
(Angles in the same segment)
But ∠TPS=∠RPS
(∵PS is the bisector of ∠TPR)
∴from (i) and (ii)
∠SRQ=∠RQS
Now in ΔSQR,
∠SRQ=∠RQS proved
∴ SQ=RS
(sides opposite to equal angles)
Q.E.D
To prove : SQ=RS
Proof : ∵PQRS is a cyclic quadrilateral
∴ Ext. ∠TPS=∠SRQ ...(i)
and ∠RPS=∠RQS...(ii)
(∵ PS is the bisector of ∠TPR)
∴ from (i) and (ii)
∠SRQ=∠RQS
Now in ΔSQR,
∴ SQ=RS
(sides opposite to equal angles)
Q.E.D
In the given figure, ABC is an isosceles triangle in which AB = AC and circle passing through B and C intersects sides AB and AC at points D and E. Prove that DE || BC.
Sol :
In the given figure,
∆ABC is an isosceles triangle in which AB = AC.
A circle passing through B and C intersects
sides AB and AC at D and E.
To prove: DE || BC
Construction : Join DE.
∵ AB = AC
∠B = ∠C (angles opposite to equal sides)
But BCED is a cyclic quadrilateral
Ext. ∠ADE = ∠C
= ∠B (∵ ∠C = ∠B)
But these are corresponding angles
DE || BC
Hence proved.
Question 15
(a) Prove that a cyclic parallelogram is a rectangle.
(b) Prove that a cyclic rhombus is a square.
Sol :
(a) ABCD is a cyclic parallelogram.
To prove: ABCD is a rectangle
Proof : ABCD is a parallelogram
∠A = ∠C and ∠B = ∠D
(opposite angles are equal)
∴ABCD is a cyclic quadrilateral
∴∠A+∠C=180∘
⇒∠A+∠A=180∘ (∵∠A=∠C)
∴∠C=∠A=90∘
Similarly, we can prove that
∠B=∠D=90∘
∴ABCD is a rectangle
(b) Given : ABCD is a cyclic rhombus
To prove : ABCD is a square
Proof : ∵ABCD is a rhombus
∴∠A=∠C and ∠B=∠D
(opposite angles are equal)
But ABCD is a cyclic quadrilateral
∴∠A+∠C=180∘ and ∠B+∠D=180∘
⇒∠A+∠A=180∘ (∵∠A=∠C)
∴∠C=∠A=90∘
Similarly we can prove that
∠B=∠D=90∘
∴ ABCD is a square.
In the given figure, chords AB and CD of the circle are produced to meet at O. Prove that triangles ODB and OAC are similar. Given that CD = 2 cm, DO = 6 cm and BO = 3 cm, area of quad. CABD
calculate AB. Also find area of quad. CABD area of Δ OAC
Sol :
In the given figure, AB and CD are chords of a circle.
They are produced to meet at O.
To prove : (i) ∆ODB ~ ∆OAC
If CD = 2 cm, DO = 6 cm, and BO = 3 cm
To find : AB and also area of the
ABCD area of ΔOAC
Construction : Join AC and BD
Proof:
(i) InΔODB and ΔOAC
Ext. ∠ODB=∠A
(Ext. angle of a cyclic quadrilateral is equal to its interior opposite angle) Similarly, ∠OBD=∠C
∠O=∠O (common)
∴Δ ODB ∼Δ OAC (AAA axiom)
(ii) ∴ODOA=OBOC
⇒6OA=36+2
⇒6OA=38
⇒OA=6×83=16
∴AB=OA−OB=16−3=13 cm
(iii) ∵ΔOAC∼ΔODB
∴ area ΔOAC area ΔODB=OC2OB2=8232=649...(i)
Subtracting 1 from both sides, we get
area ΔOAC area ΔODB−1=649−1
area ΔOAC− area ΔODB area ΔODB=64−99
area quadrilateral CABD area ΔOBD=559
and area quadrilateral CABD area ΔOAC=5564 [From (i)]
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