MATHEMATICS
MORE EXERCISES

To prove thata quadrilateral is a cyclic

Question  1

In the diagram ∠ ABC = ∠ DEC = 45°
and ∠ ACB = 30°
Prove that ABDE is a cyclic quadrilateral.
[ A 1. ]

Question  2

In the diagram QOR is the diameter of ⊙O and
ST ⊥ QOR.
Prove points P, S, T and R are concyclic.
[ A 2. ]

Question  3

In the diagram AB ǁ CD and AE = BE.
[ A 3. ]

Question  4

In the diagram ∠ A = 50°; ∠ C = 30°
and ∠ CED = 100°
[ A 4. ]

Question  5

In the diagram ∠ P = 3x°; ∠ Q = x°
∠ R = 2x° and ∠ S = 4x°
[ A 5. ]

Question  6

In the di agram∠ DAE = 34° and ∠ DCF = 146°
[ A 6. ]

Question  7

In the diagram ∠ NLM = 75°; ∠ KMN = 35°
and ∠ KNM = 70°
[ A 7. ]

Question  8

In the diagram AB = AC and ∠ ABD = x°
∠ ACD = x°
Prove ∠ADC = 90° + ½ ∠ BAC.
[ A 8. ]

Question  9

In the diagram ML and ON produced intersect at
K and LN and MO produced intersect at P.
∠ K = 42° en ∠ P = 38°
Calculate the size of ∠ M.
[ A 9. ]

Question  10

In the diagram ABCD is a parallelogram.
∠ BDE = ∠ BDC and DE = DC.
DE and BA produced cut in F. BE intersects DA
in G. FG produced intersects BD in H.

Prove
10.1   ∠ CBD = ∠DBE [ A 10.1 ]
10.2   EABD a cyclic quadrilateral.    [ A 10.2 ]
10.3   FA = FE     [ A 10.3 ]

Question  11

In the diagram PR is the diameter of circle and
PB is a tangent at P.
A is the midpoint of PB. AR and BR intersect
the circle at C and Q respectively.
Prove that
11.1   ∠ RPQ = ∠B [ A 11.1 ]
11.2   ABCQ is a cyclic quadrilateral.   [ A 11.2 ]

$$\text{∠ ABC = ∠ DEC = 45° . . . given} \\ \text{But ∠ DEC is an ext. ∠ of quad. ABDE}\\ \text{Thus ABDE is a cyclic quadrilateral.} \\$$
OR
$$\text{∠A = 180° - (∠B + ∠ E_2) . . . int.\ angles\ of\ Δ ABC } \\ \hspace*{7 mm}\mathrm{= 180° - (45° + 30°)\kern2mm\ } \\ \hspace*{7 mm}\mathrm{= 105°\kern2mm\ } \\ \hspace*{1 mm}\mathrm{∠ BDE = ∠ DEC + ∠ C . . . int.\ angles\ of\ Δ ABC\kern2mm\ } \\ \hspace*{13 mm}\mathrm{= 45° + 30°\kern2mm\ } \\ \hspace*{13 mm}\mathrm{= 75°\kern2mm\ } \\ \hspace*{6 mm}\mathrm{∠ A + ∠ BDE = 105° + 75°\kern2mm\ } \\ \hspace*{13 mm}\mathrm{= 180°\kern2mm\ } \\ \hspace*{3 mm}\mathrm{But\ they\ are\ opp. int. ∠\kern2mm\ } \\ \hspace*{1 mm}\mathrm{∴ ABDE\ is\ a\ cyclic\ quadrilateral\kern2mm\ } \\$$
[ Q 1. ]

$$\text{Punte P, S, T and R are concyclic if they lie} \\ \text{on the circumference of a circle, i.e. if PSTR is }\\ \text{a cyclic quadrilateral.} \\ \text{Thus, prove PSTR is a cyclic quadrilateral.} \\ \hspace*{2 mm}\mathrm{∠ P = 90°\ .\ .\ .\ ∠\ in\ semi-circle\kern2mm\ } \\ \hspace*{2 mm}\mathrm{∠ STR = 90°\ .\ .\ .\ ST ⊥ QR\kern2mm\ } \\ \hspace*{2 mm}\mathrm{∠ P + ∠ STR = 180°\kern2mm\ } \\ \hspace*{2 mm}\mathrm{∴ PSTR\ is\ a\ cyclic\ quadrialteral.\ \ .\ .\ .\kern2mm\ } \\ \hspace*{20 mm}\mathrm{opp. int. ∠'s suppl.\kern2mm\ } \\$$
[ Q 2. ]

$$\hspace*{4 mm}\mathrm{∠ A_1 = ∠ B_1\ .\ .\ .\ AE = BE\kern2mm\ } \\ \hspace*{4 mm}\mathrm{∠ A_1 = ∠ C_2\ .\ .\ .\ alt. ∠'s,\ AB ǁ DC\kern2mm\ } \\ \hspace*{1 mm}\mathrm{∴ ∠ B_1 = ∠ C_2\ .\ .\ .\ both\ =\ ∠ A_1\kern2mm\ } \\ \text{\hspace{5 mm}Both subtended by AD} \\ \text{∴ ABCD is a cyclic quadrilateral.} \\$$
OR
$$\hspace*{4 mm}\mathrm{∠ A_1 = ∠ B_1\ .\ .\ .\ AE = BE\kern2mm\ } \\ \hspace*{4 mm}\mathrm{∠ D_2 = ∠ B_1\ .\ .\ .\ alt. ∠'e,\ AB ǁ DC\kern2mm\ } \\ \hspace*{1 mm}\mathrm{∴ ∠ A_1 = ∠ D_2\ .\ .\ .\ both\ =\ ∠ B_1\kern2mm\ } \\ \text{\hspace{5 mm}Both subtendedn by BC} \\ \text{∴ ABCD is a cyclic quadrilateral.} \\$$
[ Q 3. ]

$$\hspace*{4 mm}\mathrm{∠ D + ∠ CED + ∠ C = 180°\ .\ .\ .\ Σ\ int.\ ∠'s\ Δ CED = 180°\kern2mm\ } \\ \hspace*{4 mm}\mathrm{∠ D = 180° - (∠ CED + ∠ C)\kern2mm\ } \\ \hspace*{11 mm}\mathrm{= 180° - (100° + 30°)\kern2mm\ } \\ \hspace*{11 mm}\mathrm{= 50°\kern2mm\ } \\ \hspace*{11 mm}\mathrm{= ∠ A\kern2mm\ } \\ \text{\hspace{5 mm}BC subtends equal angles.} \\ \text{∴ ABCD is a cyclic quadrilateral.} \\$$
[ Q 4. ]

$$\hspace*{4 mm}\mathrm{∠ P + ∠ Q + ∠ R + ∠ S = 360°\ .\ .\ .\kern2mm\ } \\ \hspace*{22 mm}\mathrm{Σ\ int.\ ∠'s\ quad.\ PQRS = 360°\kern2mm\ } \\ \hspace*{4 mm}\mathrm{3x° + x° + 2x° + 4x° = 360°\kern2mm\ } \\ \hspace*{26 mm}\mathrm{10x° = 360°\kern2mm\ } \\ \hspace*{31 mm}\mathrm{x = 36°\kern2mm\ } \\ \hspace*{1 mm}\mathrm{∠ P = 108°;\ \ ∠Q = 36°;\ \ ∠ R = 72°;\ \ ∠ S = 144°\kern2mm\ } \\ \hspace*{12 mm}\mathrm{∠ P + ∠R = 108° + 72°\kern2mm\ } \\ \hspace*{28 mm}\mathrm{= 180°\kern2mm\ } \\ \text{∴ Sum of interior angles = 180° } \\ \text{∴ PQRS is a cyclic quadripateral.} \\$$
OR
$$\hspace*{1 mm}\mathrm{∠ Q + ∠S = 36° + 144°\kern2mm\ } \\ \hspace*{18 mm}\mathrm{= 180°\kern2mm\ } \\ \text{∴ Sum of opposite interion angles = 180° } \\ \text{∴ PQRS is a cyclic quadrilateral.} \\$$
[ Q 5. ]

$$\hspace*{4 mm}\mathrm{∠ DAB + ∠ DAE = 180°\ .\ .\ .\ BAE\ a\ str.\ line\kern2mm\ } \\ \hspace*{9 mm}\mathrm{∠ DAB + 34° = 180°\kern2mm\ } \\ \hspace*{18 mm}\mathrm{∠ DAB = 146°\kern2mm\ } \\ \hspace*{30 mm}\mathrm{= ∠ DCF\kern2mm\ } \\ \text{∴ Exterior angle = opposite interior angle} \\ \text{∴ ABCD is a cyclic quadrilateral.} \\$$
OR
$$\hspace*{4 mm}\mathrm{∠ DCB + ∠ DCF = 180°\ .\ .\ .\ BCF\ a\ str.\ line\kern2mm\ } \\ \hspace*{7 mm}\mathrm{∠ DCB + 146° = 180°\kern2mm\ } \\ \hspace*{18 mm}\mathrm{∠ DCB = 34°\kern2mm\ } \\ \hspace*{30 mm}\mathrm{= ∠ DAE\kern2mm\ } \\ \text{∴ Exterior angle = interior opposite angle} \\ \text{∴ ABCD is a cyclic quadrilateral.} \\$$
[ Q 6. ]

$$\hspace*{4 mm}\mathrm{∠ MKN + ∠ KMN + ∠ KNM = 180°\ .\ .\ . \kern2mm\ } \\ \hspace*{44 mm}\mathrm{Σ\ int. ∠'s\ Δ\ KNM\ = 180°\kern2mm\ } \\ \hspace*{16 mm}\mathrm{∠ MKN + 35° + 70° = 180°\kern2mm\ } \\ \hspace*{34 mm}\mathrm{∠ MKN = 75°\kern2mm\ } \\ \hspace*{46 mm}\mathrm{= ∠ MLN\kern2mm\ } \\ \text{\hspace{5 mm}MN subtends equal angles} \\ \text{∴ KLMN is a cyclic quadrilateral.} \\$$
[ Q 7. ]

$$\text{∠ ABD = ∠ ACD = x° . . . given} \\ \text{\hspace{5 mm}AD subtends equal angles}\\ \text{∴ ABCD is a cyclic quadrilateral.}\\ \hspace*{1 mm}\mathrm{∠ ACB = ∠ ABC\ .\ .\ .\ AB = AC\kern2mm\ }\\ \hspace*{1 mm}\mathrm{2 × ∠ ACB = 180° - ∠ BAC\kern2mm\ }\\ \hspace*{7 mm}\mathrm{∠ ACB = ½(180° - ∠ BAC)\kern2mm\ }\\ \hspace*{19 mm}\mathrm{= 90° - ½ ∠ BAC\kern2mm\ }\\ \hspace*{7 mm}\mathrm{∠ ADB = ∠ ACB\ .\ .\ .\ AB\ subtends\ equal\ angles\kern2mm\ }\\ \hspace*{7 mm}\mathrm{∠ BDC = ∠ BAC\ .\ .\ .\ BC\ subtends\ equal\ angles\kern2mm\ }\\ \hspace*{7 mm}\mathrm{∠ ADC = ∠ ADB + ∠ BDC\kern2mm\ }\\ \hspace*{19 mm}\mathrm{= (90° - ½ ∠ BAC) + ∠ BAC\kern2mm\ }\\ \hspace*{19 mm}\mathrm{= 90° + ½ ∠ BAC\kern2mm\ }\\$$
[ Q 8. ]

$$\text{\hspace{1 mm}∠ KLN = ∠ NOM . . . ext. ∠ of cyc. quad. LNOM}\\ \text{∠ NOM = ∠ PNO + ∠ P . . . ext. ∠ of Δ PNO}\\ \text{∠ PNO = ∠ KNL . . . vert. opp. ∠'s}\\ \hspace*{11 mm}\mathrm{∠ K + ∠ KLN + ∠ KNL = 180°\ .\ .\ .\kern2mm\ }\\ \hspace*{50 mm}\mathrm{int. ∠'s\ ΔKLN\kern2mm\ }\\ \hspace*{2 mm}\mathrm{42° + ∠ P + ∠ PNO + ∠ KNL = 180°\kern2mm\ }\\ \hspace*{15 mm}\mathrm{42° + 42° + 2 ∠ KNL = 180°\kern2mm\ }\\ \hspace*{33 mm}\mathrm{2 ∠ KNL = 96°\kern2mm\ }\\ \hspace*{35 mm}\mathrm{∠ KNL = 48°\kern2mm\ }\\ \hspace*{4 mm}\mathrm{∠ M = ∠ KNL\ .\ .\ .\ ext.\ ∠\ of\ cyc.\ quad.\ LNOM\kern2mm\ }\\ \hspace*{11 mm}\mathrm{= 48°\kern2mm\ }\\$$
[ Q 9. ]

#### 10.1

$$\hspace*{4 mm}\mathrm{In\ Δ BDE\ and\ Δ BDC\kern2mm\ }\\ \hspace*{11 mm}\mathrm{i)\ DE = DC\ .\ .\ .\ given\kern2mm\ }\\ \hspace*{10 mm}\mathrm{ii)\ ∠ BDE = ∠ BDC\ .\ .\ .\ given\kern2mm\ }\\ \hspace*{9 mm}\mathrm{iii)\ BD = BD\ .\ .\ .\ common\ side\kern2mm\ }\\ \text{\hspace{4 mm}∴ ΔBDE ≡ Δ BDC . . . SHS}\\ \text{\hspace{3 mm}∴ ∠DBE = ∠DBC}\\$$
[ Q 10. ]

#### 10.2

$$\text{\hspace{7 mm}∠ ABD = ∠ BDC . . . alt. ∠'s, AB ǁ DC}\\ \text{\hspace{7 mm}∠ BDC = ∠ BDE . . . given}\\ \text{\hspace{3 mm}∴ ∠ ABD = ∠ BDE}\\ \hspace*{0 mm}\mathrm{Maar\ ∠ DBE = ∠ DBC\ .\ .\ .\ (10.1)\kern2mm\ }\\ \hspace*{9 mm}\mathrm{∠ DBC = ∠ ADB\ .\ .\ .\ alt. ∠'s, AD ǁ BC\kern2mm\ }\\ \hspace*{6 mm}\mathrm{∴ ∠ DBE = ∠ ADB\kern2mm\ }\\ \hspace*{0 mm}\mathrm{∠ ABE + ∠ EBD = ∠ ADE + ∠ ADB\kern2mm\ }\\ \hspace*{11 mm}\mathrm{∴ ∠ ABE = ∠ ADE\kern2mm\ }\\ \text{∴ AE subtends equal angles.}\\ \text{∴ ABDE is a cyclic quadrilateral}\\$$
[ Q 10. ]

#### 10.3

$$\text{∠ ABD = ∠ EDB . . . (10.2)}\\ \text{ΔFBD is an isosceles triangle . . . base ∠'s equal}\\ \hspace*{6 mm}\mathrm{FB = FD\kern2mm\ }\\ \hspace*{6 mm}\mathrm{FA + AB = FE + ED\kern2mm\ }\\ \hspace*{14 mm}\mathrm{AB = CD\ .\ .\ .\ opp. sides\ param.\ ABCD\kern2mm\ }\\ \hspace*{20 mm}\mathrm{= ED\ .\ .\ .\ given\kern2mm\ }\\ \hspace*{15 mm}\mathrm{FA = FE\kern2mm\ }\\$$
[ Q 10. ]

#### 11.1

$$\hspace*{1 mm}\mathrm{∠ PQR = 90°\ .\ .\ .\ angle\ in\ semi-circle\kern2mm\ }\\ \hspace*{1 mm}\mathrm{∠ PQR + ∠PQB = 180°\ .\ .\ .\ RQB str.\ line\kern2mm\ }\\ \hspace*{16 mm}\mathrm{∠ PQB = 90°\kern2mm\ }\\ \text{\hspace{6 mm}In ΔBDE and Δ BDC}\\ \hspace*{9 mm}\mathrm{∠ PQR = ∠ PQB\ .\ .\ .\ each = 90°\kern2mm\ }\\ \hspace*{9 mm}\mathrm{∠ PRQ = ∠ QPB\ .\ .\ .\ tangent\ PAB,\ chord\ PQ\kern2mm\ }\\ \hspace*{6 mm}\mathrm{∴ ∠ RPQ = ∠ B\ .\ .\ .\kern2mm\ }\\ \hspace*{25 mm}\mathrm{remaining\ ∠'s\ Δ PQR\ and\ Δ PQB\kern2mm\ }\\$$
[ Q 11. ]

#### 11.2

$$\text{\hspace{2 mm}Let RC en PQ mekaar in T sny.}\\ \text{\hspace{2 mm}In Δ CTQ and Δ PTR}\\ \text{\hspace{5 mm}∠ CTQ = ∠ PTR . . . vert. opp. ∠'s}\\ \text{\hspace{5 mm}∠ TQC = ∠ PRT . . . subtended by chord PC}\\ \text{∴ ∠ TCQ = ∠ RPT . . . }\\ \text{\hspace{16 mm}remaining angles of ΔCTQ and Δ PTR}\\ \text{∴ ∠ TCQ = ∠ B . . . ∠ RPT = ∠ B (11.1)}\\ \text{∴ ABQC is a cyclic quadrilateral . . .}\\ \text{\hspace{27 mm}ext. ∠ = opp. int. ∠}\\$$
[ Q 11. ]