1B_Ch10(1)

10.1 Angles Relating to Intersecting Lines 1B_Ch10(2) 10.1 Angles Relating to Intersecting Lines A Angles at a Point B Adjacent Angles on a Straight Line C Vertically Opposite Angles Index

10.2 Angles Relating to Parallel Lines 1B_Ch10(3) 10.2 Angles Relating to Parallel Lines A Angles Formed by Two Lines and a Transversal B Angles Formed by Parallel Lines and a Transversal Index

In the right figure, x and y are a pair of adjacent angles. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(4) Example Angles at a Point A) In the right figure, x and y are a pair of adjacent angles. 2. In the right figure, angles a, b, c and d are formed with a common vertex O and each are adjacent to the next. These angles are called angles at a point. Index

3. In the figure, a + b + c + d = 360° 【Reference: ∠s at a pt.】 10.1 Angles Relating to Intersecting Lines 1B_Ch10(5) Example Angles at a Point A) 3. In the figure, a + b + c + d = 360° 【Reference: ∠s at a pt.】 Index 10.1 Index

In (b) and (c), x and y are a pair of adjacent angles. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(6) In the following figures, determine x and y are a pair of adjacent angles. (a) x y (b) x y (c) y x In (b) and (c), x and y are a pair of adjacent angles. Key Concept 10.1.1 Index

Find r in the figure. r + 140° + 70° + 90° = 360° r + 300° = 360° 10.1 Angles Relating to Intersecting Lines 1B_Ch10(7) Find r in the figure. r + 140° + 70° + 90° = 360° ∠s at a pt. r + 300° = 360° r = 60° Index

Find the three marked angles in the figure. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(8) Find the three marked angles in the figure. 3p + 4p + 5p = 360° ∠s at a pt. 12p = 360° p = 30° Fulfill Exercise Objective Find unknown angles using ‘angles at a point’. ∴ 3p = 90°, 4p = 120°, 5p = 150° ∴ The three marked angles in the figure are 90°, 120° and 150°. Key Concept 10.1.2 Index

Adjacent Angles on a Straight Line B) 10.1 Angles Relating to Intersecting Lines 1B_Ch10(9) Example Adjacent Angles on a Straight Line B) In the right figure, AOD is a straight line. We call a, b and c adjacent angles on a straight line. 2. In the figure, a + b + c = 180°. 【Reference: adj. ∠s on st. line】 Index

Adjacent Angles on a Straight Line B) 10.1 Angles Relating to Intersecting Lines 1B_Ch10(10) Example Adjacent Angles on a Straight Line B) 3. If a + b + c = 180°, then the points A, O and B must lie on the same straight line. Index 10.1 Index

Find x in the figure. In the figure, x + 130° = 180° x = 50° 10.1 Angles Relating to Intersecting Lines 1B_Ch10(11) Find x in the figure. In the figure, x + 130° = 180° adj. ∠s on st. line x = 50° Index

In the figure, XOY is a straight line. Find . 10.1 Angles Relating to Intersecting Lines 1B_Ch10(12) In the figure, XOY is a straight line. Find . 32° + 90° +  = 180° adj. ∠s on st. line  + 122° = 180°  = 58° Fulfill Exercise Objective Find unknown angles using ‘adjacent angles on a straight line’. Index

In the figure, AOB is a straight line. Find the largest marked angle. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(13) In the figure, AOB is a straight line. Find the largest marked angle. 3z + 2z + z = 180° adj. ∠s on st. line 6z = 180° z = 30° Fulfill Exercise Objective Find unknown angles using ‘adjacent angles on a straight line’. ∠AOC is the largest angle and ∠AOC = 3z = 3 × 30° = 90° Index

(a) Express  in terms of . (b) If  = 32°, find . 10.1 Angles Relating to Intersecting Lines 1B_Ch10(14) In the figure below, a light ray SP strikes the mirror HK at the point P, then it reflects in the direction PR. Suppose ∠SPH = ∠RPK = , ∠SPR = . (a) Express  in terms of . (b) If  = 32°, find . Index

(a) Since HPK is a straight line, 10.1 Angles Relating to Intersecting Lines 1B_Ch10(15) Back to Question (a) Since HPK is a straight line,  +  +  = 180° adj. ∠s on st. line ∴  = 180° – 2 (b) Since  = 32°,  = 180° – 2 × 32° = 116° Fulfill Exercise Objective Find unknown angles using ‘adjacent angles on a straight line’. Key Concept 10.1.3 Index

∴ Yes, ACE is a straight line. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(16) The figure shows two set squares ABC and CDE, in which ∠ACB = ∠ECD = 90°. Is ACE a straight line? ∴ ∠ACB + ∠ECD = 90° + 90° = 180° ∴ Yes, ACE is a straight line. Index

It is known that ∠ACB = 60°, ∠GCF = 90° and ∠ECD = 30°. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(17) It is known that ∠ACB = 60°, ∠GCF = 90° and ∠ECD = 30°. Is BCD a straight line? A B C D E G F 60o 30o ∴ ∠ACB + ∠GCF + ∠ECD = 60° + 90° + 30° = 180° ∴ Yes, BCD is a straight line. Key Concept 10.1.4 Index

Vertically Opposite Angles C) 10.1 Angles Relating to Intersecting Lines 1B_Ch10(18) Example Vertically Opposite Angles C) In the right figure, the lines AB and CD interest at point O. We call a and b a pair of vertically opposite angles. 2. In the figure, a = b. 【Reference: vert. opp. ∠s】 Index 10.1 Index

Find x in the figure. In the figure, x = 45° 10.1 Angles Relating to Intersecting Lines 1B_Ch10(19) Find x in the figure. In the figure, x = 45° vert. opp. ∠s Index

Consider the straight line CG. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(20) In the figure, the straight lines AE, BF and CG intersect at point O, AE  CG. Find p. In the figure, ∠BOA = 4p. vert. opp. ∠s Consider the straight line CG. ∴∠COB + ∠BOA + 90° = 180° adj. ∠s on st. line p + 4p + 90° = 180° 5p = 90° p = 18° Fulfill Exercise Objective Find unknown angles using ‘vertically opposite angles’. Index

In the figure, CAD, CBE and GABF are straight lines. Find x and y. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(21) In the figure, CAD, CBE and GABF are straight lines. Find x and y. ∴ GF is a straight line. ∴ (40° – x) + 142° = 180° adj. ∠s on st. line x = 40° + 142° – 180° = 2° ∴∠ABC = 40° – x = 38° Index

GF and CD are intersecting straight lines. 10.1 Angles Relating to Intersecting Lines 1B_Ch10(22) Back to Question ∴ GF and CD are intersecting straight lines. ∴ ∠BAC = ∠DAG = 76° vert. opp. ∠s ∴ ∠BAC + ∠ABC + ∠BCA = 180° ∠sum of  ∴ 76° + 38° + 3y = 180° 114° + 3y = 180° 3y = 66° y = 22° Fulfill Exercise Objective Miscellaneous questions. Key Concept 10.1.5 Index

Angles Formed by Two Lines and a Transversal A) 10.2 Angles Relating to Parallel Lines 1B_Ch10(23) Angles Formed by Two Lines and a Transversal A) When two lines AB and CD are cut by another line TS, the line TS is called a transversal of AB and CD. Index

Angles Formed by Two Lines and a Transversal A) 10.2 Angles Relating to Parallel Lines 1B_Ch10(24) Example Angles Formed by Two Lines and a Transversal A) Each of a and e ; b and f ; c and g ; d and h is a pair of corresponding angles. Each of c and e ; d and f is a pair of alternate angles. Each of c and f ; d and e is a pair of interior angles on the same side of the transversal. Index 10.2 Index

L is a transversal of lines m1 and m2. Find all 10.2 Angles Relating to Parallel Lines 1B_Ch10(25) L is a transversal of lines m1 and m2. Find all a b c d e f g h m1 m2 L (a) corresponding angles, (b) alternate angles and (c) interior angles on the same side of the transversal in the figure. (a) a and e, b and f, c and g, d and h (b) d and f, c and e Key Concept 10.2.1 (c) d and e, c and f Index

Angles Formed by Parallel Lines and a Transversal B) 10.2 Angles Relating to Parallel Lines 1B_Ch10(26) Example Angles Formed by Parallel Lines and a Transversal B) Corresponding Angles of Parallel Lines In the figure, if AB // CD, then a = b. 【Reference: corr. ∠s, AB // CD】 Index

Angles Formed by Parallel Lines and a Transversal B) 10.2 Angles Relating to Parallel Lines 1B_Ch10(27) Example Angles Formed by Parallel Lines and a Transversal B) 2. Alternate Angles of Parallel Lines In the figure, if AB // CD, then a = b. 【Reference: alt. ∠s, AB // CD】 Index

Angles Formed by Parallel Lines and a Transversal B) 10.2 Angles Relating to Parallel Lines 1B_Ch10(28) Example Angles Formed by Parallel Lines and a Transversal B) 3. Interior Angles on the Same Side of the Transversal In the figure, if AB // CD, then a + b = 180°. 【Reference: int. ∠s, AB // CD】 Index 10.2 Index

Find a in the figure. In the figure, a = 54° 10.2 Angles Relating to Parallel Lines 1B_Ch10(29) Find a in the figure. In the figure, a = 54° corr. ∠s, AB // CD Index

Find x in the figure. ∠AEF + 60° = 180° ∠AEF = 120° ∴ x = ∠AEF = 120° 10.2 Angles Relating to Parallel Lines 1B_Ch10(30) Find x in the figure. x 60° A B C D E F ∠AEF + 60° = 180° adj. ∠s on st. line ∠AEF = 120° ∴ x = ∠AEF = 120° corr. ∠s, AB // CD Key Concept 10.2.2 Index

Find x in the figure. In the figure, x = 135° 10.2 Angles Relating to Parallel Lines 1B_Ch10(31) Find x in the figure. In the figure, x = 135° alt. ∠s, AB // CD Index

Find y in the figure. ∠BGD + 95° = 180° ∠BGD = 85° ∴ y = ∠BGD = 85° C 10.2 Angles Relating to Parallel Lines 1B_Ch10(32) Find y in the figure. 95° E A C D G B F y ∠BGD + 95° = 180° adj. ∠s on st. line ∠BGD = 85° ∴ y = ∠BGD = 85° alt. ∠s, AB // CD Key Concept 10.2.3 Index

Find x in the figure. In the figure, x + 50° = 180° ∴ x = 130° 10.2 Angles Relating to Parallel Lines 1B_Ch10(33) Find x in the figure. In the figure, x + 50° = 180° int. ∠s, PQ // RS ∴ x = 130° Index

Find the unknown angles a and b in the figure. 10.2 Angles Relating to Parallel Lines 1B_Ch10(34) Find the unknown angles a and b in the figure. a + 110° = 180° int. ∠s, AD // BE Fulfill Exercise Objective Find unknown angles associated with parallel lines. a = 70° ∠EBC = ∠DAC corr. ∠s, AD // BE b + 60° = 110° b = 50° Index

Find the unknown angles x and y in the figure. 10.2 Angles Relating to Parallel Lines 1B_Ch10(35) Find the unknown angles x and y in the figure. With the notation in the figure, x = 54° alt. ∠s, AB // CD Index

m + 120° = 180° m = 60° In CQR, m + x + y = 180° 10.2 Angles Relating to Parallel Lines 1B_Ch10(36) Back to Question m + 120° = 180° int. ∠s, AB // CD m = 60° In CQR, m + x + y = 180° ∠sum of  ∴ 60° + 54° + y = 180° y = 66° Fulfill Exercise Objective Find unknown angles associated with parallel lines. Index

10.2 Angles Relating to Parallel Lines 1B_Ch10(37) In the figure, AB, QR and CD are parallel while PQ and RS are also parallel. If ∠RSA = 57°, find ∠QPD. Index

With the notation in the figure, 10.2 Angles Relating to Parallel Lines 1B_Ch10(38) Back to Question With the notation in the figure, m + 57° = 180° int. ∠s, AB // QR m = 123° ∴ n = m alt. ∠s, PQ // RS ∴ n = 123° ∴  + n = 180° int. ∠s, QR // CD  + 123° = 180° Fulfill Exercise Objective Find unknown angles associated with parallel lines.  = 57° ∴ ∠QPD = 57° Index

Find reflex ∠TRS in the figure. 10.2 Angles Relating to Parallel Lines 1B_Ch10(39) Find reflex ∠TRS in the figure. Draw from T the line AT such that AT // PQ. Index

With the notation in the figure, 10.2 Angles Relating to Parallel Lines 1B_Ch10(40) Back to Question With the notation in the figure, y + 136° = 180° int. ∠s, PQ // AT y = 44° x + y = 2x – 36° ∴ alt. ∠s, RS // AT ∴ x + 44° = 2x – 36° 44° + 36° = 2x – x x = 80° Fulfill Exercise Objective Add auxiliary lines to find unknown angles. Reflex ∠TRS = 360° – (2x – 36°) = 360° – 2 × 80° + 36° = 236° Key Concept 10.2.4 Index

Conditions for Two Lines to be Parallel 10.3 Conditions for Parallel Lines 1B_Ch10(41) Example Conditions for Two Lines to be Parallel 1. In the figure, if a = b, then AB // CD. 【Reference: corr. ∠s equal】 Index

Conditions for Two Lines to be Parallel 10.3 Conditions for Parallel Lines 1B_Ch10(42) Example Conditions for Two Lines to be Parallel 2. In the figure, if a = b, then AB // CD. 【Reference: alt. ∠s equal】 Index

Conditions for Two Lines to be Parallel 10.3 Conditions for Parallel Lines 1B_Ch10(43) Example Conditions for Two Lines to be Parallel 3. In the figure, if a + b = 180°, then AB // CD. 【Reference: int. ∠s supp.】 Index

In the figure, is PQ // RS? ∠QBC ∴ = ∠SCD = 120° ∴ Yes, PQ // RS. 10.3 Conditions for Parallel Lines 1B_Ch10(44) In the figure, is PQ // RS? ∠QBC ∴ = ∠SCD = 120° ∴ Yes, PQ // RS. corr. ∠s equal Index

In the figure, is RS // TU? ∠MNR + 80° = 180° ∠MNR = 100° ∠MNR ∴ 10.3 Conditions for Parallel Lines 1B_Ch10(45) 100° M S N R T U 80° In the figure, is RS // TU? ∠MNR + 80° = 180° adj. ∠s on st. line ∠MNR = 100° ∠MNR ∴ = ∠NUT = 100° ∴ Yes, RS // TU. corr. ∠s equal Key Concept 10.3.1 Index

In the figure, is AB // CD? ∠AEF ∴ = ∠DFE = 60° ∴ Yes, AB // CD. 10.3 Conditions for Parallel Lines 1B_Ch10(46) In the figure, is AB // CD? ∠AEF ∴ = ∠DFE = 60° ∴ Yes, AB // CD. alt. ∠s equal Index

In the figure, is AB // CD? ∠ABC + 273° = 360° ∠ABC = 87° ∠ABC ∴ 10.3 Conditions for Parallel Lines 1B_Ch10(47) In the figure, is AB // CD? 273° A B C D 87° ∠ABC + 273° = 360° ∠s at a pt. ∠ABC = 87° ∠ABC ∴ = ∠BCD = 87° ∴ Yes, AB // CD. alt. ∠s equal Key Concept 10.3.2 Index

In the figure, is AB // DC? = 140° + 40° = 180° ∠ABC + ∠DCB ∴ 10.3 Conditions for Parallel Lines 1B_Ch10(48) In the figure, is AB // DC? = 140° + 40° = 180° ∠ABC + ∠DCB ∴ Yes, AB // DC. int. ∠s supp. Index

10.3 Conditions for Parallel Lines 1B_Ch10(49) In the figure, PQR and ABC are both straight lines, AP // BQ, ∠AQP = 2x + 18°, ∠PAQ = 30°, ∠QAB = 78° and ∠AQB = x. Is PR parallel to AC? Index

x = 30° = 2x + 18° = 2 × 30° + 18° = 78° ∠PQA ∠PQA ∴ = ∠QAB = 78° ∴ 10.3 Conditions for Parallel Lines 1B_Ch10(50) Back to Question x = 30° alt. ∠s, AP // BQ = 2x + 18° = 2 × 30° + 18° = 78° ∠PQA ∠PQA ∴ = ∠QAB = 78° ∴ Yes, PR // AC. alt. ∠s, equal Fulfill Exercise Objective Determine if two given lines are parallel. Key Concept 10.3.2 Index