15 More About Probability

11 Angles related to Lines

11 Angles related to Lines

Review Exercise 11 (p. 11.3)

1. a: round angle
b: right angle
c: reflex angle
d: acute angle
e: obtuse angle
f: straight angle

Angle / round angle / round angle / 2 right angles / 2 straight angles
Size / 90° / 180° / 180° / 360°
Type / right angle / straight angle / straight angle / round angle

3. (a)

(b)

Activity

Activity 11.1 (p. 11.7)

1. (a) (i) Yes

(ii)

(b) (i) Yes

(ii)

2. (a) a = b

(b) c = d

Activity 11.2 (p. 11.16)

1. (a) a and c, b and d
(b) a = 130°, b = 50°, c = 130°, d = 50°

2. They are equal.

Activity 11.3 (p. 11.18)

2. They are equal.

Activity 11.4 (p. 11.19)

1. (a)
(b)

Interior
angles on the same side / p and q / r and s
Their sum / /

2. Their sum equals 180°.

Classwork

Classwork (p. 11.15)

1. Corresponding angles:
s and w, t and x, u and y, v and z
Alternate angles:
s and y, v and x
Interior angles on the same side of the transversal:
s and x, v and y

2. Corresponding angles:
a and e, c and g, d and f
Alternate angles:
a and f, b and g, d and e
Interior angles on the same side of the transversal:
a and g, b and f, d and g

Quick Practice

Quick Practice 11.1 (p. 11.5)

Quick Practice 11.2 (p. 11.5)

Quick Practice 11.3 (p. 11.6)

Quick Practice 11.4 (p. 11.7)

(Ðs at a pt.)

Quick Practice 11.5 (p. 11.8)

(vert. opp. Ðs)

Quick Practice 11.6 (p. 11.9)

(vert. opp. Ðs)

(adj. Ðs on st. line)

Quick Practice 11.7 (p. 11.10)

(vert. opp. Ðs)

(adj. Ðs on st. line)

∴

Quick Practice 11.8 (p. 11.17)

(corr. Ðs, RS // PQ)

Quick Practice 11.9 (p. 11.19)

(alt. Ðs, AB // CD)

(adj. Ðs on st. line)

Quick Practice 11.10 (p. 11.21)

(int. Ðs, BA // DC)

Quick Practice 11.11 (p. 11.21)

(int. Ðs, AB // CD)

(corr. Ðs, CD // AB)

Quick Practice 11.12 (p. 11.22)

(alt. Ðs, BF // CD)

(corr. Ðs, AE // BD)

(int. Ðs, AE // BD)

Quick Practice 11.13 (p. 11.23)

∵ (vert. opp. Ðs)

∴ (int. Ðs, AC // DG)
(int. Ðs, AE // BH)

Quick Practice 11.14 (p. 11.24)

Draw a straight line CF such that CF // AB // DE.

Let ∠ACF = x1 and ∠FCD = x2.

(int.∠s, CF // AB)

(int.∠s, CF // DE)

∴

Quick Practice 11.15 (p. 11.29)

(a) ∵

∴ AB // DC (corr. ∠s equal)

(b)
i.e. Interior angles on the same side of the transversal are not supplementary.

∴ DE is not parallel to BF.

Quick Practice 11.16 (p. 11.29)

(vert. opp. Ðs)

∵

∴ AB // CD (int. Ðs supp.)

Alternative Solution

(adj. Ðs on st. line)

∵

∴ AB // CD (corr. Ðs equal)

Alternative Solution

(adj. Ðs on st. line)

∵

∴ AB // CD (alt. Ðs equal)

Quick Practice 11.17 (p. 11.30)

(adj.∠s on st. line)

(int. ∠s, CA // FE)

(∠s at a pt.)

∴ BD // FG (int. ∠s supp.)

Further Practice

Further Practice (p. 11.10)

1. (adj.∠s on st. line)

2. (∠s at a pt.)

3. (vert. opp.∠s)

(adj.∠s on st. line)

4. (adj.∠s on st. line)

(vert. opp.∠s)

Further Practice (p. 11.24)

1. (corr. ∠s, MN // XY)

(alt. ∠s, MN // XY)

2. (int. ∠s, AB // CD)

3. (corr. ∠s, HG // FE)

(vert. opp.∠s)

(int. ∠s, AB // CD)

4.
Draw a line from C to meet EF at H such that HC // FG.

(int. ∠s, AB // EF)

(corr. ∠s, HC // FG)

Draw a line FE such that FE // BC.

(corr. ∠s, FE // BC)

(adj. Ðs on st line)

Further Practice (p. 11.30)

1. (a) ∵ = 71°
∴ AC // DE (alt. ∠s equal)

(b)
i.e. unequal corresponding angles

∴ PQ is not parallel to CD.

2. (adj.∠s on st. line)

(alt. ∠s, EC // BH)

∴ CF // BG (corr. ∠s equal)

Exercise

Exercise 11A (p. 11.11)

Level 1

1. (adj. Ðs on st. line)

2. (adj. Ðs on st. line)

3. (adj. Ðs on st. line)

4. (adj. Ðs on st. line)

5. (adj. Ðs on st. line)

6. (adj. Ðs on st. line)

7. (Ðs at a pt.)

8. (Ðs at a pt.)

9. (Ðs at a pt.)

10. (Ðs at a pt.)

11. (Ðs at a pt.)

12. (Ðs at a pt.)

13. (vert. opp. Ðs)

14. (vert. opp.∠s)

15. (vert. opp.∠s)

(vert. opp.∠s)

16. (adj. Ðs on st. line)

(vert. opp. Ðs)

17. (adj. Ðs on st. line)

(vert. opp. Ðs)

18.

(vert. opp. Ðs)

(adj. Ðs on st. line)

19. ∵ a, b and c are adjacent angles on a straight line.

∴ a + b + c = 180°

∴ a = 30°, b = 60°, c = 90°

or a = 40°, b = 40°, c = 100°

(or any other reasonable answers)

20. ∵ AOB and COD are straight lines.

∴ m = r and n = s (vert. opp.∠s)

∴ m = 40°, n = 140°, r = 40°, s = 140°

or m = 50°, n = 130°, r = 50°, s = 130°

(or any other reasonable answers)

Level 2

21. (∠s at a pt.)

∴

22. (adj.∠s on st. line)

23. (a) ∠AOC = y (vert. opp.∠s)

(adj.∠s on st. line)

(b)

24. (a) (vert. opp.∠s)

(adj.∠s on st. line)

(b)

25. (a)

(∠s at a pt.)

∴

(b) ∵ ∠AOB =∠BOC = 58°

∴ OB bisects ∠AOC.

26. Let ∠AOE = x.
Then ∠BOC = 3x.
(adj.∠s on st. line)

(vert. opp.∠s)

27. (a) (adj.∠s on st. line)

(b) ∵ OB bisects ∠AOC.
∴

∴ OD bisects ÐCOE.

(d) ∵

∴ OB is perpendicular to OD.

28. (a)(adj.∠s on st. line)

(b)

∴ EOB is not a straight line.

Exercise 11B (p. 11.25)

Level 1

1. (alt. ∠s, AB // CD)

2. (int. ∠s, AB // CD)

3. (corr. ∠s, AB // CD)
(adj.∠s on st. line)

4. (vert. opp. Ðs)

(corr. Ðs, AB // CD)

5. ∠CRP = a (vert. opp. Ðs)

(int. Ðs, AB // CD)

6. (adj. Ðs on st. line)

(corr. Ðs, AB // CD)

(vert. opp. Ðs)

(adj. Ðs on st. line)

7. (alt. Ðs, CD // EF)

(corr. Ðs, AB // CD)

8. (alt. Ðs, AB // CD)

(adj. Ðs on st. line)

(alt. Ðs, CG // AF)

9. (alt. Ðs, CD // AB)

(alt. Ðs, ED // CB)

(alt. Ðs, EF // CD)

10. ∠BFE = 180° - 112° = 68° (int. ∠s, AB // EF)
(alt. ∠s, CD // EF)

11. (int. ∠s, BA // CD)

(int. ∠s, CB // DE)

(int. ∠s, FE // CD)

(∠s at a pt.)

12. a = c, b = d (corr. Ðs, AB // CD)

a + b = 180°, c + d = 180° (adj. Ðs on st. line)

∴ a = 10°, b = 170°, c = 10°, d = 170°

or a = 50°, b = 130°, c = 50°, d = 130°

(or any other reasonable answers)

13. p + q = 180° (int. Ðs, PS // QR)

p + s = 180° (int. Ðs, PQ // SR)

∴ q = s

p + q = 180° (int. Ðs, PS // QR)

q + r = 180° (int. Ðs, PQ // SR)

∴ p = r

∴ p = 40°, q = 140°, r = 40°, s = 140° or p = 120°, q = 60°, r = 120°, s = 60°

(or any other reasonable answers)

Level 2

14. (int. ∠s, DF // AC)

(corr. Ðs, DF // AC)

(adj. Ðs on st. line)

15. (alt. ∠s, AD // BC)

(int. Ðs, BC // AD)

16. (alt. ∠s, CD // AB)

(corr. Ðs, CD // AB)

(adj. Ðs on st. line)

17. (int. ∠s, FE // GB)

(adj.∠s on st. line)

(int. ∠s, BA // ED)

(∠s at a pt.)

18.

Draw EF // AB // CD.

(int. Ðs, EF // AB)

(alt. Ðs, CD // EF)

19.

Draw KE // AB // CD.

(int. Ðs, KE // AB)

(int. Ðs, CD // KE)

20.

Draw LE // AB // CD.

(int. Ðs, LE // CD)

(alt. Ðs, AB // LE)

21. ∠EPQ = y (corr. Ðs, AB // CD)

(adj. Ðs on st. line)

(corr. Ðs, AB // CD)

(adj. Ðs on st. line)

22. (alt. Ðs, AB // CD)

∵ (alt. Ðs, GH // KL)

23. (int. ∠s, AC // DE)

(alt. ∠s, DB // GE)

(alt. ∠s, DE // FH)

24. (int. ∠s, BC // ED)

(int. ∠s, AB // EF)

(∠s at a pt.)

Exercise 11C (p. 11.31)

Level 1

1. ∵

∴ AB // CD (alt. Ðs equal)

2. ∵

∴ AB // CD (int. Ðs supp.)

∵

∴ AB // CD (corr. Ðs equal)

∵

∴ AB is not parallel to CD.

∵

∴ AB // CD (int. Ðs supp.)

∵

∴ AB // CD (alt. Ðs equal)

Alternative Solution

∵

∴ AB // CD (corr. Ðs equal)

7. (∠s at a pt.)

∴ AB // DE, corr. ∠s equal

8. (adj.∠s on st. line)

∵
∴ AB // DC, int. ∠s supp.

9. ∵

∴ BE // DF, int. ∠s supp.

(adj. Ðs on st. line)

∴ AB // CD, alt. ∠s equal

10.

∴ , or ,

(or any other reasonable answers)

Level 2

11. (corr. Ðs, CE // FG)

∵

∴ AB // CD (corr. Ðs equal)

12. (corr. Ðs, EF // CD)

(adj. Ðs on st. line)

∵

∴ AB // GH (corr. Ðs equal)

Alternative Solution

(adj. Ðs on st. line)

(alt. Ðs, CD // EF)

∵

∴ AB // GH (alt. Ðs equal)

Alternative Solution

(vert. opp. Ðs)

(alt. Ðs, CD // EF)

∴ AB // GH (int. Ðs supp.)

13. (a)

∵

∴ AC // EF (int. Ðs supp.)

(b) ∵

∴ AB // CD (alt. Ðs equal)

14. (a) (int. ∠s, CD // EF)

(b) (∠s at a pt.)

∴ AB // CD (alt. ∠s equal)

15. (a)

(b) ∵

∴ CD // FE (int. Ðs supp.)

∴ AB // FE

16.

Draw a line CF such that CF // AB.

Then, ∠BCF = 66° (alt. ∠s, CF // AB)

∠FCD = 138° - 66°
= 72°

∵ ∠FCD + ∠CDE = 72° + 108°

=180°

∴ AB // DE (int. ∠s supp.)

Revision Exercise 11 (p. 11.35)

Level 1

1. (adj.∠s on st. line)

2. a = (vert. opp. Ðs)

(adj. Ðs on st. line)

3. (vert. opp.∠s)

4. (Ðs at a pt.)

5. (vert. opp.∠s)

(∠s at a pt.)

6. (vert. opp.∠s)

(adj.∠s on st. line)

7. (int. Ðs, AB // DC)

(int. ∠s, AD // BC)

8. (alt. ∠s, AB // CF)

(alt. ∠s, GH // CF)

(int. ∠s, BD // GE)

9. (int. Ðs, DC // EF)

(alt. Ðs, BC // DE)

(int. Ðs, DC // AB)

10. (alt. ∠s, AB // CF)

(adj.∠s on st. line)

11. (alt. ∠s, FG // CE)

(int. ∠s, AB // CE)

12. (alt. ∠s, CB // DE)

(int. ∠s, CA // DE)

13. (alt. ∠s, BD // CE)

∴ AB // CD (int. ∠s supp.)

14. (adj. Ðs on st. line)

(adj. Ðs on st. line)

∵ ÐABC = ÐBCD = 60°

∴ AB // CD (alt. Ðs equal)

15. (adj. Ðs on st. line)

(alt. Ðs, EC // DF)

∵ ÐABC = ÐBCD = 60°

∴ AB // CD (alt. Ðs equal)

16. (a) x = q (alt. ∠s, AC // DF)

r = y (vert. opp.∠s)

z = r (corr. ∠s, AC // DF)

z = y (z = r and r = y)

(any two of the answers)

(b) x + p = 180° (adj.∠s on st. line)

p + q = 180° (int. ∠s, AC // DF)

s + r = 180° (int. ∠s, AC // DF)

z + s = 180° (adj.∠s on st. line)

s + y = 180° (s + r = 180° and y = r)
(any two of the answers)

17.

With the notations in the figure,

ÐB = 180° – a (int. Ðs, AD // BC)

ÐB + ÐC = 180° (int. Ðs, BA // CD)

∴ a = 70°, b = 40° or a = 80°, b = 50° (or any other reasonable answers)

Level 2

18. ∠BEF = 2x (alt. ∠s, EF // AB)

(alt. ∠s, EF // CD)

19.

Construct a line XR such that it is parallel to PQ and ST.

(int. ∠s, PQ // XR)

(alt. ∠s, ST // XR)

20.

Draw AB // GE // HF // CD.

(int. Ðs, HF // CD)