11 Angles and Parallel Lines

11 Angles and Parallel Lines

11.1 Angles Related to Intersecting Lines

What Did I Learn?

☑ Adjacent angles on a straight line, angles at a point and vertically opposite angles

If AOC is a straight line, x + y + z = 360. If AOB and COD are

then a + b = 180. (Ref: s at a pt.) straight lines, then a = b
(Ref: adj. s on st. line) and x = y.
(Ref: vert. opp. s)

What Can I Do?

☑ Use the property of adjacent angles on a straight line to find unknowns.

1. In each of the following figures, AOB is a straight line. Find the unknowns.
(a) (b)

☑ Use the property of angles at a point to find unknowns.

2. Find the unknowns in the following figures.

(a) (b)

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☑ Use the property of vertically opposite angles to find unknowns.

3. In each of the following figures, AOB and COD are straight lines. Find the unknowns.
(a) (b)

4. In the figure, AOB, COD and EOF are straight lines. Find u.

Integrated Questions

5. In the figure, POQ and ROS are straight lines.

(a) Find h.
(b) Are ROT and ROU complementary angles? Explain your
answer.
(c) Is PQ perpendicular to RS? Explain your answer.

6. In the figure, AOB is a straight line.

(a) Find x and y.
(b) Is POQ a straight line? Explain your answer.

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 11 Angles and Parallel Lines

11.2 Parallel Lines

What Did I Learn?

☑ Corresponding angles
If a = b, then AB // CD. (Ref: corr. s equal)
☑ Alternate angles (alternate interior angles)
If b = c, then AB // CD. (Ref: alt. s equal)
☑ Interior angles on the same side
If c + d = 180, then AB // CD. (Ref: int. s supp.)

What Can I Do?

☑ Identify parallel lines.

1. In each of the following figures, identify a pair of parallel lines and give a reason.
(a) (b)

2. In each of the following figures, identify a pair of parallel lines and give a reason.
(a) (b)

☑ Solve problems involving the conditions for two lines being parallel.

3. In the figure, BCD is a straight line.

(a) Find ABC.
(b) Is AB parallel to EC? Explain your answer.

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4. In the figure, QR ⊥ XR and YR ⊥ ZR.

(a) Find QRZ.
(b) Is PQ parallel to RZ? Explain your answer.

5. In the figure, ABCD, EBF, FCG and HBK are straight lines.
(a) Find FBC.
(b) Is HK parallel to FG? Explain your answer.

6. In the figure, BCE and DCG are straight lines.

(a) Find BCD.
(b) Is AB parallel to DG? Explain your answer.

Integrated Questions

7. In the figure, ABC = BCD.

(a) Is AB parallel to CD? Explain your answer.
(b) Is CD parallel to EF? Explain your answer.
(c) Is AB parallel to EF? Explain your answer.

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 11 Angles and Parallel Lines

11.3 Angles Related to Parallel Lines

What Did I Learn?

☑ Corresponding angles equal

If AB // CD, then a = b. (Ref: corr. s, AB // CD)
☑ Alternate angles equal
If AB // CD, then b = c. (Ref: alt. s, AB // CD)
☑ Interior angles on the same side supplementary
If AB // CD, then c + d = 180. (Ref: int. s, AB // CD)

What Can I Do?

☑ Use the angle properties associated with parallel lines to find unknowns.

1. Find the unknowns in the following figures.

(a) (b)

(c) (d)

2. In the figure, CDE and DBF are straight lines. AB // CE. Find y.

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3. In the figure, AEB is a straight line. AB // CD. Find k.

4. In the figure, BDF and CDE are straight lines. AB // CE. Find d.

5. In the figure, ABC is a straight line. AC // DE. Find m.

6. In the figure, CDE is a straight line. AB // CE.

(a) Find h.
(b) Find k.

7. In the figure, ABC, DEFG, BEI and FBH are straight lines.
AC // DG.
(a) Find x.
(b) Find y.

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 11 Angles and Parallel Lines

☑ Find unknowns from more than one pair of parallel lines.

8. In the figure, PQR and STU are straight lines. SQ // TR and

PR // SU. Find f.

9. In the figure, BDF and CDE are straight lines. AB // CE and

BF // EG.
(a) Find u.
(b) Find v.

☑ Find unknowns by adding suitable lines.

10. In the figure, AB // DE. Find a.

11. In the figure, AB // CD. Find b.

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12. In the figure, AB // EF.

(a) Suppose that CG is a straight line such that AB // CG. Find DCG.
(b) Find y.

Integrated Questions

13. In the figure, AB // CD // FE and BC // ED.

(a) Find x.
(b) Is DP parallel to EQ? Explain your answer.

Exam Practice

14. In the figure, BCD and ECF are straight lines. AB // EF and
DCF = 62. Find k.

Refer to TSA 2019 9ME4 Q33

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 11 Angles and Parallel Lines

11.4 Angles Related to Triangles

What Did I Learn?

☑ Sum of interior angles of a triangle

a + b + c = 180. (Ref:  sum of )
☑ Exterior angle of a triangle
d = a + b. (Ref: ext.  of )

What Can I Do?

☑ Use the properties of interior angles of triangles to find unknowns.

1. Find the unknowns in the following figures.

(a) (b)

2. In the figure, BAG, BCF and DACE are straight lines. Find b.

☑ Use the properties of exterior angle of triangles to find unknowns.

3. Find the unknowns in the following figures.

(a) (b)

4. In the figure, ABCD is a straight line. Find k.

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☑ Use the properties of angles of triangles to find unknowns.

5. In the figure, IH // KL. HK and IL intersect at J. Find x and y.

6. In the figure, AB // DE. AB produced and CD intersect at F. Find m.

Integrated Questions

7. In the figure, P = 32, R = 17 and S = 61. Find k.

Exam Practice

8. In the figure, A = 108. Find x.

Refer to TSA 2018 9ME1 Q43

9. In the figure, AED and BCE are straight lines. BAE = 64,
ABE = 28 and CDE = 43. Find x and y.

Refer to TSA 2017 9ME3 Q32

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 11 Angles and Parallel Lines

11.5 Introduction to Geometric Proofs

What Did I Learn?

☑ Geometric theorems
(1) adj. s on st. line (2) s at a pt. (3) vert. opp. s  Section 11.1

(4) corr. s equal (5) alt. s equal (6) int. s supp.  Section 11.2

(7) corr. s, AB // CD (8) alt. s, AB // CD (9) int. s, AB // CD  Section 11.3

(10)  sum of  (11) ext.  of   Section 11.4

What Can I Do?

☑ Use the conditions for angles related to intersecting lines to perform proofs.

1. In the figure, ABC is a straight line. Prove that BF ⊥ BD.

2. Prove that COF is a straight line in the figure.

3. In the figure, AOB, COD and EOF are straight lines.

Prove that a + b = 90.

☑ Perform proofs of lines being parallel.

4. In the figure, PQ produced and RS intersect at U.

Prove that PQ // ST.

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☑ Use the conditions for parallel lines to perform proofs.

5. In the figure, DF bisects CDE. AB // CD and BC // DE.

Prove that b = 2a.

6. In the figure, ADB and BFC are straight lines. DE // FG.

Prove that c = a + b.

☑ Use the conditions for angles of triangles to perform proofs.

7. In the figure, ABC is a straight line. Prove that DB ⊥ AC.

Integrated Questions

8. In the figure, C is a point on DE such that AD // BC.

Prove that AB // DE.

9. In the figure, BC ⊥ CE. Prove that AB // DE.

Exam Practice

10. In the figure, AFB, CGD and EFG are straight lines.
Prove that AB // CD.

Refer to TSA 2019 9ME3 Q46

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