6. Lines and Angles
- 1. NUMBER SYSTEMS
- 2. A POLYNOMIALS
- 3. COORDINATE GEOMETRY
- 4. LINEAR EQUATIONS IN TWO VARIABLES
- 5. INTRODUCTION TO EUCLID'S GEOMETRY
- 6. LINES AND ANGLES
- 7. TRIANGLES
- 8. QUADRILATERALS
- 9. CIRCLES
- 10. HERON'S FORMULA
- 11. SURFACE AREAS AND VOLUMES
- 12. STATISTICS
- 13. APPENDIX 1: PROOFS IN MATHEMATICS
- 14. APPENDIX 2: INTRODUCTION TO MATHEMATICAL MODELLING
6.1 Introduction - Learning Objectives
6.2 Basic Terms and Definitions
6.3 Intersecting Lines and Non-intersecting Lines
6.4 Pairs of Angles
Theorem 6.1:
If two lines intersect each other, then the vertically opposite angles are equal.
Example 1:
Given ∠POR : ∠ROQ = 5 : 7, find all angles at intersection point.
Example 2:
If ∠POS = x and OR, OT bisect angles POS and SOQ respectively, find ∠ROT.
Example 3:
Prove that ∠POQ + ∠QOR + ∠SOR + ∠POS = 360°.
Exercise 6.1
- Find ∠BOE and reflex ∠COE given ∠AOC + ∠BOE = 70°, ∠BOD = 40°.
- Find ∠c given ∠POY = 90° and a : b = 2 : 3.
- Prove ∠PQS = ∠PRT when ∠PQR = ∠PRQ.
- Prove AOB is a line given x + y = w + z.
- Prove ∠ROS = ½(∠QOS − ∠POS).
- Draw figure and find ∠XYQ and reflex ∠QYP given ∠XYZ = 64°.
6.5 Lines Parallel to the Same Line
Theorem 6.6:
Lines which are parallel to the same line are parallel to each other.
Example 4:
Given ∠MXQ = 135° and ∠MYR = 40°, find ∠XMY using parallel lines.
Example 5:
Prove PQ || RS using bisectors of corresponding angles and their equality.
Example 6:
Find x, y, z from diagram using angle relationships and parallel lines.
Exercise 6.2
- Find x given y : z = 3 : 7 and AB || CD || EF.
- Find ∠AGE, ∠GEF and ∠FGE given ∠GED = 126°.
- Find ∠QRS given ∠PQR = 110°, ∠RST = 130° and PQ || ST.
- Find x and y given ∠APQ = 50°, ∠PRD = 127°, AB || CD.
- Prove AB || CD using ray reflection across parallel mirrors PQ and RS.