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6-C Angles, parallel lines and transversals KEY CONCEPTS Parallel lines are two or more lines that are simple translations of each other. The distance between parallel lines is the same across their entire lengths; they never intersect each other. The train tracks in the picture are parallel to each other; they only appear to intersect at the horizon. Transversal Parallel lines The term ‘transverse’ means ‘crossways’. A line that intersects with a pair of parallel lines is called a transversal. The road crossing the two parallel lines in the diagram represents a transversal. When a transversal cuts a set of parallel lines, a number of angles are created. The table below shows how pairs of angles are related. Angle identification 204 Relationship Corresponding angles are positioned on the same side of the transversal and are either both above or both below the parallel lines; think of them as F-shaped. ∠a = ∠b Co-interior angles are positioned ‘inside’ the parallel lines, on the same side of the transversal; think of them as C-shaped. Angles are supplementary: ∠a + ∠b = 180°. Alternate angles are positioned ‘inside’ the parallel lines on alternate sides of the transversal; think of them as Z-shaped. ∠a = ∠b Vertically opposite angles are created when two lines intersect. The angles opposite each other are equal in size; think of them as X-shaped. ∠a = ∠a ∠b = ∠b Maths XPRESS 8 Example a a b b a a b b a a b b a a b b EXAMPLE 1 Use the diagram at right to answer the questions below. a b c d a Which angle is vertically opposite c? e f g h b Which angle is alternate to d? c Which angle is corresponding to g? d Which angle is co-interior to f ? e Which of the above pairs are equal and which are supplementary? WRITE a Vertically opposite means that the angle is opposite the intersection from c. They are X-shaped. a c d e f g h b b Alternate angles are in a Z shape. a c d e f g h c is corresponding to g. b d Co-interior angles are in a C shape. a c d e f g h e is alternate to d. b c Corresponding angles are in an F shape. a c d e f g h b is vertically opposite c. d is co-interior to f. b e 1 Colour all the angles equal to a in blue: • d is vertically opposite a • e is corresponding to a • h is vertically opposite e. 2 a c d e f g h b Colour all the angles equal to b in red: • c is vertically opposite b • f is corresponding to b • h is vertically opposite f . Angle relationships formed between parallel lines and a transversal can help us evaluate the size of unknown angles by: identifying the relationship between a known angle and an unknown angle using your knowledge of which pair of angles is either equal or supplementary. Chapter 6 Geometry 205 EXAMPLE 2 Use the diagram to calculate the value of the pronumerals. Provide a reason for each answer. a 63o b 81o c d e f 74o WRITE 1 Angles a and 63° are supplementary so they sum to 180°. Write an equation and solve for a. 2 Angles b and 63° are alternate, and alternate angles are equal. Angles b, 81° and c are supplementary so they sum to 180°. Write an equation and solve for c. 3 5 Angles c and d are alternate and so are equal. The triangle contains angles c, e and 74°. The sum of the interior angles in a triangle is 180°. 6 Angles f and 74° are alternate and so are equal. 4 a + 63° = 180° a = 180° − 63° a = 117° b = 63° (supplementary) (alternate) 63° + 81° + c = 180° (supplementary) 144° + c = 180° c = 180° − 144° c = 36° d = c = 36° (alternate) c + e + 74° = 180° (interior angles of a triangle) 36° + e + 74° = 180° 110° + e = 180° e = 180° − 110° e = 70° f = 74° (alternate) LEARNING EXPERIENCE ‘Identify the angle’ race Equipment: string, scissors, Blu Tack, stopwatch, paper, pen 1 Set up the classroom with four rows of tables. 2 Form groups of four students and collect three pieces of string and a blob of Blu Tack for each person. 3 Each group lines up along one of the rows, with one person per table. Each person uses the string and Blu Tack to model one of the four types of angles represented in Key concepts. Use blobs of Blu Tack to show the positions of the angles. (Use a small amount of Blu Tack to hold the string in place.) 4 Remove one of the blobs of Blu Tack showing the position of the angles. A competitor from another group completes the problem by putting the blob of Blu Tack back in the appropriate position. Group members who have modelled an angle stand behind their angle during the race. 5 Now the race begins! A group of four lines up, with one group member at the top of each row. When the timekeeper says ‘Start’, timing starts and does not finish until the last group member has completed all their problems. When the competitor reaches a model, the student who constructed the model tells the competitor the kind of angle they need to make with the second blob of Blu Tack. 6 Groups take turns to race through the tables and make the models. Points are based on speed and accuracy. Each second is worth 1 point and every incorrect answer is worth an additional 5 points. The group with the fewest points wins. 206 Maths XPRESS 8