2. Year 9 Geometrical Reasoning Alternate Angles on Parallel Lines

3. Can we show the pair of PARALLEL LINES and one pair of CORRESPONDING ANGLES. g g CORRESPONDING ANGLES on PARALLEL LINES are EQUAL so we can label the angles to show this.

4. Now can you describe the angles x and y? Angle x is ALTERNATE to angle y. g g x y

5. We call x and y ALTERNATE ANGLES because they are formed by the transversal (red line) and the top line and the transversal and the bottom line. However, they are on ALTERNATE (opposite) sides of the transversal. We call x and y ALTERNATE ANGLES because they are formed by the transversal (red line) and the top line and the transversal and the bottom line. However, they are on ALTERNATE (opposite) sides of the transversal. x y

6. Can we say anything about the comparative sizes of x and y? We want to prove that pairs of ALTERNATE ANGLES on PARALLEL LINES are EQUAL. g g x y Can we say anything about the comparative sizes of x and y? …we must be precise in our reasoning. Can we say anything about the comparative sizes of x and y? …we must be precise in our reasoning.

7. g g x y First consider the marked angles on this line.

8. g g x y Now let’s make the diagram a little simpler…

9. What can you say about x and g? 180 degrees g x x + g = 180 (Adjacent angles on a straight line add up to 180). x + g = 180 (Adjacent angles on a straight line add up to 180).

10. g g x y We can do the same with angles y and g.

11. 180 degrees g y We can do the same with angles y and g. y + g = 180 (Adjacent angles on a straight line add up to 180). y + g = 180 (Adjacent angles on a straight line add up to 180).

12. g g x y y + g = 180 (Adjacent angles on a straight line add up to 180). y + g = 180 (Adjacent angles on a straight line add up to 180). x + g = 180 (Adjacent angles on a straight line add up to 180). x + g = 180 (Adjacent angles on a straight line add up to 180).

13. g g x y x + g = 180 y + g = 180 What can we deduce? x + g = 180 y + g = 180 What can we deduce? Angle g is common to both equations so x and y must be EQUAL.

14. g g x y PROVED: Pairs of ALTERNATE ANGLES on PARALLEL LINES must be EQUAL. PROVED: Pairs of ALTERNATE ANGLES on PARALLEL LINES must be EQUAL.

15. x x PROVED: Pairs of ALTERNATE ANGLES on PARALLEL LINES must be EQUAL. PROVED: Pairs of ALTERNATE ANGLES on PARALLEL LINES must be EQUAL.

16. One more thing… ALTERNATE ANGLES are sometimes referred to as “Z” ANGLES… One more thing… ALTERNATE ANGLES are sometimes referred to as “Z” ANGLES…