Lines and Angles

This chapter introduces pairs of related angles. Two angles whose sum is 90 degrees are complementary, like 30 degrees and 60 degrees, and two angles whose sum is 180 degrees are supplementary, like 110 degrees and 70 degrees. Adjacent angles have a common vertex and a common arm but no common interior points. A linear pair is a pair of adjacent angles whose non-common arms are opposite rays, so they always add up to 180 degrees. When two lines intersect, the vertically opposite angles formed are equal. A transversal is a line that intersects two or more lines at distinct points. It forms eight angles, which are grouped as corresponding angles, alternate interior angles, alternate exterior angles and interior angles on the same side of the transversal. When the two lines are parallel, each pair of corresponding angles is equal, each pair of alternate interior angles is equal, and interior angles on the same side add up to 180 degrees. These facts can also be used in reverse to check whether two lines are parallel.

Detailed Worksheet: Lines and Angles Section A - Definitions (10 marks) 1. Define complementary angles and supplementary angles with one example of each. (2 marks) 2. What are adjacent angles? State the three conditions they must satisfy. (2 marks) 3. What is a linear pair? Why is the sum of a linear pair always 180 degrees? (2 marks) 4. What are vertically opposite angles? State the property they satisfy. (2 marks) 5. What is a transversal? How many angles are formed when a transversal cuts two lines? (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the complement of (i) 20 degrees (ii) 63 degrees (iii) 57 degrees, and the supplement of (i) 105 degrees (ii) 87 degrees (iii) 154 degrees. (3 marks) 7. Two angles are complementary, and one is 12 degrees more than the other. Find both angles. Also find the angle that is equal to its own complement. (3 marks) 8. Two supplementary angles are in the ratio 2 : 3. Find them. Also find the angle that is equal to its supplement. (3 marks) 9. Two lines intersect and one of the angles formed is 40 degrees. Find the other three angles and name the property used for each. (3 marks) 10. Two parallel lines are cut by a transversal. One of the interior angles on the same side is 75 degrees. Find the other interior angle on that side, the corresponding angle of 75 degrees and its alternate interior angle. (3 marks) Section C - Diagrams (10 marks) 11. Draw two parallel lines l and m cut by a transversal p. Label the eight angles 1 to 8 and list (i) two pairs of corresponding angles (ii) two pairs of alternate interior angles (iii) two pairs of alternate exterior angles (iv) two pairs of interior angles on the same side. (4 marks) 12. Draw a pair of adjacent angles that form a linear pair, with one angle 125 degrees. Mark and find the other angle. (3 marks) 13. Draw two intersecting lines and mark the two pairs of vertically opposite angles. If one angle is 3x and its vertically opposite angle is 2x + 30 degrees, find x. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. State whether the following are true or false and justify: (i) Two obtuse angles can be supplementary. (ii) Two acute angles can be complementary. (iii) Two right angles are supplementary. (iv) If two adjacent angles are supplementary, they form a linear pair. (v) An angle and its complement can both be acute. (5 marks) 15. In a figure, lines l and m are cut by a transversal t. The corresponding angles formed are 70 degrees and 75 degrees. (i) Are l and m parallel? Justify. (ii) If instead alternate interior angles were 65 degrees each, what would you conclude? (iii) If the interior angles on the same side were 110 degrees and 70 degrees, are the lines parallel? (iv) State the rule you used in each case. (5 marks) 16. Look at a railway track, a zebra crossing and a ruled notebook. (i) Identify parallel lines and transversals in each. (ii) Why must the rails of a railway track be parallel? (iii) In a ladder, the rungs are parallel and the side bar acts as a transversal; if one rung makes an angle of 80 degrees with a side bar, what angle does the next rung make with it at the corresponding position? (iv) Name the angle property used. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat labelled figures with a ruler and give the reason for every angle you find.