Fundamental Geometric Concepts

• Angle- a figure formed by two rays with same initial point. The initial point is its vertex and the two rays are its arms or sides.
                     
• Type of angle:
  Acute angle -an angle whose measure lies between 0° and 90°.
  Right angle -an angle whose measure is 90°.
  Obtuse angle -an angle whose measure lies between 90° and 180°.
  Straight angle -an angle whose measure is 180°.
  Reflex angle -an angle whose measure lies between 180° and 360°.
• Types of angles:
  Adjacent angles -two angles that have a common vertex, a common arm and their other arms lie on either side of the common arm.
  Linear pair -two adjacent angles whose exterior arms are in a straight line. Thus if two adjacent angles form a linear pair, then their sum is 180°.
  Complementary angles -two angles whose sum of measures is 90°.
  Supplementary angles -two angles whose sum of measures is 180°.
• Sum of angles at a point = 360°.
• Sum of angles at a point on a straight line = 180°.
• If the sum of two adjacent angles is 180°, then their exterior arms are in a straight line.
• If two straight lines intersect, then vertically opposite angles are equal.
• Transversal -a line that interests two (or more) lines in a plane at distinct points.
• Properties of angles associated with parallel lines
  If a transversal meets two parallel lines, then:
  corresponding angles are equal.
  alternate angles are equal.
  co-interior angles are supplementary.
• Conditions of parallelism
  If two lines are cut by a transversal such that a pair of:
  corresponding angles is equal, then the lines are parallel.
  alternate angles is equal, then the lines are parallel.
  co-interior angles is supplementary, then the lines are parallel.
• If two lines are parallel to a third line, then the lines are themselves parallel.

Exercise

1. Two angles are complementary and one angle is 10° less than three times the other, find the angles.
2. Two supplementary angles are in the ratio 2:7, find the complement of the smaller angle.
3. From the following diagram, find the value of x and hence complete the following:
   (i) AOC =...
   (ii) DOE =...
               
   
4. In the figure below, AB and CD are straight lines. Find
   (i) x, y and z if x = p and q = 80°
   (ii) p if y : z = 2 : 3
   (iii) z if p : q : x = 2 : 3 : 1
   (iv) x if y = 40° and q = 2p +10°.
                   
5. Find the measure of each lettered angle in the following figures :
   (i)      (ii)
   (iii)
6. Find the value of x from the following sketches :
   (i) (ii)
7. In the following figure, find the value of x so that the lines AB and CD may be parallel.
                    

Answers

1. 25°, 65°                2. 40°, 140°, 50°
3. 35°     (i) 125°       (ii) 109°
4. (i) x = 50°, y = 50°, z = 130°     (ii) 72°  (iii) 120°   (iv) 50°
5. (i) x = 42°, y = 63°, z = 75°, p = 138°
    (ii) x = 75°, y = 15°    (iii) x = 145°,y = 55°, z = 125°
6. (i) 38   (ii) 23      7. 37°