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Geometry

LINES AND ANGLES

  • Angles are formed at the point of intersection of two lines.
  • Angles can be defined according to their sizes:
    • Acute angle: This angle is less than 90°
    • Right angle: This angle is equal to 90°
    • Obtuse angle: This angle is between 90° and 180°, i.e. This angle is greater than 90° but less than 180°
    • Reflex angle: This angle is between 180° and 360°, i.e. This angle is greater than 180° and less than 360°
    • Complementary angles: These are angles that sum up to 90°.
    • Supplementary angles: These are angles that sum up to 180°
  • Perpendicular lines are lines that intersect at right angles; 90°.
    • The orientation of the lines do not matter, as long as the angle between them is 90°, they are perpendicular.
  • A line that cuts across two or more parallel lines is called a transversal.
  • Using the diagram below to define more angles:

    • Angles at a point add up to 360°. Therefore, a + b + c + d = 360° and e + f + g + h = 360°
    • Angles on a straight line add up to 180°. Therefore, a + b = c + d = a + d = b + c = 180° and
      e + f = g + h = e + h = f + g = 180°
    • Vertically opposite angles are equal. Therefore, a = c, b = d, e = g, f = h.
    • Alternate angles are equal. Therefore, c = e, and d = f
    • Corresponding angles are equal. Therefore, a = e, b = f, c = g, and d = h.
    • Two interior angles on the same side of the transversal add up to 180°.
      Therefore, d + e = c + f = 180°
    • NOTE: There might be more than one way to get a particular angle. (Check out Example 1a)
POLYGONS
  • A polygon is a plane figure that is bounded by many straight lines.
  • A regular polygon has all sides and angles equal.
  • A quadrilateral is a four-sided polygon. Examples, square, rhombus, kite, rectangle, etc.
  • Other examples of polygons:
    • A pentagon has five sides
    • A decagon has ten sides
  • The interior angles of an n-sided polygon would add up to (2n – 4) × 90° or (n – 2) × 180°
  • The exterior angles of a polygon would add up to 360°
Examples
  1. Determine the angles a, b, c in the following:
      • To get c:
        • 62° + c + 55° = 180° [Sum of angles on a straight line]
        • ∴ c = 63°
      • To get b:
        • First step:
          • b = 55° [Vertically opposite angles]
        • Second step:
          • b + 62° + c = 180° [Sum of angles on a straight line]
          • b + 62° + 63° = 180°
          • ∴ b = 55°
      • To get a:
        • First step:
          • a + b = 180° [Sum of angles on a straight line]
          • a + 55° = 180°
          • ∴ a = 125°
        • Second step:
          • a = 62° + c [Vertically opposite angles]
          • a = 62° + 63°
          • ∴ a = 125°
      • To get a:
        • 65° + a + 47° = 180° [Sum of angles on a straight line]
        • ∴ a = 68°
      • To get b:
        • a + b = 180° [Interior angles on the same side]
        • 68° + b = 180°
        • ∴ b = 112°
      • To get c:
        • a = Angle DEC [Corresponding angles}
        • c = Angle DEC [Alternate angles]
        • ∴ c = a = 68°
  2. Find the values of all unknown angles below:

    • To get a:
      • a = 40° [Vertically opposite angles]
    • To get d:
      • d = 55° [Vertically opposite angles]
    • To get c:
      • c + 55° = 180° [Angles on a straight line]
      • ∴ c = 125°
    • To get b:
      • b + 40° = 180°
      • ∴ b = 140°
    • To get e:
      • c + e = 180° [Interior angles on the same side]
      • 125° + e = 180°
      • ∴ e = 55°
    • To get f:
      • Sum of interior angles in a 3-sided polygon = (3 – 2) × 180° = 180°
      • ∴ d + a + f = 180°
      • 55° + 40° + f = 180°
      • ∴ f = 85°
    • To get h:
      • h = f [vertically opposite angles]
      • ∴ h = 85°
    • To get g:
      • g + e + f = 180° [Angles on a straight line]
      • g + 55° + 85° = 180°
      • ∴ G = 40°
    • To get k:
      • h + k = 180° [Angles on a straight line]
      • 85° + k = 180°
      • ∴ k = 95°
    • To get m:
      • g = m [Alternate angles]
      • ∴ m = 40°
    • To get p:
      • m = p [Vertically opposite angles]
      • ∴ p = 40°
    • To get n:
      • h + m + n = 180° [Sum on interior angles in a 3-sided polygon]
      • 85° + 40° + n = 180°
      • ∴ n = 55°
    • To get q:
      • n = q
      • ∴ q = 55°
  3. Determine the value of x in the figures below:
      • Sum of interior angles in a 5-sided polygon = (5 – 2) × 180° = 540°
      • ∴ 120° + 70° + 45° + 45° + x = 540°
      • ∴ x = 260°
      • Sum of interior angles of a 6-sided polygon = (6 – 2) × 180° = 720°
      • ∴ 130° + 120° + 100° + y + 130° + 140° = 720°
      • ∴ y = 100°
      • y + x = 180° [Angles on a straight line]
      • 100° + x = 180°
      • ∴ x = 80°
  4. The interior angles of a polygon add up to 1980°. How many sides has the polygon?
    • Sum of interior angles of a n-sided polygon = (n – 2) × 180°
    • ∴ (n – 2) × 180° = 1980°
    • n – 2 = 11
    • ∴ n = 13
  5. Find y in the diagram below.

    • Sum of exterior angles of a polygon = 360°
    • ∴ 5y + 2y + 4y + 3y + 7y + 3y = 360°
    • ∴ 24y = 360°
    • ∴ y = 15°
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