GEOMETRY(i)

 

Contents:

 

Angles & Lines:

The angles around a point add up to 360.

The angles along a straight line add up to 180.

 

The angles in a triangle add up to 180.

The angles in a quadrilateral add up to 360.

 

An angle of 90 is a right angle, lines meeting at a right angle are perpendicular. An angle smaller than 90 is acute, an angle greater than 90 is obtuse.

The opposite angles of a cross are equal, remembering that they all add up to 360. Lines which have the same direction are known as parallel. A line that crosses two paralell lines is known as a transversal. The angles on either side of this are equal, known as alternate angles. The angles on the same side of the transversal are equal, known as corresponding angles.

   

Example: Find all the angles (from the diagram above) if one of the angles b was measured as 135.

Answer: Using (straight line 180and opposite angles on a cross section); all the angles b are 135 and all the angles a are 45.

Example: Find all the missing angles (from the diagram below).

Answer: b is found by opposite angles on a crossover being identical: 35. c is found by angles on a straight line totalling 180, 180- 105= 75. a is found either by 180- b + c = 70. or by transferring angles b and c to the top line, where f = 180- b + c = 70, and angle a = (crossover) f = 70.

Top of PageTop of Angles & Lines

Polygons: A polygon is a plane figure with straight sides. The names of some polygons are: 3 sided...Triangle; 4 sided...quadrilateral; 5 sided...pentagon; 6 sided...hexagon; 7 sided...heptagon; 8 sided...octagon; 9 sided...nonegon; 10 sided...decagon.

Triangles: 3 sided polygon. These are known in 3 different categories:

Scalene; 3 sides are different length, 3 angles are different sizes, no reflection possible - no symmetry. (Acute or Obtuse; possible Right Angled).

Isosceles; 2 sides are equal length, 2 angles are equal in size, there is a single line of symmetry. (Acute or Obtuse; possible Right Angled).

Equilateral; 3 sides are equal length, 3 angles are equal in size (60), there are 3 lines of symmetry from all 3 points. (Acute only, cannot be Right Angled).

All triangles have internal angles which total to 180. If the triangle was to sit on a straight line the external angle would be the sum of the opposite two sides (a straight line also has a total angle of 180).

Diagrams of triangles may show a bar across two or three sides to denote equal lengths, or letters in the corners to denote the angles. If these letters are the same then it is to be assumed the angles are the same size. The angles may be given a letter on the outside of each corner to indicate which angle is being examined ie. angle PQR means angle Q is being examined.

Exercise:

Top of Page Back to TrianglesTop of Polygons

Quadrilaterals: 4 sided polygon. Known a square, rectangle, rhombus, parallelogram, trapezium, kite. All these shapes have a pair or both pair of sides in parallel. If any of these shapes are split from opposite corner it results in two triangles being formed of sum of angles 180. Therefore the angles sum of two triangles is 360.

A square and a rectangle both have four angles of 90. A rhombus has all four sides equal in length and opposite corners of equal angle. A parallelogram also has opposite corners of equal angle. A trapezium has only one pair of paralell sides and no assumption can be made of the angles. A kite has its adjacent sides equal in length, therefore the two opposite angles where the different lengths meet are equal.

Exercise:

Top of Page Back to QuadrilateralsTop of Polygons

Regular Polygons: A regular polygon is one where all the sides and angles are identical in length and degree.The sum of the exterior angles for any sided polygon of any length is 360. Each external angle is 360n (where n is the number of sides of any length). The sum of the interior angles is calculated by the formula: 180n - 360. Each regular polygon has n internal isoscles triangles, (base is the outside edge),and many lines of symmetry.

Example using a regular six sided hexagon:

Angle A + a = 180
Angle B + b = 180
Angle C + c = 180
Angle D + d = 180
Angle E + e = 180
Angle F + f = 180
 

 

Sum of internal and external angles = 1080. 1080- external = 720. Each internal angle = 720n (7206) = 120and each external angle = 360n (3606) = 60.

 

Irregular Polygons:

Irregular polygons are any shapes with many straight sides of no regular length. The sum of the exterior angles for any sided polygon of any length is 360. Each external angle is 360n (where n is the number of sides of any length). The sum of the interior angles is calculated by the formula: 180x (n - 2). The polygon is split into triangles using full diagonals, each triangle equals 180, and there are always two less triangles than given sides.

Diagram shows 7 sides and 5 triangles.

External angles: A B C D E F G.

Internal angles: a b c d e f g            

Triangles: v w x y z.

Top of PageTop of Regular PolygonsBack to Polygons

©Mathstutor.com  2001-2005