Angles

 

When two lines meet at a point, the space between them is called an angle. Angles are measured in degrees and there are 360 degrees in one revolution.

The symbol for degrees is a small zero as a superscript i.e. a small zero to the right and to the top of the number of degrees. E.g. 10 degrees is written 10o.

 

There are names to describe angles, based on their size.

Right Angle:

A quarter of a revolution is a quarter of 3600, i.e. 90o

An angle of 90o is known as a right angle.

Acute Angle:

An acute angle is less than 90o

Obtuse Angle:

An obtuse angle is greater than 90o but less than 180o.

Reflex Angle:

A reflex angle is greater than 180o but less than 360o.

Straight Angle:

Half a revolution is 180o. An angle of 180o is known as a straight angle.

Revolution:

A revolution is 360o. The angles around a point have a sum of 360o and make a revolution.

Exercise: Write down the value of the following pronumerals and give a brief reason in each case:

Answers:         (i)         a = 60o Sum of angles in a straight angle = 180o.

(ii)                b = 65o Sum of angles in a right angle = 90o

(iii)               c = 120o Sum of angles in a revolution = 360o

 

More About Angles

Adjacent angles:

Angles that are next to each other are called adjacent angles.

Adjacent angles have a common vertex and a common side between them.

Complementary angles:

Two angles that add up to 90o are said to be complementary.

The complement of an angle is the angle that must be added to it to make 90o.

The complement of 30o is 60o because 60o has to be added to 30o to get 90o.

Supplementary angles:

Two angles that add up to 180o are said to be supplementary.

The supplement of an angle is the angle that must be added to it to make 180o.

The complement of 60o is 120o because 120o has to be added to 60o to get 180o.

 

Exercise: Complete the following table to show the complement and supplement of the angles in column 1.

Angleo

Complemento

Supplemento

43

 

 

27

 

 

18

 

 

65

 

 

73

 

 

82

 

 

47

 

 

59

 

 

38

 

 

51

 

 

Answers:         

Angleo

Complemento

Supplemento

43

47

137

27

63

153

18

72

162

65

25

115

73

17

107

82

8

98

47

43

133

59

31

121

38

52

142

51

39

129

 

POLYGONS

 

A polygon is a closed figure with straight sides. Most have names that indicate the number of sides.

3 sides: Triangle

4 sides: Quadrilateral

5 sides: Pentagon

6 sides: Hexagon

7 sides: Heptagon

8 sides: Octagon

9 sides: Nonagon

10 sides:           Decagon

 

A polygon with all sides and angles equal is known as a regular polygon.

 

The interior angles of a polygon have a sum of (n – 2)180o where n = number of sides.

The exterior angles of a polygon have a sum of 360o regardless of the number of sides.

In the above diagram of a pentagon:

ao + bo + co + do + eo (sum of interior angles)  = (5 – 2)180o = 3 x 180o  = 540o.

fo + go + ho + io + jo (sum of exterior angles) = 360o.

 

Exercise:

  1. What is the sum of the (i) interior angles and (ii) exterior angles in a heptagon?
  2. What is the value of each (i) interior angle and (ii) exterior angle in a regular octagon?
  3. An Australian 50 cent coin is in the shape of a regular dodecagon (12 sides)

(i)                  What is the sum of the interior angles of a 50 cent coin?

(ii)                What is the sum of the exterior angles of a 50 cent coin?

(iii)               What is the value of each interior angle of a 50 cent coin?

(iv)              What is the value of each exterior angle of a 50 cent coin?

4. Why isn’t a rhombus a regular quadrilateral?

 

Answers:

1. (i) 900o   (ii) 360o    

2. (i) 135o   (ii) 45o

3. (i) 1800o   (ii) 360o   (iii) 150o   (iv) 30o

4. A regular polygon is one in which all the sides and angles are equal. A regular quadrilateral is a square. A rhombus does not have all angles equal.

PARALLEL LINES

 

Two straight lines are parallel if they will not meet regardless of how far they are produced (extended) in either direction.

Transversal:

A transversal is a straight line that cuts two or more parallel lines

Alternate angles are equal:.

The above diagrams show two pairs of alternate angles.

Corresponding angles are equal:

The above diagrams show three pairs of corresponding angles.

Co-interior opposite angles are supplementary:

The above diagrams show two pairs of co-interior opposite angles, usually just called co-interior angles. Co-interior angles are supplementary i.e. they add up to 180o.

 

Exercise:

1. Write down the value of each pronumerals in the following diagrams.

2. This question refers to the following diagram.

Write down      (i) 2 pairs of alternate angles

                        (ii) 2 pairs of corresponding angles

                        (iii) 2 pairs of co-interior angles

Answers

Q.1.

a

b

c

d

e

f

g

(i)

60

60

120

120

60

60

120

(ii)

50

130

130

50

50

130

130

(iii)

72

108

108

72

108

108

72

Q.2.     (i) c & f,   d & e          

            (ii) a & e, c & g, b & f, d & h

            (iii)c & e, d & f

TRIANGLES

 

A triangle is a closed figure consisting of three straight lines.

The sum of the angles in a triangle is 180o.

 

A triangle with all three sides equal is called an equilateral triangle. It also has all three angles equal. Each angle = 60o.

           

 

A triangle with two equal sides is called an isosceles triangle.

The base angles (those opposite the equal sides) of an isosceles triangle are equal.

                       

Conversely if two angles of a triangle are equal then the sides opposite those angles are equal and the triangle is isosceles.

 

The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

           

Exercise: Calculate the values of the pronumerals in the following triangles. The triangles are not drawn to scale.

 

Answers: a = 110o,  b = 80o,   c = 45o,   d = 45o.   e = 20o,   f = 15o,   g = 45o

 

CONGRUENT TRIANGLES

 

Congruent triangles are equal in all respects. Each side the first triangle is equal to the corresponding side in the second triangle and each angle of the first triangle is equal to the corresponding angle in the second triangle.

Congruency has the symbol  and we say triangle 1 triangle 2.

Triangles are congruent if they satisfy the following conditions:

 

If three sides of one triangle are equal to three sides another triangle then the triangles are congruent. (SSS)

             DABC  DDEF

 

If two sides and an included angle of one triangle are equal to two sides and an included angle of another triangle then the triangles are congruent. (SAS)

             DABC  DDEF

 

If two angles and a side of one triangle are equal to two angles and the corresponding side of another triangle then the triangles are congruent. (AAS)

             DABC  DDEF

 

If the hypotenuse and one side of a right angled triangle are equal to the hypotenuse and one side of another right angled triangle then the triangles are congruent. (RHS)

             DABC  DDEF

Exercise: State which pairs of the following triangles are congruent. They are not drawn to scale.

 Answers: A & E, F & H. Not B & G, not C & D

GEOMETRY I

 

Q.1.  Find the value of the pronumerals in the following diagrams. Give a reason for your answer in each case.

Q.2. Find the values of the pronumerals in the following diagram.

                       

Q.3.  Find the value of the pronumerals in the following diagrams. Give a reason for your answer in each case.

        

 

Q.4.     Show that AB is parallel to CD.

                       

ANSWERS

Q.1.     a = 74 (alternate angles are equal); b = 60 (co-interior angles are supplementary); c = 54 (alternate angles are equal)

Q.2.     a = 70, b = 70, c = 110, d = 110, e = 70, f = 110.

Q.3.     (i) a = 60 (exterior angle of triangle = sum of interior opposite angles

130 = 70 + a    a = 60)

            b = 50 (BCD is a straight angle = 180o  b = 180 – 130 = 50)

 (ii)       a = 50 (angle sum of D = 180o) b = 50  (BAC = 90o)

            c = 40 (angle sum of DBAC = 180o)     e = 140 (DCE straight line)

Q.4.     DCB = 120o (sum of angles in DBCD = 180o)

            ABC = 120o (given)

            Since DCB = ABC and are alternate then AB || CD

GEOMETRY II

 

Q.1.     Find the values of the pronumerals a & b and hence show AB||ED.

Q.2.     Find the values of the pronumerals a, b & c and hence show that DAFE is isosceles.

Q.3.     Triangles ABC and DBC are isosceles. Find the values of the pronumerals a, b & c, giving reasons for your answers.

ANSWERS

 

Q.1.     a = 100 (sum of angles in DDEC = 180o)

            b = 50 (sum of angles in DABC = 180o)

            Since ABC = EDC (each 50o) and these are complementary angles then ED || AB

Q.2.     ACB = 80o (BCD is a straight angle = 180o)

            a = 50o (angle sum of DABC = 180o)

            AEF = ABC = 50o (corresponding angles) b = 50

            c = 80 (angle sum of DAEF = 180o)

            Since a = b = 50o then DAEF is isosceles with EF = AF (base angles equal)

Q.3.     BCD = 30o (base angles of isosceles triangle are equal)

            a = 120 (angle sum of DDBC = 180o)

            b = 360 – 120 = 240 (angle sum at a point = 360o)

            BCA = 30 + 20 = 50o (base angles of isosceles triangle are equal)

            c = 80 (angle sum of DABC = 180o)