G3 Geometry : Angles Formulae

Triangles - 3 sides, sum interior angles = 180^o

     Sum Angles of a triangle = 180^o
      ∠a + ∠b + ∠c = 180^o       

     The three angles inside a triangle always add up to 180°

(1) Equilateral  Triangle

        (a) All sides are of equal length.
       (b) Sum of Angles = 180^o       
       (c) Each angle = 180 / 3 = 60 ^o  
                 

(2) Isosceles Triangle

        (a) 2 sides of equal length (parallel black lines)

        (b) 2 angles are of equal degree
                 ∠a = ∠b
        (c) All angles add to 180^o

(3) Scalene Triangle

        (a) No equal sides, each side is of different length

        (b) No equal angles, each angle is different

        (c) All angles add to 180^o

Right-angle triangle         

      (a) one of the angle = 90^o

     (b) The two sides that form the right angle are the base and height

OTHER ANGLES FORMULAE

Turns/Rotation

~ Associate the amount of turning(rotation) with an amount measured in degree.

~ Turns can be clockwise or anti-clockwise.

   3/4 turn = 270^o.

Adjacent Angles on a Straight Line
    ∠a + ∠b = 180^o
    ∠c + ∠d = 180^o .

   << Adjacent ~ next to >>

Vertically Opposite Angles

Opposite angles at the point where 2 straight lines cross 
-> Vertically opposite angles are equal to each other.

       ∠a = ∠c
       ∠b = ∠d 

<< Opposite ~ facing    👉👈>>

Corresponding Angles (F)
The 2 arrows on the straight lines indicate that the lines are parallel to each other.

(1)   a and b are called corresponding angles and are equal to each other.

                        ∠a = ∠b

                        ∠c = ∠d

(2) Identify corresponding angles (F): parallel line, F (in any position), 
both angles ‘below’ the two F parallel lines and the ‘intersecting’ line

  << Correspond ~ equivalent 👉👉>>  << Remember equivalent fractions? 😉>>

Alternate Angles (Z)

The 2 arrows on the straight lines indicate that the lines are parallel to each other.

 (1) a and b are called alternate angles and are equal to each other.

                                                ∠a = ∠b

 (2) Identify alternate angles (Z) : parallel line, ‘Z’ and angles on both points of Z.

     << Alternate ~ occurs in turn  >>

Relating Different Angles

 Vertically Opposite               Corresponding    Alternate

            ∠c = ∠g                          ∠b = ∠g                   ∠a = ∠b

            ∠j = ∠k                           ∠d = ∠h                   ∠c = ∠d

           ∠e = ∠h                           ∠c = ∠p                   ∠p = ∠h

What are the other angles?

Quadrilateral  - 4 sides, Sum interior angles = 360^o

Parallelogram

(a) Opposite lines are parallel and equal

            AB is parallel to CD, AD is parallel to BC

            AB = CD , AD = BC

(b) Opposite angles are equal

            ∠a = ∠c, ∠b = ∠d

(c) Supplementary Angles ∠a + ∠d = 180^o

                                             ∠b + ∠c = 180^o

Trapezium

   (a) Line AB is parallel to CD

  (b) ∠a + ∠d = 180^o, ∠c + ∠b = 180 ^o

Rhombus

(a) Diagonal line AC and BD forms a right angle at the centre. 

(b) Line AB parallel to CD, Line AD parallel to BC

(c)  AB = BC = CD = AC

(d)  ∠a = ∠c , ∠d = ∠b

(e)  ∠a + ∠b = 180^o, ∠c + ∠d = 180^o

Note:

<<< tri ~ 3 , quad ~ 4, cir ~ round >>>

METHOD TO SOLVE GEOMETRY QUESTIONS

     Step1: Underline and Mark the keywords

     Step2: Double Mark Required angles

     Step3: Solve using Given Data (from step1&2) and Formulae

Example

In the figure, RSTU is a rhombus and SVU is an isosceles triangle. UV = VS, ∠STU = 72^o and ∠SVY = 44^o.

Find ∠UST and ∠TSV.

Step1: Underline and Mark the keywords

            - Rhombus, all sides equal, ‘mark’ all side with a single line

            - Isosceles triangle, mark 2 sides with 2 dashes and 2 angles. 

Step2: Double Mark Required angles

Step3: Solve using Given Data (from step1&2) and Formulae

          (1)   To find ∠UST

          What is given? ∠UST is within the triangle UST (isosceles triangle, and 72 ^o)

          Formula: Isosceles triangle, and sum angles of triangle

72^o + ∠TUS +∠UST = 180 ^o

∠TUS = ∠UST (isosceles triangle)

2∠UST = 180 ^o – 72 ^o

  ∠UST = 108 ^o / 2

                 = 54 ^o

           (2)  To find ∠TSV

          What is given? < isosceles triangle SUV, and 44 ^o)

          Formula: Isosceles triangle, and sum angles of triangle

44 ^o + VUS +USV = 180 ^o

∠VUS = ∠USV (isosceles triangle)

2USV = 180 ^o – 44 ^o

  ∠UST = 136 ^o / 2

                 = 68 ^o

              ∠TSV = 68 ^o – 54 ^o

                        = 14 ^o

Example

            

        Find x

    Step1: lines to indicate equilateral ∆

    Step2: double Mark ‘X’

    Step3: What is the formula that  can be used?

∆ABC is equilateral ∆.

Each angle = 60^o (angles in a triangle)

                X = 60^o

Example

        

    Find x

     Step1: lines to indicate isosceles ∆, 1 angle = 35 ^o

     Step2: Double Mark ‘X’

     Step3: Formula to use?

           (Isosceles angle -> 2 equal angles,angles in a ∆=180^o)

CAB = ACB = 35^o (isosceles triangle)

               35^o + 35^o + x = 180^o (angles in a ∆)

               X = 180^o – 70^o = 110^o  

Example:

           

   Find x

   < Formula? -  Angles at a point >           

   90^o + 140^o + 54^o + x = 360^o (angles at a point)

   x = 360^o – 90^o – 140^o – 54 ^o
      =  74^o                 

SUMMARY:
Triangles - 3 sides, sum interior angles = 180^o

Angles

Quadrilateral  - 4 sides, Sum interior angles = 360^o

Related Angles

          

~~~~ END ~~~~ :)