Triangles

What is a similar figure? In geometry, similar figures are shapes that have the exact same shape, but not necessarily the same size. Think of a photograph of yourself: whether it is printed as a small passport-size photo or a large wall poster, your features look exactly the same because the proportions are preserved. Only the size changes!

In contrast, congruent figures are identical twins: they have both the same shape and the same size. Therefore, all congruent figures are similar, but all similar figures are not congruent.

For two polygons to be similar, they must satisfy two strict conditions:

• Their corresponding angles must be equal.
• Their corresponding sides must be in the same ratio (proportional).

Criteria for Similarity of Triangles Instead of checking all three angles and all three sides every time, we can use short-cut rules to prove that two triangles are similar:

1. AAA (Angle-Angle-Angle) Criteria: If all three corresponding angles of two triangles are equal, their corresponding sides will automatically be proportional, making the triangles similar. (Even if just two angles are equal, the third must be equal due to the angle sum property. This is often called the AA criteria). 2. SSS (Side-Side-Side) Criteria: If the three corresponding sides of two triangles are in the same ratio, then their corresponding angles will automatically be equal, making them similar. 3. SAS (Side-Angle-Side) Criteria: If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, the triangles are similar.

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DIAGRAM 1: CONGRUENT VS SIMILAR TRIANGLES

    Congruent Triangles (Same Shape & Size)
         /\                 /\
        /  \               /  \
     5 /____\ 5         5 /____\ 5
         6                  6
  
    Similar Triangles (Same Shape, Scaled Size)
         /\
        /  \                   /\
     3 /____\ 3               /  \
         4                 6 /____\ 6
                                8

DIAGRAM 2: AA SIMILARITY CRITERION

     Triangle ABC               Triangle DEF
          /\                         /\
         /  \                       /  \
        / 60 \                     / 60 \
       /      \                   /      \
      /40    80\                 /40    80\
     /__________\               /__________\
     
     Since Angle A = Angle D, Angle B = Angle E, and Angle C = Angle F,
     Triangle ABC ~ Triangle DEF by AAA Similarity.

DIAGRAM 3: REAL-LIFE APPLICATION (SHADOW PUPPET SCALE)

       Light Source
           O
          / \
         /   \     Object (Hand)
        /     \_______|_
       /       \      |
      /         \     v
     /___________\________ [Wall Shadow: Scaled Up & Similar]

The Basic Proportionality (Thales) theorem: a line drawn parallel to one side of a triangle divides the other two sides in the same ratio. Its converse also holds.

What is the Pythagoras Theorem? The Pythagoras Theorem is one of the most famous and useful rules in geometry. It applies exclusively to right-angled triangles (triangles where one angle is exactly 90 degrees).

The theorem states: In a right-angled triangle, the square of the length of the hypotenuse (the longest side, directly opposite the 90-degree angle) is equal to the sum of the squares of the lengths of the other two sides (often called the base and the height).

Mathematical Equation:

$$\text{Base}^2 + \text{Height}^2 = \text{Hypotenuse}^2$$

Or simply:

$$a^2 + b^2 = c^2$$

(where $c$ is the length of the hypotenuse, while $a$ and $b$ are the other two sides).

What is the Converse of the Pythagoras Theorem? The word "converse" means looking at a rule backward. The Converse of the Pythagoras Theorem states: If a triangle has three sides such that the square of the longest side equals the sum of the squares of the other two sides, then the triangle must be a right-angled triangle. The 90-degree angle will always be located directly opposite that longest side.

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DIAGRAM 1: THE RIGHT-ANGLED TRIANGLE STRUCTURE

         |\
         | \
         |  \   Hypotenuse (c) -> Longest side, opposite to 90°
  Side 1 |   \
     (a) |    \
         |_____\
          Side 2 (b)
          [Note the square box corner indicating the 90° angle]

DIAGRAM 2: VISUAL PROOF OF SQUARES (3-4-5 Triangle)

             /\
            /  \ 5
         4 /____\
            3
            
     Side 1 squared: 4 * 4 = 16 blocks
     Side 2 squared: 3 * 3 = 9 blocks
     Hypotenuse squared: 5 * 5 = 25 blocks
     Notice that: 16 + 9 = 25!

DIAGRAM 3: REAL-WORLD USE (LADDER ON A WALL)
         | \
    Wall |  \ Ladder (c)
     (a) |   \
         |____\
          Ground (b)