8.2 Triangles, Polygons & The Pythagorean Theorem

Triangles, Polygons & The Pythagorean Theorem

Quick Summary: Triangles and polygons form the core of plane geometry. Every triangle has an interior angle sum of exactly $180^\circ$. For any $n$-sided convex polygon, the interior angle sum is given by $S = (n - 2) \times 180^\circ$. For right-angled triangles, the Pythagorean Theorem states that the sum of the squared leg lengths equals the square of the hypotenuse ($a^2 + b^2 = c^2$), which directly generates the 2D distance formula on the coordinate plane.

Understanding polygon properties and right-triangle relationships allows test-takers to navigate geometric word problems, solve for missing dimensions, and calculate real-world distances.

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Triangle Classifications & Fundamental Theorems

Triangles are three-sided polygons classified both by their angles and by their side lengths.

   Triangle Classifications by Angles and Sides
   
   By Angles:
     Acute (all < 90°)          Right (one = 90°)         Obtuse (one > 90°)
           /\                         │\                         /\
          /  \                        │ \                       /  ───
         /____\                       └───\                    /______\
   
   By Sides:
     Equilateral (3 equal)      Isosceles (2 equal)       Scalene (0 equal)
           /\                         /\                         /\
         / == \                     / || \                      /  \ 
        /______\                   /______\                    /____\ 
        (All 60°)                (Base angles equal)         (All sides different)

1. The Triangle Angle Sum Theorem

The interior angles of any triangle in Euclidean geometry sum to exactly $180^\circ$:

$$\angle A + \angle B + \angle C = 180^\circ$$

2. The Exterior Angle Theorem

An exterior angle of a triangle is formed by extending one of its sides outward. The measure of an exterior angle is always equal to the sum of the two remote (non-adjacent) interior angles:

$$\angle d = \angle a + \angle b$$

               Exterior Angle Theorem Visualized
                               A
                              /\
                             /  \
                            /    \
                           /      \
                          / a    b \
                        B/──────────\C─────────► D
                                       d (Exterior)
                         ∠d = ∠a + ∠b

3. The Triangle Inequality Theorem

For any valid triangle with side lengths $a$, $b$, and $c$, the sum of any two side lengths must be strictly greater than the third side:

$$a + b > c, \quad a + c > b, \quad \text{and} \quad b + c > a$$

Example: Can sides of length $4\text{ cm}, 5\text{ cm},$ and $10\text{ cm}$ form a triangle? No, because $4 + 5 = 9$, which is not greater than $10$.

Multi-Sided Polygons: Interior & Exterior Angles

A polygon is a closed two-dimensional figure made of straight line segments joined end-to-end. A polygon is regular if all sides are congruent and all interior angles are equal.

Interior Angle Sum Formula

Every $n$-sided polygon can be divided into $(n - 2)$ non-overlapping triangles by drawing diagonals from a single vertex. Because each triangle contains $180^\circ$, the total interior angle sum $S$ is:

$$S = (n - 2) \times 180^\circ$$

   Decomposing Polygons into Triangles from One Vertex
   
     Triangle (n=3)        Quadrilateral (n=4)         Pentagon (n=5)
          /\                    ┌─────────┐                /\
         /  \                   │ 1  /  2 │               / 1\ 2
        / 1  \                  │   /     │              /____\───3
       /______\                 └─────────┘             /_________\
      1 Triangle                2 Triangles             3 Triangles
      1 × 180° = 180°           2 × 180° = 360°         3 × 180° = 540°

Polygon Angle Properties Table

Exterior Angle Property: For any convex polygon, regardless of the number of sides, the sum of all exterior angles (one per vertex) is always $360^\circ$.

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The Pythagorean Theorem

For any right triangle with perpendicular legs of length $a$ and $b$, and hypotenuse of length $c$ (the side opposite the $90^\circ$ right angle):

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

                     The Pythagorean Anatomy
                                 |
                                 |\
                                 | \  Hypotenuse (c)
                          Leg a  |  \ (Opposite the 90° angle)
                                 |   \ (Always longest side)
                                 |┌───\
                                 └───--
                                  Leg b

Finding Missing Side Lengths

• Solving for Hypotenuse ($c$): $$c = \sqrt{a^2 + b^2}$$
• Solving for a Missing Leg ($a$ or $b$): $$a = \sqrt{c^2 - b^2} \quad \text{or} \quad b = \sqrt{c^2 - a^2}$$

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Recognizing triples and their scalar multiples ($k \cdot a, k \cdot b, k \cdot c$) eliminates tedious square-root calculations on the HiSET.

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The 2D Coordinate Distance Formula

The distance $d$ between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the Cartesian plane is a direct application of the Pythagorean Theorem:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

          Coordinate Distance as a Right Triangle
             y
             │           (x₂, y₂)
             │             •
             │            /│
             │        d  / │ Δy = (y₂ - y₁)
             │          /  │
             │ (x₁, y₁)•───┘
             │          Δx = (x₂ - x₁)
             └──────────────────────── x
             d² = (Δx)² + (Δy)²

Worked Example: Distance on the Grid

Find the distance between points $A(-3, 4)$ and $B(5, -2)$.

1. Identify coordinates: $x_1 = -3, y_1 = 4$ and $x_2 = 5, y_2 = -2$.
2. Calculate horizontal and vertical changes: $$\Delta x = 5 - (-3) = 8$$ $$\Delta y = -2 - 4 = -6$$
3. Substitute into formula: $$d = \sqrt{(8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$$

The distance is $10$ units.