Step 2: The Quadrilateral Taxonomy, Angle Sum ($360^\circ$) & Parallelograms
A quadrilateral is a 4-sided polygon whose internal angles always sum to four right angles ($360^\circ$).
1. Angle Sum Theorem of a Quadrilateral
- Theorem: The sum of all four interior angles of a quadrilateral is strictly $360^\circ$:
$$\mathbf{\angle A + \angle B + \angle C + \angle D = 360^\circ}$$
- Proof: Draw diagonal $AC$. $\Delta ABC$ has $\angle B + \angle 1 + \angle 2 = 180^\circ$. $\Delta ADC$ has $\angle D + \angle 3 + \angle 4 = 180^\circ$. Adding both yields $\angle A + \angle B + \angle C + \angle D = 360^\circ$. $\blacksquare$
2. The Complete Parallelogram Theorems Matrix
A quadrilateral $ABCD$ is a Parallelogram ($ABCD$ is a $\parallel\text{gm}$) if:
- Both pairs of opposite sides are parallel ($AB \parallel CD, AD \parallel BC$).
- Theorem 1: A diagonal divides it into two congruent triangles ($\Delta ABC \cong \Delta CDA$).
- Theorem 2: Opposite sides are equal ($AB = CD$ and $AD = BC$).
- Theorem 3: Opposite angles are equal ($\angle A = \angle C$ and $\angle B = \angle D$).
- Theorem 4: Consecutive angles are supplementary ($\angle A + \angle B = 180^\circ, \angle B + \angle C = 180^\circ$).
- Theorem 5: Diagonals bisect each other ($OA = OC, OB = OD$).
3. Golden Sufficient Condition for a Parallelogram
[!IMPORTANT] A quadrilateral is a parallelogram if one pair of opposite sides is both equal AND parallel:
$$AB = CD \quad \text{and} \quad AB \parallel CD \implies ABCD \text{ is a parallelogram!}$$