Essential Theorems
Below are the criteria and proofs for parallelograms \(ABCD\):
- Theorem I & II: A diagonal divides a parallelogram into two congruent triangles (by ASA). Consequently, opposite sides and angles are equal.
- Theorem III: If each pair of opposite sides of a quadrilateral is equal, it is a parallelogram.
Logic: SSS Congruence \(\rightarrow\) Alternate interior angles are equal \(\rightarrow\) Sides are parallel.
- Theorem IV & V: In a parallelogram, opposite angles are equal. Conversely, if opposite angles are equal, the shape is a parallelogram.
- Theorem VI & VII: In a parallelogram, the diagonals of a parallelogram bisect each other. Conversely, If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Logic: Use ASA congruence on triangles formed by the intersection point \(O\) (e.g., \(\Delta AOD \cong \Delta COB\)).
- Theorem VIII: One Pair of Equal & Parallel Sides: A quadrilateral is a parallelogram if one pair of opposite sides is both equal and parallel.
Logic: Joining a diagonal creates two congruent triangles (ASA), proving the second pair of sides is also parallel.
Mid-point Theorem (Theorem IX & X)
This theorem describes the properties of a line segment connecting the mid-points of two sides of any triangle.- Theorem IX: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.
Proof: By constructing a line parallel to one side, you create congruent triangles (AAS) to prove the resulting quadrilateral is a parallelogram.
- Theorem X (Converse): A line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
Logic: Form a parallelogram via construction to prove the remaining segments are equal using CPCT.
Practical Applications
Specific shapes have unique behaviors when diagonals are introduced:Rectangle to Square: If a diagonal of a rectangle bisects its vertex angles, the rectangle must be a square.
Reasoning: If a diagonal bisects the angles, the resulting triangles are isosceles, meaning all four sides of the rectangle are equal.
Reasoning: If a diagonal bisects the angles, the resulting triangles are isosceles, meaning all four sides of the rectangle are equal.
Diagonal Property of Squares: In a square, diagonals bisect each of the vertex angles.
Important!
CPCT - Corresponding Parts of Congruence Triangles