Proofs with Quadrilaterals

Writing proofs for quadrilaterals involves using established geometric properties, theorems, and triangle congruence to verify the characteristics of shapes like parallelograms, rectangles, rhombuses, and trapezoids. In Grade 10, you will typically encounter two main types of proofs: coordinate geometry proofs and formal geometric proofs.

Coordinate Geometry Proofs

Coordinate proofs use the $xy$-plane to prove geometric properties. You will heavily rely on three formulas:

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (to prove segments are congruent)
• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ (to prove lines are parallel or perpendicular)
• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ (to prove diagonals bisect each other)

Example: Given vertices $A(0,0)$, $B(4,0)$, $C(5,3)$, and $D(1,3)$, prove that $ABCD$ is a parallelogram.

To prove a quadrilateral is a parallelogram, we can show that both pairs of opposite sides are parallel by finding their slopes.

• Slope of $AB = \frac{0 - 0}{4 - 0} = 0$

• Slope of $CD = \frac{3 - 3}{1 - 5} = 0$ Since their slopes are equal, $AB \parallel CD$.

• Slope of $AD = \frac{3 - 0}{1 - 0} = 3$

• Slope of $BC = \frac{3 - 0}{5 - 4} = 3$ Since their slopes are equal, $AD \parallel BC$.

Because both pairs of opposite sides are parallel, quadrilateral $ABCD$ is a parallelogram.

Formal Geometric Proofs

Formal proofs (often written in two-column or paragraph format) use axioms, definitions, and theorems. A common strategy is to draw a diagonal to create two triangles, then use triangle congruence postulates (SSS, SAS, ASA, AAS, or HL) to prove parts are equal via CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Example: Prove that the diagonals of a rectangle are congruent.

Given: Rectangle $ABCD$ with diagonals $AC$ and $BD$. Prove: $AC \cong BD$

Proof Steps:

1. $ABCD$ is a rectangle. (Given)
2. $AD \cong BC$ (Opposite sides of a rectangle are congruent).
3. $\angle ADC$ and $\angle BCD$ are right angles. (Definition of a rectangle).
4. $\angle ADC \cong \angle BCD$ (All right angles are congruent).
5. $DC \cong CD$ (Reflexive Property of Congruence).
6. $\triangle ADC \cong \triangle BCD$ (Side-Angle-Side / SAS Congruence Postulate, using steps 2, 4, and 5).
7. $AC \cong BD$ (CPCTC - Corresponding parts of congruent triangles are congruent).

By mastering coordinate formulas and triangle congruence, you can confidently prove the properties of any given quadrilateral.