**Introduction to Properties of Quadrilaterals**

A four-sided geometric figure that lies in a single plane is called a *quadrilateral* . There are several classifications, and various formulas that apply to each.

**Parallelograms**

**Parallelogram Diagonals**

Suppose we have a parallelogram defined by four distinct points *P, Q, R* , and *S* . Let *D* be a straight line segment connecting *P* and *R* as shown in Fig. 9-22A. Then *D* is a *minor diagonal* of the parallelogram, and the triangles defined by *D* are congruent:

Δ *PQR* ≅ Δ *RSP*

Let *E* be a line segment connecting *Q* and *S* (Fig. 9-22B). Then *E* is a *major diagonal* of the parallelogram, and the triangles defined by *E* are congruent:

Δ *QRS* ≅ Δ *SPQ*

**Fig. 9-22.** Triangles defined by the minor diagonal (A) or the major diagonal (B) of a parallelogram are congruent.

**Bisection of Parallelogram Diagonals**

Suppose we have a parallelogram defined by four distinct points *P, Q, R* , and *S* . Let *D* be the straight diagonal connecting *P* and *R* ; let *E* be the straight diagonal connecting *Q* and *S* (Fig. 9-23). Then *D* and *E* bisect each other at their intersection point *T* . In addition, the following pairs of triangles are congruent:

Δ *PQT* ≅ Δ *RST*

Δ *QRT* ≅ Δ *SPT*

The converse of the foregoing is also true: if we have a plane quadrilateral whose diagonals bisect each other, then that quadrilateral is a parallelogram.

**Fig. 9-23.** The diagonals of a parallelogram bisect each other.

**Rectangles**

Suppose we have a parallelogram defined by four distinct points *P, Q, R* , and *S* . Suppose any of the following statements is true:

∠ *PQR* = 90°

∠ *QRS* = 90°

∠ *RSP* = 90°

∠ *SPQ* = 90°

Then all four interior angles measure 90°, and the parallelogram is a *rectangle* : a four-sided plane polygon whose interior angles are all congruent (Fig. 9-24). The converse of this is also true: if a quadrilateral is a rectangle, then any given interior angle has a measure of 90°.

**Fig. 9-24.** If a parallelogram has one right interior angle, then the parallelogram is a rectangle.

**Rectangle Diagonals**

Imagine a parallelogram defined by four distinct points *P, Q, R* , and *S* . Let *D* be the straight diagonal connecting *P* and *R* ; let *E* be the straight diagonal connecting *Q* and *S* . Let the length of *D* be denoted by *d* ; let the length of *E* be denoted by *e* (Fig. 9-25). If *d* = *e* , then the parallelogram is a rectangle. The converse is also true: if a parallelogram is a rectangle, then *d* = *e* . A parallelogram is a rectangle if and only if its diagonals have equal lengths.** **

**Fig. 9-25.** The diagonals of a rectangle have equal length.

**Rhombus Diagonals**

Imagine a parallelogram defined by four distinct points *P, Q, R* , and *S* . Let *D* be the straight diagonal connecting *P* and *R* ; let *E* be the straight diagonal connecting *Q* and *S* . If *D* is perpendicular to *E* , then the parallelogram is a *rhombus* , which is a four-sided plane polygon whose sides are all equally long (Fig. 9-26). The converse is also true: if a parallelogram is a rhombus, then *D* is perpendicular to *E* . A parallelogram is a rhombus if and only if its diagonals intersect at a right angle.

**Fig. 9-26.** The diagonals of a rhombus are perpendicular.

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