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By — McGraw-Hill Professional
Updated on Oct 25, 2011

## 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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