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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Every quadrilateral has certain properties, depending on the “species.” Here are some useful facts concerning these four-sided plane figures.

### Sum Of Measures Of Interior Angles

No matter what the shape of a quadrilateral, as long as all four sides are straight line segments of positive and finite length, and as long as all four vertices lie in the same plane, the sum of the measures of the interior angles is 360° (2 π rad). Figure 3-7 shows an example of an irregular quadrilateral. The interior angles are denoted w, x, y , and z . In this example, angle w measures more than 180° ( π rad). If you use your imagination, you might call this type of quadrilateral a “boomerang,” although this is not an official geometric term.

Fig. 3-7 . In any plane quadrilateral, the sum of the measures of the interior angles w, x, y , and z is 360° (2 π rad).

## Parallelogram Diagonals

Suppose we have a parallelogram defined by four points P, Q, R , and S . Let D be a line segment connecting P and R as shown in Fig. 3-8A. Then D is a minor diagonal of the parallelogram, and the following triangles defined by D are congruent:

Fig. 3-8 . Triangles defined by the minor diagonal ( A ) or the major diagonal ( B ) of a parallelogram are congruent.

Let E be a line segment connecting Q and S (Fig. 3-8B). Then E is a major diagonal of the parallelogram, and the following triangles defined by E are congruent:

## Bisection Of Parallelogram Diagonals

Suppose we have a parallelogram defined by four points P, Q, R , and S . Let D be the diagonal connecting P and R ; let E be the diagonal connecting Q and S (Fig. 3-9). Then D and E bisect each other at their intersection point T . In addition, the following pairs of triangles are congruent:

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. 3-9 . The diagonals of a parallelogram bisect each other.

## Rectangle

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

∠QRS = 90° = π /2 rad

∠RSP = 90° = π /2 rad

∠SPQ = 90° = π /2 rad

∠PQR = 90° = π /2 rad

Then all four interior angles are right angles, and the parallelogram is a rectangle : a four-sided plane polygon whose interior angles are all congruent. The converse of this is also true: If a quadrilateral is a rectangle, then any given interior angle is a right angle. Figure 3-10 shows an example of a parallelogram PQRS in which ∠QRS = 90° = π /2 rad. Because one angle is a right angle and opposite pairs of sides are parallel, all four of the angles must be right angles.

Fig. 3-10 . If a parallelogram has one right interior angle, then the parallelogram is a rectangle.

## Rectangle Diagonals

Suppose we have a parallelogram defined by four points P, Q, R , and S . Let D be the diagonal connecting P and R ; let E be the diagonal connecting Q and S . Let the length of D be denoted by d ; let the length of E be denoted by e (Fig. 3-11). If d = e , then the parallelogram is a rectangle. The converse is also true: if a parallelogram is a rectangle, then d = e . Thus, a parallelogram is a rectangle if and only if its diagonals have equal lengths.

Fig. 3-11 . The diagonals of a rectangle have equal length.

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