Facts about Quadrilaterals Help
Facts about Quadrilaterals - Interior Angles
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.
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:
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.
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.
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.
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