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3 Special Parallelograms: Rectangle, Rhombus, and Square

3 Special Parallelograms: Rectangle, Rhombus, and Square

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represent the set of squares.



5.3B Principles Involving Properties of the Special Parallelograms

PRINCIPLE



1:



 A rectangle, rhombus, or square has all the properties of a parallelogram.



PRINCIPLE



2:



 Each angle of a rectangle is a right angle.



PRINCIPLE



3:



 The diagonals of a rectangle are congruent.



Thus in rectangle ABCD in Fig. 5-14,

PRINCIPLE



4:



.



 All sides of a rhombus are congruent.



Fig. 5-14



Fig. 5-15

PRINCIPLE



5:



 The diagonals of a rhombus are perpendicular bisectors of each other.



Thus in rhombus ABCD in Fig. 5-15,

PRINCIPLE



6:



7:



are



bisectors of each other.



 The diagonals of a rhombus bisect the vertex angles.



Thus in rhombus ABCD,

PRINCIPLE



and



bisects



 The diagonals of a rhombus form four congruent triangles.



Thus in rhombus ABCD,

PRINCIPLE



8:



.



.



 A square has all the properties of both the rhombus and the rectangle.



By definition, a square is both a rectangle and a rhombus.



5.3C Diagonal Properties of Parallelograms, Rectangles, Rhombuses, and

Squares

Each check in the following table indicates a diagonal property of the figure.



5.3D Proving that a Parallelogram is a Rectangle, Rhombus, or Square

Proving that a Parallelogram is a Rectangle

The basic or minimum definition of a rectangle is this: A rectangle is a parallelogram having one

right angle. Since the consecutive angles of a parallelogram are supplementary, if one angle is a right

angle, the remaining angles must be right angles.

The converse of this basic definition provides a useful method of proving that a parallelogram is a

rectangle, as follows:

PRINCIPLE



9:



 If a parallelogram has one right angle, then it is a rectangle.



Thus if ABCD in Fig. 5-16 is a



and



= 90°, then ABCD is a rectangle.



Fig. 5-16

PRINCIPLE



10: If a parallelogram has congruent diagonals, then it is a rectangle.



Thus if ABCD is a



and



, then ABCD is a rectangle.



Proving that a Parallelogram is a Rhombus

The basic or minimum definition of a rhombus is this: A rhombus is a parallelogram having two

congruent adjacent sides.

The converse of this basic definition provides a useful method of proving that a parallelogram is a

rhombus, as follows:

PRINCIPLE



11: If a parallelogram has congruent adjacent sides, then it is a rhombus.



Thus if ABCD in Fig. 5-17 is a



, then ABCD is a rhombus.



and



Fig. 5-17



Proving that a Parallelogram is a Square

PRINCIPLE



12: If a parallelogram has a right angle and two congruent adjacent sides, then it is a

square.



This follows from the fact that a square is both a rectangle and a rhombus.

SOLVED PROBLEMS



5.7 Applying algebra to the rhombus

Assuming ABCD is a rhombus, find x and y in each part of Fig. 5-18.



Fig. 5-18



Solutions

(a) Since

, 3x – 7 = 20 or x = 9. Since

equilateral, and so y = 20.



is equiangular it is



, 5y + 6 = y + 20 or y = 3 . Since



(b) Since



, x = y + 20



or x = 23 .

(c) Since



bisects



, 4x – 5 = 2x + 15 or x = 10. Hence, 2x + 15 = 35 and



= 2(35°)



= 70°. Since



and



are supplementary, y + 70 = 180 or y = 110.



5.8 Proving a special parallelogram problem



PROOF:



5.9 Proving a special parallelogram problem stated in words

Prove that a diagonal of a rhombus bisects each vertex angle through which it passes.

Solution



PROOF:



5.4 Three or More Parallels; Medians and Midpoints

5.4A Three or More Parallels

PRINCIPLE



1:



 If three or more parallels cut off congruent segments on one transversal, then they

cut off congruent segments on any other transversal.

in Fig. 5-19 and segments a and b of transversal



Thus if



segments c and d of transversal



are congruent, then



are congruent.



Fig. 5-19



5.4B Midpoint and Median Principles of Triangles and Trapezoids

PRINCIPLE



Thus in



2:



 If a line is drawn from the midpoint of one side of a triangle and parallel to a

second side, then it passes through the midpoint of the third side.

in Fig. 5-20 if M is the midpoint of



and



, then N is the midpoint of



.



PRINCIPLE



 If a line joins the midpoints of two sides of a triangle, then it is parallel to the

third side and its length is one-half the length of the third side.



3:



, if M and N are the midpoints of



Thus in



and



, then



and MN = AC.



Fig. 5-20



Fig. 5-21

PRINCIPLE



 The median of a trapezoid is parallel to its bases, and its length is equal to onehalf of the sum of their lengths.



4:



is the median of trapezoid ABCD in Fig. 5-21 then



Thus if



, and m =



(b + b′).

PRINCIPLE



 The length of the median to the hypotenuse of a right triangle equals one-half the

length of the hypotenuse.



5:



Thus in rt.



in Fig. 5-22, if



is the median to hypotenuse



, then CM = AB; that is,



.



Fig. 5-22

PRINCIPLE



Thus if



 The medians of a triangle meet in a point which is two-thirds of the distance from

any vertex to the midpoint of the opposite side.



6:

,



, and



are medians of



in Fig. 5-23, then they meet in a point G which is



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