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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

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 ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

3 Special Parallelograms: Rectangle, Rhombus, and Square

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