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