Download Definition: Rectangle A rectangle is a parallelogram in which all four

Definition: Rectangle
A rectangle is a parallelogram in
which all four angles are congruent.
A
B
D
C
A parallelogram in
which all four angles are congruent
is a rectangle. (reverse of definition)
A
B
D
C
Each pair of angles must be
congruent and supplementary, so
they must all be right angles.
A
B
D
C
This quadrilateral is given as having
four right angles.
A
B
D
C
The four right angles allow us to prove
that the opposite sides are parallel, since
congruent and supplementary angles on
the same side of a transversal prove lines
parallel.
A
B
D
C
A quadrilateral with four right angles
must be a rectangle.
A
B
D
C
This parallelogram is given with one
known right angle.
A
B
D
C
Since opposite angles of a parallelogram
are congruent, we can mark angle B as a
right angle.
A
B
D
C
Since angles on the same side of a
transversal of two parallel lines are
supplementary, we can mark the other two
angles as right angles.
A
B
D
C
A parallelogram with a right angle is a
rectangle.
A
B
D
C
This parallelogram is given with
congruent diagonals,
.
A
B
D
C
Opposite sides of a parallelogram are
congruent.
A
B
D
C
Triangle ADC is congruent to triangle
BCD by SSS.
A
B
D
C
Angle ADC is congruent to angle BCD by
CPCTC.
A
B
D
C
The parallel sides of the parallelogram
mean that the two congurent angles are
also supplementary. The both must be
right angles.
A
B
D
C
Opposite angles of a parallelogram are
congruent, so all four of the angles are
right angles, and the parallelogram is a
rectangle.
A
B
D
C
If a parallelogram has congruent
diagonals, it is a rectangle.
Definition: Rhombus
A rhombus is a parallelogram in
which all four sides are congruent.
A
D
B
C
A parallelogram in
which all four sides are congruent is
a rhombus. (reverse of definition)
A
D
B
C
This parallelogram is
given with two consecutive sides
congruent.
A
D
B
C
Opposite sides of a parallelogram
are congruent, so all four sides are
congruent.
A
D
B
C
A parallelogram with two
consecutive sides congruent is a
rhombus.
A
D
B
C
This parallelogram has a diagonal
that bisects two of the angles.
A
D
B
C
Since the opposite angles of a
parallelogram are congruent, we
know that all four of the angles
marked in red are congruent.
A
D
B
C
Triangles ABC and CDA are
isosceles because the base angles are
congruent. The triangles are
congruent by ASA.
A
D
B
C
All four sides are congruent, so the
parallelogram is a rhombus.
A
D
B
C
If either diagonal of a parallelogram
bisects two angles of the
parallelogram, then it is a rhombus.
A
D
B
C
Here is a quadrilateral with
diagonals that are perpendicular
bisectors of each other.
A
B
M
D
C
Four congruent right triangles,
DMC, DMA, BMA, and BMC are
formed. The are congruent by SAS.
A
B
M
D
C
The four sides of the quadrilateral
are congruent by CPCTC.
A
B
M
D
C
The quadrilateral is a parallelogram,
because both pairs of opposite sides
are congruent. It is a rhombus
because all four sides are congruent.
A
B
M
D
C
A quadrilateral in which the two
diagonals are perpendicular bisectors
of each other is a rhombus.
Definition: Square
A square is a parallelogram in which
all four sides and all four angles are
congruent.
If a quadrilateral is both a rectangle
and a rhombus, then it is a square.
Definition: Kite
A kite is a quadrilateral with two distinct
pairs of congruent adjacent sides.
Definition: Kite
A kite is a quadrilateral with two
pairs of congruent adjacent sides.
A quadrilateral with two distinct
pairs of congruent adjacent sides is a kite.
(reverse of definition)
A quadrilateral with two pairs of
congruent adjacent sides is a kite.
(reverse of definition)
B
A
M
D
C
Here is a quadrilateral in which one of the
diagonals is the perpendicular bisector of
the other.
B
A
M
D
C
Triangle AMB is congruent to triangle
AMC by SAS. Triangle DMC is
congruent to triangle DMB by SAS.
B
A
M
D
C
AB is congruent to AC, and DB is
congruent to DC by CPCTC.
The quadrilateral is a kite, because two
distinct pairs of adjacent sides are
congruent.
B
A
M
D
C
If exactly one diagonal of a quadrilateral
is the perpendicular bisector of the other,
then the quadrilateral is a kite.
If one or both of the diagonals of a
quadrilateral is the perpendicular bisector
of the other, then the quadrilateral is a kite.
Definition: Isosceles trapezoid
An isosceles trapezoid is a trapezoid
in which the two sides that are not
parallel are congruent. Those two
sides are called the legs.
A trapezoid in which the two sides
that are not parallel are congruent is
an isosceles trapezoid.
A
D
B
C
Here is a trapezoid in which the
lower two base angles are congruent.
E
A
D
B
C
Since DA and CB are not parallel, the
lines containing the segments must
intersect. We can extend the segments
until they intersect at point E.
E
A
D
B
C
Angle EAB is congruent to angle EDC,
because they are corresponding angles.
Angle EBA is congruent to angle ECD,
because they are corresponding angles.
E
A
D
B
C
EA is congruent to EB, because "if
angles, then sides." ED is congruent to
EC for the same reason.
E
A
D
B
C
AD is congruent to BC by subtraction.
A
D
B
C
If either the upper or lower two base
angles of a trapezoid are congruent,
then the trapezoid is isosceles.
Ways to prove a quadrilateral is a rectangle.
A parallelogram in which all four angles
are congruent is a rectangle.
Ways to prove a quadrilateral is a rectangle.
A parallelogram in which all four
angles are congruent is a rectangle.
A quadrilateral with four right angles
must be a rectangle.
Ways to prove a quadrilateral is a rectangle.
A parallelogram in which all four
angles are congruent is a rectangle.
A quadrilateral with four right angles
must be a rectangle.
A parallelogram with a right angle is a
rectangle.
Ways to prove a quadrilateral is a rectangle.
A parallelogram in which all four
angles are congruent is a rectangle.
A quadrilateral with four right angles
must be a rectangle.
A parallelogram with a right angle is a
rectangle.
If a parallelogram has congruent
diagonals, it is a rectangle.
Ways to prove a quadrilateral is a rhombus.
A parallelogram in which all four
sides are congruent is a rhombus.
Ways to prove a quadrilateral is a rhombus.
A parallelogram in which all four
sides are congruent is a rhombus.
A parallelogram with two consecutive
sides congruent is a rhombus.
Ways to prove a quadrilateral is a rhombus.
A parallelogram in which all four
sides are congruent is a rhombus.
A parallelogram with two consecutive
sides congruent is a rhombus.
If either diagonal of a parallelogram
bisects two angles of the
parallelogram, then it is a rhombus.
Ways to prove a quadrilateral is a rhombus.
A parallelogram in which all four
sides are congruent is a rhombus.
A parallelogram with two consecutive
sides congruent is a rhombus.
If either diagonal of a parallelogram
bisects two angles of the
parallelogram, then it is a rhombus.
A quadrilateral in which the two
diagonals are perpendicular bisectors
of each other is a rhombus.
Ways to prove a quadrilateral is a square.
If a quadrilateral is both a rectangle
and a rhombus, then it is a square.
Ways to prove a quadrilateral is a kite.
A quadrilateral with two distinct
pairs of congruent adjacent sides is a kite.
A quadrilateral with two pairs of
congruent adjacent sides is a kite.
Ways to prove a quadrilateral is a kite.
A quadrilateral with two distinct
pairs of congruent adjacent sides is a kite.
A quadrilateral with two pairs of
congruent adjacent sides is a kite.
If exactly one diagonal of a quadrilateral
is the perpendicular bisector of the other,
then the quadrilateral is a kite.
If one or both of the diagonals of a
quadrilateral is the perpendicular bisector
of the other, then the quadrilateral is a kite.
Ways to prove a quadrilateral is an
isosceles trapezoid.
A trapezoid in which the two sides
that are not parallel are congruent is
an isosceles trapezoid.
Ways to prove a quadrilateral is an
isosceles trapezoid.
A trapezoid in which the two sides
that are not parallel are congruent is
an isosceles trapezoid.
If either the upper or lower two base
angles of a trapezoid are congruent,
then the trapezoid is isosceles.