Rhombuses Or Rhombi
What makes a quadrilateral a rhombus?
Rhombuses Or Rhombi
A rhombus is an
equilateral
parallelogram.
– All sides are
congruent
Rhombus Corollary
A quadrilateral is a
rhombus if and only
if it has four
congruent sides.
Rectangles
What makes a quadrilateral a rectangle?
Rectangles
A rectangle is an
equiangular
parallelogram.
• All angles are
congruent
Example 1
What must each angle of a rectangle
measure?
Rectangle Corollary
A quadrilateral is a
rectangle if and only
if it has four right
angles.
Squares
What makes a quadrilateral a square?
Squares
A square is a regular
parallelogram.
• All angles are
congruent
• All sides are
congruent
Square Corollary
A quadrilateral is a
square if and only if
it is a rhombus and
a rectangle.
Properties of Rhombuses, Rectangles, and
Squares
Objectives:
1. To discover and use properties of
rhombuses, rectangles, and squares
2. To find the area of rhombuses,
rectangles, and squares
Example 2
Below is a concept map showing the
relationships between some members of the
parallelogram family. This type of concept
map is known as a Venn Diagram. Fill in the
missing names.
Example 2
Below is a concept map showing the
relationships between some members of the
parallelogram family. This type of concept
map is known as a Venn Diagram.
Example 3
For any rhombus QRST, decide whether the
statement is always or sometimes true.
Draw a sketch and explain your reasoning.
1. Q  S
2. Q  R
Example 4
For any rectangle ABCD, decide whether the
statement is always or sometimes true.
Draw a sketch and explain your reasoning.
1. AB  CD
2. AB  BC
Example 5
Classify the special quadrilateral. Explain
your reasoning.
Diagonal Theorem 1
A parallelogram is a rectangle if and only if
its diagonals are congruent.
Example 6
The previous theorem is a biconditional.
Write the two conditional statements that
must be proved separately to prove the
entire theorem.
Example 7
You’ve just had a new door installed, but it
doesn’t seem to fit into the door jamb
properly. What could you do to determine
if your new door is rectangular?
Diagonal Theorem 2
A parallelogram is a rhombus if and only if its
diagonals are perpendicular.
Diagonal Theorem 3
A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite
angles.
Example 8
Prove that if a parallelogram has
perpendicular diagonals, then it is a
rhombus.
Given: ABCD is a
parallelogram;
AC  BD
Prove: ABCD is a
rhombus
Use Properties of Trapezoids and Kites
Objectives:
1. To discover and use properties of
trapezoids and kites
2. To find the area of trapezoids and kites
Trapezoids
What makes a quadrilateral a trapezoid?
Trapezoids
A trapezoid is a
quadrilateral with
exactly one pair of
parallel opposite
sides.
Trapezoid Parts
• The parallel sides
are called bases
• The non-parallel
sides are called
legs
• A trapezoid has two
pairs of base angles
Example 1
Find the value of x.
C
B
100
x
A
D
Trapezoid Theorem 1
If a quadrilateral is a trapezoid, then the
consecutive angles between the bases are
supplementary.
C
B
y
x
A
t
r
D
If ABCD is a trapezoid, then x + y = 180° and r + t = 180°.
Midsegment
A midsegment of a
trapezoid is a
segment that
connects the
midpoints of the
legs of a trapezoids.
Isosceles Trapezoid
An isosceles trapezoid is a trapezoid with
congruent legs.
Trapezoid Theorem 2
If a trapezoid is isosceles, then each pair of
base angles is congruent.
Trapezoid Theorem 3
A trapezoid is isosceles if and only if its
diagonals are congruent.
T
i
Trapezoid Theorem 4
The midsegment of a
trapezoid is parallel
to each base and its
length is one half
the sum of the
lengths of the
bases.
If
Example 2
Find the measure of each missing angle.
Example 3
For a project, you must cut an 11” by 14”
rectangular piece of poster board.
Knowing how poorly you usually wield a
pair of scissors, you decide to do some
measuring to make sure your board is truly
rectangular. Thus, you measure the
diagonals and determine that they are in
fact congruent. Is your board rectangular?
Example 4
Find the value of x.
Kites
What makes a quadrilateral a kite?
Kites
A kite is a
quadrilateral that
has two pairs of
consecutive
congruent sides,
but opposite sides
are not congruent.
Angles of a Kite
You can construct a kite by joining two
different isosceles triangles with a common
base and then by removing that common
base.
Two isosceles triangles can form one kite.
Angles of a Kite
Just as in an
isosceles triangle,
the angles between
each pair of
congruent sides are
vertex angles. The
other pair of angles
are nonvertex
angles.
Kite Theorem 1
If a quadrilateral is a kite, then the nonvertex
angles are congruent.
Kite Theorem 2
If a quadrilateral is a kite, then the diagonal
connecting the vertex angles is the
perpendicular bisector of the other
C
diagonal.
B
E
A
and CE  AE.
D
Kite Theorem 3
If a quadrilateral is a kite, then a diagonal
bisects the opposite non-congruent vertex
C
angles.
D
B
A
If ABCD is a kite, then BD bisects B and D.
Example 5
Quadrilateral DEFG is
a kite. Find mD.
Example 6
Find the measures of each side of kite
PQRS. Write your answers in simplest
radical form.
Example 7
Example 8
Example 9