15-4: Rhombuses, Rectangles, and Squares

15-4: 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
Assignment:
SpringBoard:
• P. 220: 17, 20-22
• P. 222: 8-13
Purple Geometry Book:
• P. 723: 5, 18, 25, 27, 40
• P. 732: 8, 11, 15, 20,
28, 36
• Challenge Problems
1
You
able
will
be
to
discover
and
use
properties
of
rhombuses,
rectangles,
and
squares
Rhombuses Or Rhombi
What makes a quadrilateral a rhombus?
Rhombuses Or Rhombi
A rhombus is
an equilateral
parallelogram.
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.
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.
Square Corollary
A quadrilateral is a
square if and only if
it is a rhombus and
a rectangle.
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.
Investigation 1
We know that the
diagonals of
parallelograms bisect
each other. The
diagonal of rectangles
and rhombuses have
a few other properties
we will discover using
Geogebra.
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.
Constructions!
Use your compass and straightedge, combined with the properties of
rhombuses, rectangles, and squares, to construct one of each special
parallelogram.
Golden Rectangle
Example 8
Prove that if a parallelogram has
perpendicular diagonals, then it is a
rhombus.
Given: ABCD is a
parallelogram;
𝐴𝐵 ⊥ 𝐵𝐷
Prove: ABCD is a
rhombus
Example 9
Given: 𝑚∠𝑆𝐶𝑅 ≠ 𝑚∠𝐶𝑆𝐼
Prove: Parallelogram 𝑅𝐼𝑆𝐶 is not a rectangle
Objective 2
You will be able to find the area of
rhombuses, rectangles, and
squares
Example 10: SAT
In the figure, a small
square is inside a
larger square.
What is the area, in
terms of x, of the
shaded region?
Example 11
In the diagram at the
right, MRVU 
SPTV. Let the area
of MRVU equal A.
Show that A = bh.
Rhombus Area
Since a rhombus is a
parallelogram, we
could find its area
by multiplying the
base and the
height.
A  bh
Rhombus Area
However, you’re not
always given the
base and height, so
let’s look at the two
diagonals. Notice
that d1 divides the
rhombus into 2
congruent triangles.
Ah, there’s a couple of
triangles in there.
1
A  bh
2
Rhombus Area
So find the area of
one triangle, and
then double the
result.
1

A  2 b  h
2

1 
1
A  2  d1  d 2 
2 
2
1
1

A  2  d1  d 2   d1  d 2
2
4

Ah, there’s a couple of
triangles in there.
1
A  bh
2
1
A  d1  d 2
2
Polygon Area Formulas
Example 12
Find the area of the shaded region.
1.
2.
3.
Example 13
If the length of each diagonal of a rhombus is
doubled, how is the area of the rhombus
affected?
15-4: 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
Assignment:
SpringBoard:
• P. 220: 17, 20-22
• P. 222: 8-13
Purple Geometry Book:
• P. 723: 5, 18, 25, 27, 40
• P. 732: 8, 11, 15, 20,
28, 36
• Challenge Problems