GEOMETRY: Chapter 6
6.4: Properties of Rhombuses,
Rectangles, and Squares
Rhombus Corollary
A quadrilateral is a rhombus if and only if it
has four congruent sides.
A B C D is a rhom bus if and only if A B  B C  C D  A D .
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 533.
Rectangle Corollary
A quadrilateral is a rectangle if and only if
it has four right angles.
A B C D is a rectangle if and only if  A ,  B ,  C ,
and  D are right angles.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 533.
Square Corollary
A quadrilateral is a square if and only if it is
a rhombus and a rectangle.
A B C D is a square if and only if A B  B C  C D  A D
and  A ,  B ,  C , and  D are right angles.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 533.
Ex.1:
For any rectangle ABCD, decide whether the
statement is always or sometimes true. Draw a
sketch and explain your reasoning.
a . AB  C D
b . AB  BC
Ex.1:
For any rectangle ABCD, decide whether the
statement is always or sometimes true. Draw a
sketch and explain your reasoning.
a . AB  C D
b . AB  BC
ANSWERS:
a. always. All rectangles are parallelograms, and
opposite sides of parallelogram are congruent.
b. Sometimes; AB is congruent to BC provided the
rectangle ABCD is a square. But not all
rectangles are squares.
Ex. 2:
Classify the special quadrilateral. Explain your
reasoning.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 534.
Ex. 2:
Classify the special quadrilateral. Explain your
reasoning.
Answer: rhombus; it is a parallelogram because
opposite angles are congruent. Since a pair of
adjacent sides are congruent, all four sides are
congruent.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 534.
Theorem 6.11
A parallelogram is a rhombus if and only if
its diagonals are perpendicular.
A B C D is a rhom bus if and only if A C  B D .
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 535.
Theorem 6.12:A parallelogram is a rhombus if
and only if each diagonal bisects a pair of
opposite angles.
A B C D is a rh o m b u s if an d o n ly if
A C b isects  B C D an d  B A D
an d B D b isects  A B C an d  A D C .
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 535.
Theorem 6.13
A parallelogram is a rectangle if and only if
its diagonals are congruent.
A B C D is a rectangle if and only if A C  B D .
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 535.
Ex. 3:
Sketch a square EFGH. List everything that
you know about it.
Ex. 3:
Sketch a square EFGH. List everything that
you know about it.
Ex. 4: You are building a case with glass shelves for
collectibles.
a. Given the shelf measurements in the diagram, can
you assume that the shelf is a square? Explain.
b. You measure the diagonals; they are both 33.94
inches. What can you conclude about the shape?
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 536.
Ex. 4: You are building a case with glass shelves for
collectibles.
a. Given the shelf measurements in the diagram, can
you assume that the shelf is a square? Explain.
b. You measure the diagonals; they are both 33.94
inches. What can you conclude about the shape?
Answer: a) No. It has four congruent sides so it is a
rhombus. However, we do not know whether the
angles are right angles.
b) It is a square.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 536.
6.4, p. 351, #3-21 all, 33-38 all