Download Geometry 6_4 Properties of Rhombuses_ Rectangles_ and Squares

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.
ABCD is a rhombus if and only if AB  BC  CD  AD.
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.
ABCD 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.
ABCD is a square if and only if AB  BC  CD  AD
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  CD
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  CD
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.
ABCD is a rhombus if and only if AC  BD.
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.
ABCD is a rhombus if and only if
AC bisects BCD and BAD
and BD bisects ABC and ADC.
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.
ABCD is a rectangle if and only if AC  BD.
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