Post on 27-Dec-2015
transcript
5.10 Prop. of Rhom., Rect., and Squares
Rhombus CorollaryA quadrilateral is a rhombus if and only if it has
four congruent _______.
.AD CD BC AB ABCD is a rhombus if and only if
sidesA
D
B
C
5.10 Prop. of Rhom., Rect., and Squares
Rectangle CorollaryA quadrilateral is a rectangle if and only if
it has four _____________.
angles.right are D and C, B, A, ABCD is a rectangle if and only if
right angles
A
D
B
C
5.10 Prop. of Rhom., Rect., and Squares
Square CorollaryA quadrilateral is a square if and only if it is a
________ and a __________.
and AD CD BC AB ABCD is a square if and only if
rhombus rectangleA
D
B
C
angles.right are D and C, B, A,
5.10 Prop. of Rhom., Rect., and Squares
Example 1 Use properties of special quadrilateralsUse properties of special quadrilaterals
For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.
a. By definition, a rhombus is a parallelogram with four congruent _______.
VS a. R
V
S
Tsidesparallelogram
By Theorem 5.19, opposite angles of a parallelogram are ___________.congruent V.S So, The statement is _________ true.always
5.10 Prop. of Rhom., Rect., and Squares
Example 1 Use properties of special quadrilateralsUse properties of special quadrilaterals
For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.
b. If rhombus RSTV is a _______, then all four angles are congruent right angles
VT b.
square
Because not all rhombuses are also ___________,squares________. a is RSTV if VT So
the statement is ___________ true.sometimes
R
V
S
Tsquare
5.10 Prop. of Rhom., Rect., and Squares
Example 2 Classify special quadrilateralsClassify special quadrilaterals
Classify the special quadrilateral. Explain your reasoning.
The quadrilateral has four congruent _____.sidesOne of the angles is not a ____________,right angle
o127
so the rhombus is not also a ________.squareBy the Rhombus Corollary, the quadrilateral is a __________.rhombus
5.10 Prop. of Rhom., Rect., and SquaresCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1.1. For any square CDEF, is it For any square CDEF, is it alwaysalways or or sometimessometimes
true thattrue that ? Explain your reasoning.DECD C
F
D
E
By the Square Corollary, all the sides are congruent.
Therefore, it is always true.
5.10 Prop. of Rhom., Rect., and SquaresCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2.2. A quadrilateral has four congruent sides and four A quadrilateral has four congruent sides and four
congruent angles. Classify the quadrilateral.congruent angles. Classify the quadrilateral.C
F
D
E
By the Square Corollary, all the sides are congruent and all angles are right angles.The quadrilateral is a square.
o3604 xo90x
x x
xx
5.10 Prop. of Rhom., Rect., and Squares
Theorem 5.26A parallelogram is a rhombus if and only if its
diagonals are _____________.
.________
perpendicular
ABCD is a rhombus if and only if
AC BD
A
D
B
C
5.10 Prop. of Rhom., Rect., and Squares
Theorem 5.27A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite sides.
____ and____ bisects AC ABCD is a rhombus if and only if
BCD
A
D
B
C
BAD
.____ and____ bisects BD ABC ADC
5.10 Prop. of Rhom., Rect., and Squares
Theorem 5.28A parallelogram is a rectangle if and only if
its diagonals are __________.congruent
A
D
B
C
.________ ABCD is a rectangle if and only if
AC BD
5.10 Prop. of Rhom., Rect., and Squares
Example 3 List properties of special parallelogramsList properties of special parallelogramsSketch rhombus FGHJ. List everything you know about it.
By definition, you need to draw a figure with the following properties:
• The figure is a _______________.parallelogramsides
Because FGHJ is a parallelogram, it has these properties:
F
J
G
H
Solution
• The figure has four congruent _______.
• opposite sides are ________ and ___________.parallel congruent• opposite angles are __________. Consecutive angles are _________________.
congruentsupplementary
• diagonals ________ each other.bisectBy Theorem 5.26, the diagonals of FGHJ are _______________.perpendicular
By Theorem 5.27, each diagonals bisect a pair of _____________.opposite angles
5.10 Prop. of Rhom., Rect., and Squares
Example 4 Solve a real-world problemSolve a real-world problemFraming You are building a frame for a painting. The measurements of the frame are shown at the right.a. The frame must be a rectangle. Given
the measurements in the figure, can you assume that it is? Explain.
SolutionNo, you cannot. The boards on opposite sides are the same length, so they form a ______________. parallelogramBut you do not know whether the angles are ______________. right angles
5.10 Prop. of Rhom., Rect., and Squares
Example 4 Solve a real-world problemSolve a real-world problemFraming You are building a frame for a painting. The measurements of the frame are shown at the right.b. You measure the diagonals of the
frame. The diagonals are about 25.6 inches. What can you conclude about the shape of the frame?
SolutionBy Theorem 5.28, the diagonals of a rectangle are __________. congruentThe diagonals of the frame are __________, so the frame forms a ______________.
congruentrectangle
5.10 Prop. of Rhom., Rect., and SquaresCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 3. Sketch rectangle WXYZ.
List everything you know about it.The drawing of WXYZ is a parallelogram with four right angles.
W
Z
X
YIt has these characteristics:
• opposite sides are parallel and congruent.• opposite angles are congruent and consecutive angles are supplementary.• diagonals bisect each other and are congruent.
5.10 Prop. of Rhom., Rect., and SquaresCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4. Suppose the diagonals of the frame
in Example 4 are not congruent. Could the frame still be a rectangle?
By Theorem 5.28, a rectangle must have congruent diagonals.