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.

5.10 Prop. of Rhom., Rect., and Squares

Pg. 331, 5.10 #2-24 even, 25