properties of squares and rectangles

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10.1

LEARNINg goALS

In 2010, a technology company unveiled what was billed at the time as the world’s largest plasma TV. This monster of a TV measured 12 2 __ 3 feet across the diagonal

(152 inches long) and displayed nearly 9 million pixels in HD and, if necessary, 3D.

Just two years earlier, the same company released a 103-inch version (89.3 inches long) which you could buy for a mere $70,000.

How much would you estimate the 152-inch TV cost if the company set its prices based on the total viewing areas for their TVs?

KEy TERM

• Perpendicular/Parallel Line TheoremIn this lesson, you will:

• Prove the Perpendicular/Parallel Line Theorem .

• Construct a square and a rectangle .• Determine the properties of a square

and rectangle .• Prove the properties of a square and

a rectangle .• Solve problems using the properties of a

square and a rectangle .

Squares and RectanglesProperties of Squares and Rectangles

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742 Chapter 10 Properties of Quadrilaterals

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Problem 1 Know the Properties or Be Square!

A quadrilateral is a four-sided polygon . A square is a quadrilateral with four right angles and all sides congruent .

Quadrilaterals have different properties that are directly related to the measures of their interior angles and their side lengths . Perpendicular lines and right angles are useful when proving properties of certain quadrilaterals .

1. Ramira says that if two lines are perpendicular to the same line, then the two lines are parallel to each other . Is Ramira correct? If she is correct, complete the proof to justify the reasoning, or state why she is not correct .

Given: 1 ' 3; 2

' 3

Prove: 1 i 2

�1

�2

�3

1 2

43

5 6

87

Remember, in proofs, start with the Given statement or statements.

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10.1 Properties of Squares and Rectangles 743

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The Perpendicular/Parallel Line Theorem states: “If two lines are perpendicular to the same line, then the two lines are parallel to each other.”

2. Draw a square with two diagonals. Label the vertices and the intersection of the diagonals. List all of the properties you know to be true.

3. Use ___

AB to construct square ABCD with diagonals ___

AC and ___

BD intersecting at point E.

BA

A diagonal of a polygon is a

line segment that connects two non-adjacent

vertices.

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4. Prove the statement DAB CBA .

Given: Square ABCD with diagonals ___

AC and ___

BD intersecting at point E

Prove: DAB CBA

5. Do you have enough information to conclude ___

AC ___

BD ? Explain your reasoning .

D C

BA

E

You have just proven a

property of a square: that its diagonals are congruent. You can now use this property as a

valid reason in future proofs.

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6. Prove the statement ___

DA i ___

CB and ___

DC i ___

AB .

Given: Square ABCD

Prove: ___

DA i ___

CB and ___

DC i ___

AB

7. If a parallelogram is a quadrilateral with opposite sides parallel, do you have enough information to conclude square ABCD is a parallelogram? Explain your reasoning .

D C

BA

You have just proven another

property of a square: that its opposite sides are parallel. You can now use this property

as a valid reason in future proofs.

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8. Prove the statement ___

DE ___

BE and ___

CE ___

AE .

Given: Square ABCD with diagonals ___

AC and ___

BD intersecting at point E

Prove: ___

DE ___

BE and ___

CE ___

AE

9. Do you have enough information to conclude the diagonals of a square bisect each other? Explain your reasoning .

D C

BA

E

You have just proven another

property of a square: that its diagonals bisect each other. You can now use this property

as a valid reason in future proofs.

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10. Prove that the diagonals of a square bisect the vertex angles . Use square ABCD in Question 8 .

11. Prove that the diagonals of a square are perpendicular to each other . Use square ABCD in Question 8 .

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Problem 2 The Rectangle

A rectangle is a quadrilateral with opposite sides congruent and all angles congruent .

1. Draw a rectangle with two diagonals . Label the vertices and the intersection of the two diagonals . List all of the properties you know to be true .

2. Use ___

RE to construct rectangle RECT with diagonals ___

RC and ___

ET intersecting at point A . Do not construct a square .

ER

A square is also a

rectangle. But, don’t draw a

square.

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3. Prove the statement RCT ETC .

Given: Rectangle RECT with diagonals ___

RC and ___

ET intersecting at point A

Prove: RCT ETC

4. Do you have enough information to conclude ___

RT ___

EC ? Explain your reasoning .

R

A

E

CT

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5. Describe how you could prove the second pair of opposite sides of the rectangle are congruent .

6. Do you have enough information to conclude rectangle RECT is a parallelogram? Explain your reasoning .

7. Do you have enough information to conclude the diagonals of a rectangle are congruent? Explain your reasoning .

8. Do you have enough information to conclude the diagonals of a rectangle bisect each other? Explain your reasoning .

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Problem 3 Using Tools

1. Ofelia is making a square mat for a picture frame . How can she make sure the mat is a square using only a ruler?

2. Gretchen is putting together a bookcase . It came with diagonal support bars that are to be screwed into the top and bottom on the back of the bookcase . Unfortunately, the instructions were lost and Gretchen does not have the directions or a measuring tool . She has a screwdriver, a marker, and a piece of string . How can Gretchen attach the supports to make certain the bookcase will be a rectangle and the shelves are parallel to the ground?

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3. Matsuo knows this birdhouse has a rectangular base, but he wonders if it has a square base .

a. What does Matsuo already know to conclude the birdhouse has a rectangular base?

b. What does Matsuo need to know to conclude the birdhouse has a square base?

4. Consider each LED television shown .

29.4°y

x 27˝

32°y

x30˝

a. Determine the dimensions of the 270 LED TV .

A TV screen is

always measured in inches from corner

to corner on a diagonal.

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b. Determine the dimensions of the 300 LED TV .

c. Compare the viewing area of each size television screen .

d. What property of a rectangle would be helpful when locating the center point of the television screen?

e. What property of a rectangle would be helpful when determining the perimeter of the television screen?

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5. Sketch a square . Label the midpoint of each side of the square .

a. Determine the polygon formed by connecting the midpoints of each side of a square and justify your conclusion .

x

y

Consider the sketch of your square so your

algebraic justifications will include x- and

y-intercepts.

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b. If the same process was repeated one more time by connecting the midpoints of each side of the polygon determined in part (a), describe the polygon that would result .

Be prepared to share your solutions and methods .

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properties of squares and rectangles

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