Rectangles, Rhombi, and Squares

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1 4 hat ou ll Learn ou ll learn to identify and use the properties of rectangles, rhombi, and squares. Rectangles, Rhombi, and Squares In previous lessons, you studied the properties of quadrilaterals and parallelograms. Now you will learn the properties of three other special types of quadrilaterals: rectangles, rhombi, and squares. The following diagram shows how these quadrilaterals are related. hy It s Important arpentry arpenters use the properties of rectangles when they build rectangular decks. See Exercise 46. Quadrilateral Parallelogram opposite sides parallel opposite sides congruent Rectangle parallelogram with 4 right angles Rhombus parallelogram with 4 congruent sides Square parallelogram with 4 congruent sides and 4 right angles Notice how the diagram goes from the most general quadrilateral to the most specific one. ny four-sided figure is a quadrilateral. ut a parallelogram is a special quadrilateral whose opposite sides are parallel. The opposite sides of a square are parallel, so a square is a parallelogram. In addition, the four angles of a square are right angles, and all four sides are equal. rectangle is also a parallelogram with four right angles, but its four sides are not equal. oth squares and rectangles are special types of parallelograms. The best description of a quadrilateral is the one that is the most specific. Lesson 4 Rectangles, Rhombi, and Squares 327

Re rld al o Example Identify the parallelogram that is outlined in the painting at the right. rt Link Parallelogram has four right angles, but the four sides are not congruent. It is a rectangle.

2 Re rld al o Example Identify the parallelogram that is outlined in the painting at the right. rt Link Parallelogram has four right angles, but the four sides are not congruent. It is a rectangle. our Turn a. Identify the parallelogram. iana Ong, lue, Red, and ellow Faces Rhombi is the plural of rhombus. Rectangles, rhombi, and squares have all of the properties of parallelograms. In addition, they have their own properties. Materials: Step dot paper ruler protractor raw a rhombus on isometric dot paper. raw a square and a rectangle on rectangular dot paper. Label each figure as shown below X 2 3 X X Step 2 Measure and X for each figure. Step 3 Measure 9, 0,, and 2 for each figure. Step 4 Measure through for each figure. Try These. For which figures are the diagonals congruent? 2. For which figures are the diagonals perpendicular? 3. For which figures do the diagonals bisect a pair of opposite angles? 32 hapter Quadrilaterals

3 The results of the previous activity can be summarized in the following theorems. ords Theorem 0 Models and Symbols The diagonals of a rectangle are congruent. The diagonals of a rhombus are perpendicular. 2 Each diagonal of a rhombus bisects a pair of opposite angles m m 2, m 3 m 4, m 5 m 6, m 7 m square is defined as a parallelogram with four congruent angles and four congruent sides. This means that a square is not only a parallelogram, but also a rectangle and a rhombus. Therefore, all of the properties of parallelograms, rectangles, and rhombi hold true for squares. Examples 2 Find X in square X if 4. square has all of the properties of a rectangle, and the diagonals of a rectangle are congruent. So, X is congruent to, and X 4. 3 O X Find m OX in square X. square has all the properties of a rhombus, and the diagonals of a rhombus are perpendicular. Therefore, m OX 90. our Turn b. Name all segments that are congruent to O in square X. Explain your reasoning. c. Name all the angles that are congruent to XO in square X. Explain your reasoning. Lesson 4 Rectangles, Rhombi, and Squares 329

heck for Understanding ommunicating Mathematics. raw a quadrilateral that is a rhombus but not a rectangle. rectangle rhombus square 2. ompare and contrast the definitions of rectangles and squares.

4 heck for Understanding ommunicating Mathematics. raw a quadrilateral that is a rhombus but not a rectangle. rectangle rhombus square 2. ompare and contrast the definitions of rectangles and squares. Eduardo says that every rhombus is a square. Teisha says that every square is a rhombus. ho is correct? Explain your reasoning. 3. Getting Ready Guided Practice hich quadrilaterals have each property? Sample: ll angles are right angles. Solution: square, rectangle 4. The opposite angles are congruent. 5. The opposite sides are congruent. 6. ll sides are congruent. Example Identify each parallelogram as a rectangle, rhombus, square, or none of these. 7. Examples 2 & 3 Use square FNRM or rhombus STPK to find each measure. 9. R. m FN 3. P Example F 24 mm N S T M 2. TP 4. m KTP 6 cm M R K cm P 0 cm 5. Sports asketball is played on a court that is shaped like a rectangle. Name two other sports that are played on a rectangular surface and two sports that are played on a surface that is not rectangular. Exercises Practice. Identify each parallelogram as a rectangle, rhombus, square, or none of these hapter Quadrilaterals 7..

5 Homework Help For Exercises See Examples 2, 3 Extra Practice See page 740. Use square SQUR or rhombus LMP to find each measure EQ SU m SEQ m SQE EU RQ m SQU m RUE P m LMP m P P M m ML L m LPM S E R L pplications and Problem Solving M ft 2 7 ft 5 ft P Exercises The Venn diagram shows relationships among some quadrilaterals. Use the Venn diagram to determine whether each statement is true or false. Every square is a rhombus. Every rhombus is a square. Every rectangle is a square. Every square is a rectangle. ll rhombi are parallelograms. Every parallelogram is a rectangle. 6 in. U Exercises hich quadrilaterals have diagonals that are perpendicular? Q Quadrilaterals Parallelograms Rhombi Rectangles Squares 45. lgebra The diagonals of a square are (x ) feet and 3x feet. Find the measure of the diagonals. 46. arpentry carpenter is starting to build a rectangular deck. He has laid out the deck and marked the corners, making sure that the two longer lengths are congruent, the two shorter lengths are congruent, and the corners form right angles. In addition, he measures the diagonals. hich theorem guarantees that the diagonals are congruent? 47. ritical Thinking Refer to rhombus PLN. a. lassify PL by its sides. b. lassify PEN by its angles. c. Is PEN EL? Explain your reasoning. P L E N Lesson 4 Rectangles, Rhombi, and Squares 33

Mixed Review etermine whether each quadrilateral is a parallelogram. State yes or no. If yes, give a reason for your answer. (Lesson 3) 4. 49. 50. etermine whether each statement is true or false.

6 Mixed Review etermine whether each quadrilateral is a parallelogram. State yes or no. If yes, give a reason for your answer. (Lesson 3) etermine whether each statement is true or false. (Lesson 2) 5. If the measure of one angle of a parallelogram is known, the measures of the other three angles can be found without using a protractor. 52. The diagonals of every parallelogram are congruent. 53. The consecutive angles of a parallelogram are complementary. Standardized Test Practice ata Update For the latest information on homes with televisions, visit: Extended Response rite the converse of this statement. (Lesson 4) If a figure is a rectangle, then it has four sides. 55. Multiple hoice If x represents the number of homes with televisions in allas, which expression represents the number of homes with televisions in tlanta? (lgebra Review) x 22 x 22 x 2035 x 2035 Homes with Televisions (thousands) rea New ork Los ngeles hicago Philadelphia San Francisco oston allas-ft. orth ashington, tlanta etroit Homes Source: Nielsen Media Research Quiz 2 > Lessons 3 and 4 etermine whether each quadrilateral is a parallelogram. State yes or no. If yes, give a reason for your answer. (Lesson 3). 2. Refer to rhombus TLE. (Lesson 4) 3. Name all angles that are congruent to IE. 4. Name all segments congruent to I E. 5. Name all measures equal to E. I T 332 hapter Quadrilaterals E L

Exterior Angle Theorem

Exterior Angle Theorem 7 Exterior ngle Theorem What You ll Learn You ll learn to identify exterior angles and remote interior angles of a triangle and use the Exterior ngle Theorem. Why It s Important Interior Design Designers