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Page 1 of 6 6.4 Goal Use properties of special types of parallelograms. Key Words Rhombuses, Rectangles, and Squares In this lesson you will study three special types of parallelograms. A rhombus is a parallelogram with four congruent sides. rhombus • rhombus • rectangle • square A rectangle is a parallelogram with four right angles. rectangle A square is a parallelogram with four congruent sides and four right angles. EXAMPLE 1 square Use Properties of Special Parallelograms In the diagram, ABCD is a rectangle. A B a. Find AD and AB. 5 b. Find maA, maB, maC, and maD. D C 8 Solution a. By definition, a rectangle is a parallelogram, so ABCD is a parallelogram. Because opposite sides of a parallelogram are congruent, AD 5 BC 5 5 and AB 5 DC 5 8. b. By definition, a rectangle has four right angles, so maA 5 maB 5 maC 5 maD 5 908. Use Properties of Special Parallelograms P 1. In the diagram, PQRS is a rhombus. Find QR, RS, and SP. 6 P R S 6.4 Rhombuses, Rectangles, and Squares 325 Page 2 of 6 Student Help COROLLARIES STUDY TIP Rhombus Corollary The corollaries allow you to show that a quadrilateral is a rhombus, rectangle, or square without first showing that it is a parallelogram. Words If a quadrilateral has four congruent sides, then it is a rhombus. Symbols If AB &* c BC &* c CD &* c AD &*, then ABCD is a rhombus. A B D C Rectangle Corollary Words If a quadrilateral has four right angles, then it is a rectangle. Symbols If maA 5 maB 5 maC 5 maD 5 908, then ABCD is a rectangle. A B D C Square Corollary Words If a quadrilateral has four congruent sides and four right angles, then it is a square. Symbols IStudent Help EXAMPLE If AB &* c BC &* c CD &* c AD &* and maA 5 maB 5 maC 5 maD 5 908, then ABCD is a square. 2 ICLASSZONE.COM MORE EXAMPLES More examples at classzone.com A B D C Identify Special Quadrilaterals Use the information in the diagram to name the special quadrilateral. 3 5 Solution The quadrilateral has four right angles. So, by the Rectangle Corollary, the quadrilateral is a rectangle. Because all of the sides are not the same length, you know that the quadrilateral is not a square. Identify Special Quadrilaterals Use the information in the diagram to name the special quadrilateral. 2. 326 Chapter 6 Quadrilaterals 3. 4 4 4 4 Page 3 of 6 THEOREM 6.10 Words Symbols EXAMPLE In rhombus ABCD, AC &* ∏ BD &*. 3 C B The diagonals of a rhombus are perpendicular. A D Use Diagonals of a Rhombus ABCD is a rhombus. Find the value of x. B 608 C x8 E A Student Help LOOK BACK To review the Corollary to the Triangle Sum Theorem, see p. 180. D Solution By Theorem 6.10, the diagonals of a rhombus are perpendicular. Therefore, aBEC is a right angle, so TBEC is a right triangle. By the Corollary to the Triangle Sum Theorem, the acute angles of a right triangle are complementary. So, x 5 90 2 60 5 30. THEOREM 6.11 Words The diagonals of a rectangle are congruent. Symbols In rectangle ABCD, AC &* c BD &*. A B D C Carpentry EXAMPLE 4 Use Diagonals of a Rectangle a. You nail four pieces of wood together 4 ft to build a four-sided frame, as shown. What is the shape of the frame? b. The diagonals measure 7 ft 4 in. and 7 ft 2 in. Is the frame a rectangle? 6 ft 6 ft Solution DOORS If a screen door is not rectangular, you can use a piece of hardware called a turnbuckle to shorten the longer diagonal until the door is rectangular. a. The frame is a parallelogram because both 4 ft pairs of opposite sides are congruent. b. The frame is not a rectangle because the diagonals are not congruent. 6.4 Rhombuses, Rectangles, and Squares 327 Page 4 of 6 Use Diagonals Find the value of x. 4. rhombus ABCD 5. rectangle EFGH F B 6. square JKLM G K L x8 x8 A C 12 x8 x E H D J M 6.4 Exercises Guided Practice Vocabulary Check Skill Check 1. What is the name for a parallelogram with four congruent sides? List all of the properties that must be true for the quadrilateral. 2. Parallelogram A. All sides are congruent. 3. Rectangle B. All angles are congruent. 4. Rhombus C. The diagonals are congruent. 5. Square D. Opposite angles are congruent. P 6. PQRS is a rectangle. The length of &* is 12. Find PR and PT. QS R T P S Practice and Applications Extra Practice Using Properties Find the measures. See p. 686. 7. rhombus ABCD E B A Example 1: Example 2: Example 3: Example 4: 328 Exs. 7–9 Exs. 10–12 Ex. 22 Ex. 13 Chapter 6 Quadrilaterals F 9. square WXYZ Z C 4 Homework Help 8. rectangle EFGH D W 3 H G Y X AB 5 __?__ maE 5 __?__8 maW 5 __?__8 BC 5 __?__ maF 5 __?__8 YZ 5 __?__ AD 5 __?__ maG 5 __?__8 XY 5 __?__ Page 5 of 6 Careers Using Corollaries Use the information in the diagram to name the special quadrilateral. 10. 11. 12. 5 5 5 5 13. Making a Chair If you FURNITURE DESIGNERS use geometry, trigonometry, and artistic skills to create designs for furniture. Career Links measure the diagonals of the chair frame as shown and find that they are congruent, can you conclude that the frame is rectangular? If not, what other information do you need? Explain your reasoning. Sorting Quadrilaterals In Exercises 14–17, list each quadrilateral for which the statement is true. CLASSZONE.COM parallelogram square rhombus rectangle 14. It has four right angles. 15. Opposite sides are congruent. 16. Diagonals bisect each other. 17. Diagonals are perpendicular. Use Properties of Quadrilaterals EXAMPLE PQRS is a rectangle. Find the value of x. P P 2x x15 S R Solution PR 5 SQ Diagonals of a rectangle are congruent. 2x 5 x 1 5 Substitute 2x for PR and x 1 5 for SQ. x55 Subtract x from each side. Using Algebra Find the value of x. 18. rhombus KLMN K 19. square ABCD A L 5x 8 B 20. rectangle EFGH F 3x E N x12 M D G 6x J 3x 1 9 H C 6.4 Rhombuses, Rectangles, and Squares 329 Page 6 of 6 21. Logical Reasoning In g JKLM, aJ is a right angle. Explain why g JKLM is a rectangle. 22. Using Theorems Find the value of R S x in rhombus QRST. x 5 P 4 P T Challenge GHJK is a square with diagonals intersecting at L. Given that GH 5 2 and GL 5 Ï2 w, complete the statement. Ï2 24. maKLJ 5 __?__ L 25. maHJG 5 __?__ 26. Perimeter of THJK 5 __?__ Standardized Test Practice H 2 G 23. HK 5 __?__ K J 27. Multiple Choice In g KLMN, KL 5 LM. What is maN? A X C X 308 B 458 X D Cannot be determined X 908 28. Multiple Choice In rhombus ABCD, AB 5 7x 2 3 and CD 5 25. What is the value of x? F X H X Mixed Review 3 A G 4 X J 25 X 7 D 25 7x 2 3 B C Finding Angle Measures Find the measure of the numbered angle. (Lesson 3.4) 29. 30. 31. 2 3 358 1 Algebra Skills 1208 Finding Ratios Find the ratio of the length to the width for the rectangle. Write the ratio in simplest form. (Skills Review, p. 660) 32. 33. 34. 7 in. 5 cm 9 cm 330 Chapter 6 Quadrilaterals 12 m 14 in. 18 m