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ALGEBRA Quadrilateral ABCD Is a Rhombus. Find Each Value Or Measure. 1. If , Find . SOLUTION

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ALGEBRA Quadrilateral ABCD Is a Rhombus. Find Each Value Or Measure. 1. If , Find . SOLUTION 6-5 Special Parallelograms: Rhombi, Squares ALGEBRA Quadrilateral ABCD is a rhombus. 3. PROOF Write a two-column proof to prove that if Find each value or measure. ABCD is a rhombus with diagonal . 1. If , find . SOLUTION: A rhombus is a parallelogram with all four sides SOLUTION: congruent. So, Then, is an Given: ABCD is a rhombus with diagonal . isosceles triangle. Therefore, Prove: If a parallelogram is a rhombus, then each diagonal Proof: bisects a pair of opposite angles. So, Statements(Reasons) . 1. ABCD is a rhombus with diagonal . (Given) 2. (Diag. of rhombus bisects ) Therefore, 3. (Refl. Prop.) ANSWER: 4. (Def. of rhombus) 32 5. (SAS) 6. (CPCTC) 2. If AB = 2x + 3 and BC = x + 7, find CD. ANSWER: SOLUTION: Given: ABCD is a rhombus with diagonal . A rhombus is a parallelogram with all four sides Prove: congruent. So, Proof: Statements(Reasons) So, AB = 2(4) + 3 = 11. 1. ABCD is a rhombus with diagonal . (Given) 2. (Diag. of rhombus bisects ) CD is congruent to AB, so CD = 11. 3. (Refl. Prop.) 4. (Def. of rhombus) ANSWER: 5. (SAS) 11 6. (CPCTC) eSolutions Manual - Powered by Cognero Page 1 6-5 Special Parallelograms: Rhombi, Squares 4. GAMES The checkerboard below is made up of 64 COORDINATE GEOMETRY Given each set congruent black and red squares. Use this of vertices, determine whether QRST is a information to prove that the board itself is a square. rhombus, a rectangle, or a square. List all that apply. Explain. 5. Q(1, 2), R(–2, –1), S(1, –4), T(4, –1) SOLUTION: First, graph the quadrilateral. SOLUTION: Because each side of the board is 8 squares in length and each of the squares is congruent, the lengths of all four sides of the board are equal. Because we know that each of the four quadrilaterals that form the corners of the board are squares, we know that the measure of the angle of each vertex of the board is 90. Therefore, the board is a square. If the diagonals of the parallelogram are congruent, ANSWER: then it is a rectangle. Use the Distance Formula to Sample answer: Because each side of the board is 8 find the lengths of the diagonals. squares in length and each of the squares is congruent, the lengths of all four sides of the board are equal. Because we know that each of the four quadrilaterals that form the corners of the board are squares, we know that the measure of the angle of each vertex of the board is 90. Therefore, the board So, the parallelogram is a rectangle. Check whether is a square. the two diagonals are perpendicular. QS has a slope of . RT has a slope of . These slopes are opposite reciprocals. The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and perpendicular to each other, the parallelogram is a rectangle, a rhombus, and a square. ANSWER: Rectangle, rhombus, square; consecutive sides are , all sides are . eSolutions Manual - Powered by Cognero Page 2 6-5 Special Parallelograms: Rhombi, Squares 6. Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2) ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. SOLUTION: First, graph the quadrilateral. 7. If AB = 14, find BC. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, Therefore, BC = 14. If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to ANSWER: find the lengths of the diagonals. 14 8. If , find . SOLUTION: The diagonals are not congruent. So, the A rhombus is a parallelogram with all four sides parallelogram is not a rectangle. Check whether the congruent. So, Then, is an two diagonals are perpendicular. isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So, Therefore, ANSWER: The diagonals are not perpendicular. So, it is not a 27 rhombus either. Since it is neither a rectangle nor a 9. If AP = 3x – 1 and PC = x + 9, find AC. rhombus, it cannot be a square either. SOLUTION: ANSWER: The diagonals of a rhombus bisect each other. None; the diagonals are not congruent or 3x – 1 = x + 9 perpendicular. 2x = 10 x = 5 Therefore, AC = 2(5 + 9) = 28. ANSWER: 28 eSolutions Manual - Powered by Cognero Page 3 6-5 Special Parallelograms: Rhombi, Squares 10. If DB = 2x – 4 and PB = 2x – 9, find PD. SOLUTION: The diagonals of a rhombus bisect each other. So, SOLUTION: Therefore, PD = PB = 2(7) – 9 = 5. Given: ANSWER: Prove: WXYZ is a rhombus. 5 11. If , find . SOLUTION: In a rhombus, consecutive interior angles are supplementary. Proof: Statements(Reasons) 1. (Given) 2. WXYZ is a . (Both pairs of opp. sides are .) Each pair of opposite angles of a rhombus is 3. WXYZ is a rhombus. (If one pair of consecutive congruent. So, sides of a are , the is a rhombus.) ANSWER: ANSWER: 95 Given: 12. If , find x. Prove: WXYZ is a rhombus. SOLUTION: The diagonals of a rhombus are perpendicular to each other. ANSWER: Proof: Statements(Reasons) 35 1. (Given) CONSTRUCT ARGUMENTS Write a two- 2. WXYZ is a . (Both pairs of opp. sides are .) column proof. 3. WXYZ is a rhombus. (If one pair of consecutive 13. Given: sides of a are , the is a rhombus.) Prove: WXYZ is a rhombus. 14. Given: QRST is a parallelogram. eSolutions Manual - Powered by Cognero Page 4 6-5 Special Parallelograms: Rhombi, Squares Prove: QRST is a square. Proof: Statements(Reasons) 1. QRST is a parallelogram; . (Given) 2. QRST is a rectangle. (If the diags of a are , the is a rectangle.) 3. is a right angle. (Def of rt. ) 4. (Def. of perpendicular) SOLUTION: 5. QRST is a rhombus. (If the diags of a Given: QRST is a parallelogram. are , is a rhombus.) 6. QRST is a square. (Thm. 6.2, if a quadrilateral is a Prove: QRST is a square. rectangle and a rhombus, then it is a square.) 15. Given: ABCD is a parallelogram. Prove: ABCD is a rhombus. Proof: Statements(Reasons) 1. QRST is a parallelogram; . (Given) 2. QRST is a rectangle. (If the diags of a are , SOLUTION: the is a rectangle.) Given: ABCD is a parallelogram. 3. is a right angle. (Def of rt. ) 4. (Def. of perpendicular) Prove: ABCD is a rhombus. 5. QRST is a rhombus. (If the diags of a are , is a rhombus.) 6. QRST is a square. (Thm. 6.2, if a quadrilateral is a rectangle and a rhombus, then it is a square.) ANSWER: Given: QRST is a parallelogram. Prove: QRST is a square. Proof: Statements (Reasons) 1. .ABCD is a parallelogram, (Given) 2. (CPCTC) 3. bisects (Def. of ∠ bisector) eSolutions Manual - Powered by Cognero Page 5 6-5 Special Parallelograms: Rhombi, Squares 4. (CPCTC) Prove: ABHF is a rhombus. 5. bisects (Def. of ∠ bisector) 6. ABCD is a rhombus (If one diagonal of a parallelogram bisects a pair of opp. ∠s, then the parallelogram is a rhombus) ANSWER: Given: ABCD is a parallelogram. Proof: Prove: ABCD is a rhombus. Statements (Reasons) 1. ACDH and BCDF are parallelograms; . (Given) 2. (Def. of ) 3. (Trans. Prop) 4. (Def. of ) 5. AC = HD (Def. of segs.) 6. AC = AB + BC, HD = HF + FD (Seg. Add. Proof: Post.) Statements (Reasons) 7. AC – HD = AB + BC – HF – FD (Subt. Prop.) 1. .ABCD is a parallelogram, 8. AB = HF (Subst.) (Given) 9. (Def. of segs.) 2. (CPCTC) 10. (Subst.) 3. bisects (Def. of ∠ bisector) 11. ABFH is a rhombus. (Def. of rhombus) 4. (CPCTC) ANSWER: 5. bisects (Def. of ∠ bisector) Given: ACDH and BCDF are parallelograms; 6. ABCD is a rhombus (If one diagonal of a parallelogram bisects a pair of opp. ∠s, then the . parallelogram is a rhombus) Prove: ABHF is a rhombus. 16. Given: ACDH and BCDF are parallelograms; . Prove: ABFH is a rhombus. Proof: Statements (Reasons) 1. ACDH and BCDF are parallelograms; . (Given) SOLUTION: 2. (Def. of ) Given: ACDH and BCDF are parallelograms; 3. (Trans. Prop) . 4. (Def. of ) eSolutions Manual - Powered by Cognero Page 6 6-5 Special Parallelograms: Rhombi, Squares 5. AC = HD (Def. of segs.) 18. MODELING A landscaper has staked out the area 6. AC = AB + BC, HD = HF + FD (Seg. Add. for a square garden as shown. She has confirmed Post.) that each side of the quadrilateral formed by the 7. AC – HD = AB + BC – HF – FD (Subt. Prop.) stakes is congruent and that the diagonals are 8. AB = HF (Subst.) perpendicular. Is this information enough for the 9. (Def. of segs.) landscaper to be sure that the garden is a square? 10. (Subst.) Explain your reasoning. 11. ABFH is a rhombus. (Def. of rhombus) 17. ROADWAYS Main Street and High Street intersect as shown in the diagram. Each of the crosswalks is the same length. Classify the quadrilateral formed by the crosswalks. Explain your reasoning. SOLUTION: Because the four sides of the quadrilateral are congruent and the diagonals are perpendicular, the figure is either a square or a rhombus. To be sure that the garden is a square, she also needs to confirm that the diagonals are congruent. SOLUTION: ANSWER: The measure of the angle formed between the two streets is 29, and vertical angles are congruent, so the no; Sample answer: Because the four sides of the measure of one angle of the quadrilateral is 29. quadrilateral are congruent and the diagonals are Because the crosswalks are the same length, the perpendicular, the figure is either a square or a sides of the quadrilateral are congruent. Therefore, rhombus.
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