Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name ________________________________________ Date ___________________ Class __________________ LESSON 6-4 Reteach Properties of Special Parallelograms A rectangle is a quadrilateral with four right angles. A rectangle has the following properties. Properties of Rectangles If a quadrilateral is a rectangle, then it is a parallelogram. If a parallelogram is a rectangle, then its diagonals are congruent. Since a rectangle is a parallelogram, a rectangle also has all the properties of parallelograms. A rhombus is a quadrilateral with four congruent sides. A rhombus has the following properties. Properties of Rhombuses If a quadrilateral is a rhombus, then it is a parallelogram. If a parallelogram is a rhombus, then its diagonals are perpendicular. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Since a rhombus is a parallelogram, a rhombus also has all the properties of parallelograms. ABCD is a rectangle. Find each length. 1. BD ________________________ 3. AC ________________________ 2. CD _________________________ 4. AE _________________________ KLMN is a rhombus. Find each measure. 5. KL ________________________ 6. m∠MNK _________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-30 Holt McDougal Geometry Name ________________________________________ Date ___________________ Class __________________ LESSON 6-4 Reteach Properties of Special Parallelograms continued A square is a quadrilateral with four right angles and four congruent sides. A square is a parallelogram, a rectangle, and a rhombus. Show that the diagonals of square HJKL are congruent perpendicular bisectors of each other. Step 1 Show that HK ≅ JL . HK = JL = (6 − 0) ( 4 − 2) 2 2 + ( 4 − 2 ) = 2 10 2 + ( 0 − 6 ) = 2 10 2 HK = JL = 2 10, so HK ≅ JL . Step 2 Show that HK ⊥ JL . 4−2 1 = slope of HK = 6−0 3 slope of JL = 0−6 = −3 4−2 Since the product of the slopes is −1, HK ⊥ JL . Step 3 Show that HK and JL bisect each other by comparing their midpoints. midpoint of HK = (3, 3) midpoint of JL = (3, 3) Since they have the same midpoint, HK and JL bisect each other. The vertices of square ABCD are A(−1, 0), B(−4, 5), C(1, 8), and D(4, 3). Show that each of the following is true. 7. The diagonals are congruent. _________________________________________________________________________________________ 8. The diagonals are perpendicular bisectors of each other. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-31 Holt McDougal Geometry 6-5 CONDITIONS FOR SPECIAL Reteach 1. 13 in. PARALLELOGRAMS 2. 5 in. 3. 13 in. 4. 6.5 in. 5. 28 6. 50° 7. AC = BD = 2 17 , so AC ≅ BD . 1 , 4 so AC ⊥ BD . The mdpts. of AC and BD 8. Slope of AC = 4 and slope of BD = − are at (0, 4), so AC and BD bisect each other. Challenge 1. Practice A 1. 3. 5. 7. 9. 11. 12. rhombus 2. rectangle; rhombus 4. rhombus 6. sides 8. parallelogram 10. rhombus rectangle; rhombus perpendicular diagonals rectangle congruent rectangle Practice B 1. Possible answer: To know that the reflecting pool is a parallelogram, the congruent sides must be opposite each other. If this is true, then knowing that one angle in the pool is a right angle or that the diagonals are congruent proves that the pool is a rectangle. 2. Not valid; possible answer: you need to know that . AC ⊥ BD . 1. 8 yd 2. 25.3 ft 3. possible answer: you need to know that AC and BD bisect each other. 4. valid 5. Not valid; possible answer: you need to know that AD || BC . 3. 106° 4. 2.6 m 6. rectangle, rhombus, square Problem Solving 5. 13 in. by 12 1 in. 4 26 ; 1 −5; 5 6. B 7. F 26 7. rhombus Reading Strategies 1. no 2. yes 3. yes 4. no 5. no 2;3 2 1; −1 Practice C 6. They are all polygons, and they all have 4 sides. 1. Parallelogram and rhombus; possible answer: in a square or a rectangle, the interior angles must measure 90°. Therefore the longest side of the triangle formed by two sides and a diagonal must be the diagonal. 7. All 4 angles would have to be right angles. 8. All 4 sides would have to be congruent. 2. rhombus 3. x 3 4. 60° and 120° 5. 3 6. 1 7. 1 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A64 Holt McDougal Geometry