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Lesson Notes 6.4 Special Parallelograms Diagonals have special properties in the rectangle and rhombus. Recall: Rhombus: Rectangle: If a parallelogram is: A rhombus Each diagonal bisects two angles m1 m2 and m3 m4 A rhombus The diagonals are perpendicular BD AC A rectangle The diagonals are congruent AC = BD Guided Problem Solving Given the rectangle to the right, Find the missing angles. Property Conclusion 4 is supplementary to ___________ m4 = 180 - ______ =______ The diagonals of a rectangle are ______________ _________ = _________ The diagonals of a parallelogram ________________ each other. BM = ___________ And AM=______ The bisected segments are congruent BM = _____ BMC is an isosceles triangle, so by the Isosceles Triangle Theorem. (use this theorem for any rectangle) Triangle Angle Sum, m1 m2 40 180 3 and 2 are complementary m1 m _________ m1 m2 ______ m3 90 _____ Note: Parallelogram -> bisects ___________ and Rectangle -> has congruent ___________ Therefore, the segments of the diagonal are _____________________ to each other. For each parallelogram, (a) choose the best name, and then (b) find the measures of the numbered angles. 1. 2. 3. 4. 5. 6. HIJK is a rectangle. Find the value of x and the length of each diagonal. 10. and 11. and For each rhombus, (a) find the measures of the numbered angles, and then (b) find the area. 14. 15. 16. To determine if a parallelogram is a rectangle or rhombus. Given a parallelogram…. If this is true Theorem 6-12 One diagonal __________ two angles of the parallelogram Theorem 6-6 Diagonals of a parallelogram are __________________ Theorem 6-7 Diagonals of a parallelogram are __________________ diagram algebra <XWE=<ZWE and <XEY=<ZEY OR Then WXYZ is a rhombus XZ is perpendicular to YW Then WXYZ is a rhombus XZ=WY Then WXYZ is a rectangle Determine whether the quadrilateral can be a parallelogram. If not, write impossible. 17. One pair of opposite sides is parallel, and the other pair is congruent. Explain your answer. 18. Opposite angles are congruent and supplementary, but the quadrilateral is not a rectangle. The parallelograms below are not drawn to scale. Can the parallelogram have the conditions marked? If not, write impossible. Explain your answer. 7. 8. 9.