6-5 Study Guide and Intervention (continued) Rhombi and Squares Conditions for Rhombi and Squares The theorems below can help you prove that a parallelogram is a rectangle, rhombus, or square. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus. If a quadrilateral is both a rectangle and a rhombus, then it is a square. Example: Determine whether parallelogram with vertices ABCD 1), is a a or a A(–3, B(1, C(5, D(1, rhombus, rectangle, square. –3), –3), –7) = = = 8 AC = = = 8 BD The diagonals are the same length; the figure is a rectangle. Slope of = = = 0 The line is horizontal. Slope of = = = The line is vertical. undefined Since a horizontal and vertical line are perpendicular, the diagonals are perpendicular. Parallelogram is a square which is also a rhombus and a rectangle. ABCD Exercises each set of vertices, determine whetherABCD is a or List all that apply. Explain. rhombus, rectangle, square. 1. 2), 4), 2), 0) 2. 1), 3), 1), A(0, B(2, C(4, D(2, A(–2, B(–1, C(3, D(2, –1) 3. 2), 4. 0), 3), A(–2, B(0, C(2, D(0, A(–3, B(–1, C(5, D(3, –1), –1), –4) –1), –4) 5. PROOF Write a two-column proof. Parallelogram RSTU. ≅ Prove: is a rhombus. RSTU
6-5 Study Guide and Intervention (continued) Rhombi and Squares Conditions for Rhombi and Squares The theorems below can help you prove that a parallelogram is a rectangle, rhombus, or square. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus. If a quadrilateral is both a rectangle and a rhombus, then it is a square. Example: Determine whether parallelogram ABCD with vertices A(–3, –3), B(1, 1), C(5, –3), D(1, –7) is a rhombus, a rectangle, or a square. AC = = = 8 BD = = = 8 The diagonals are the same length; the figure is a rectangle. Slope of = = = 0 The line is horizontal. Slope of = = = undefined The line is vertical. Since a horizontal and vertical line are perpendicular, the diagonals are perpendicular. Parallelogram ABCD is a square which is also a rhombus and a rectangle. Exercises each set of vertices, determine whetherABCD is a rhombus, rectangle, or square. List all that apply. Explain. 1. A(0, 2), B(2, 4), C(4, 2), D(2, 0) 2. A(–2, 1), B(–1, 3), C(3, 1), D(2, –1) 3. A(–2, –1), B(0, 2), C(2, –1), D(0, –4) 4. A(–3, 0), B(–1, 3), C(5, –1), D(3, –4) 5. PROOF Write a two-column proof. Parallelogram RSTU. ≅ Prove: RSTU is a rhombus.
Rhombi_and_Squares.docx