Then/Now You determined whether quadrilaterals were parallelograms and/or rectangles. Recognize and apply the properties of rhombi and squares. Determine.

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Then/Now You determined whether quadrilaterals were parallelograms and/or rectangles. Recognize and apply the properties of rhombi and squares. Determine whether quadrilaterals are rectangles, rhombi, or squares.

Vocabulary rhombus square

Concept 1

Example 1B Use Properties of a Rhombus B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

Example 1B Use Properties of a Rhombus Answer: x = 4 WX  WZBy definition, all sides of a rhombus are congruent. WX=WZDefinition of congruence 8x – 5=6x + 3Substitution 2x – 5=3Subtract 6x from each side. 2x=8Add 5 to each side. x=4Divide each side by 4.

Example 1A A.m  CDB = 126 B.m  CDB = 63 C.m  CDB = 54 D.m  CDB = 27 A. ABCD is a rhombus. Find m  CDB if m  ABC = 126.

Example 1B A.x = 1 B.x = 3 C.x = 4 D.x = 6 B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

Concept 3

Concept

Example 2 Is there enough information given to prove that ABCD is a rhombus? Given:ABCD is a parallelogram. AD  DC Prove:ADCD is a rhombus

Example 2 A.Yes, if one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus. B.No, you need more information.

Example 4 Classify Quadrilaterals Using Coordinate Geometry Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain. UnderstandPlot the vertices on a coordinate plane.

Example 4 Classify Quadrilaterals Using Coordinate Geometry PlanIf the diagonals are perpendicular, then ABCD is either a rhombus or a square. The diagonals of a rectangle are congruent. If the diagonals are congruent and perpendicular, then ABCD is a square. SolveUse the Distance Formula to compare the lengths of the diagonals. It appears from the graph that the parallelogram is a rhombus, rectangle, and a square.

Example 4 Classify Quadrilaterals Using Coordinate Geometry Use slope to determine whether the diagonals are perpendicular.

Example 4 Classify Quadrilaterals Using Coordinate Geometry Since the slope of is the negative reciprocal of the slope of the diagonals are perpendicular. The lengths of and are the same, so the diagonals are congruent. Answer:ABCD is a rhombus, a rectangle, and a square. CheckYou can verify ABCD is a square by using the Distance Formula to show that all four sides are congruent and by using the Slope Formula to show consecutive sides are perpendicular.

Example 4 A.rhombus only B.rectangle only C.rhombus, rectangle, and square D.none of these Determine whether parallelogram EFGH is a rhombus, a rectangle, or a square for E(0, –2), F(–3, 0), G(–1, 3), and H(2, 1). List all that apply.