Lesson 6 – 5 Rhombi and Squares

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Presentation transcript:

Lesson 6 – 5 Rhombi and Squares Geometry Lesson 6 – 5 Rhombi and Squares Objective: Recognize and apply the properties of rhombi and squares. Determine whether quadrilaterals are rectangles, rhombi, or squares.

Rhombus What is the definition of a rhombus? A parallelogram with all four sides congruent.

Properties of Rhombus Theorem 6.15 If a parallelogram is a rhombus, then its diagonals are perpendicular.

Properties of Rhombus Theorem 6.16 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

The diagonals of rhombus FGHJ intersect at K The diagonals of rhombus FGHJ intersect at K. Use the given info to find each value. 98 49 82 49

If GH = x + 9 and JH = 5x – 2, find x. If FK = 5 and FG = 13, find KJ. 5x – 2 = x + 9 9y - 5 13 4x = 11 x + 9 x = 2.75 5 6y + 7 (FK)2 + (GK)2 = (FG)2 5x - 2 52 + (GK)2 = 132 (GK)2 = 144 GK = 12 6y + 7 = 9y - 5 12 = 3y 4 = y

Square What is the definition of a square? A parallelogram with four congruent sides and four right angles.

Summary: Flow chart Quadrilateral Parallelogram Rectangle Rhombus Square Square has all of the properties of both rectangles and rhombi.

Summary: Venn Diagram Parallelograms Rhombi Rectangles 4 congruent sides Rectangles 4 right angles Squares 4 right angles & 4 congruent sides

Conditions for Rhombi and Squares Theorem 6.17 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Conditions for Rhombi and Squares Theorem 6.18 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Conditions for Rhombi and Squares NEW! Theorem 6.19 If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.

Conditions for Rhombi and Squares Theorem 6.20 If a quadrilateral is both a rectangle and a rhombus, then it is a square.

Is the figure a rectangle? Are the diagonals congruent? Determine whether parallelogram JKLM with vertices J (-7, -2) K(0, 4) L (9, 2) and M (2, -4) is a rhombus, a rectangle, or a square. List all that apply. Explain. Is the figure a rectangle? Are the diagonals congruent? The figure is not a rectangle. If its not a rectangle, then its not a square. Is the figure a rhombus? Can either check that 2 consecutive sides are congruent or that the slope of the diagonals are perpendicular. Slope of KM = -4 Slope of JL = 1/4 Parallelogram JKLM is a Rhombus

Is the figure a rectangle? Given J (5, 0) L (-3, -14) K (8, -11) M (-6, -3), determine whether parallelogram JKLM is a rhombus, rectangle, or square. List all that apply. Explain. Is the figure a rectangle? The figure is a rectangle. Is the figure a square? Are the diagonals perpendicular? Slope of JL = 7/4 Slope of KM = -4/7 The figure is a square. Since it’s a square it is also a rhombus. The figure is a rhombus, rectangle, and a square.

Homework Pg. 431 1 – 6 all, 8 – 14 E, 22 – 30 E, 48, 52 – 60 E