Lesson 7.4 Properties of Special Parallelograms Essential Question: What are the properties of the diagonals of rectangles, rhombuses, and squares? Parallelogram • A quadrilateral with both pairs of opposite sides parallel. January 30, 2017 3 Rhombus • Parallelogram with four congruent sides. January 30, 2017 4 Rectangle • A parallelogram with four right angles. January 30, 2017 5 Square • A parallelogram with four congruent sides and four right angles. • A rectangle with four congruent sides. • A rhombus with four right angles. January 30, 2017 6 How these relate Polygons Quadrilaterals Rhombuses Squares Parallelograms Rectangles January 30, 2017 7 Learning the relations Polygons Quadrilaterals Rhombuses Squares Parallelograms Parallelograms Rectangles Rhombuses Some Quadrilaterals are Parallelograms. All Parallelograms are quadrilaterals. January 30, 2017 8 Learning the relations Polygons Quadrilaterals Rhombuses Squares Parallelograms Parallelograms Rectangles Rhombuses Some Quadrilaterals are Rectangles. All Rectangles are Parallelograms… …and quadrilaterals, too. January 30, 2017 9 Learning the relations All Rhombuses are Parallelograms… Polygons Quadrilaterals Parallelograms Parallelograms Rhombuses Squares Some Quadrilaterals are Rhombuses. Rectangles Rectangles …and quadrilaterals, too. January 30, 2017 10 Sometimes, Always, Never Polygons Quadrilaterals Parallelograms Rhombuses Squares Parallelograms A Parallelogram is a Rhombus. Sometimes Rectangles Rhombuses Squares January 30, 2017 11 Sometimes, Always, Never Polygons A Square is a Rectangle. Quadrilaterals Parallelograms Rhombuses Squares Parallelograms Always Rectangles Rhombuses Squares January 30, 2017 12 Sometimes, Always, Never Polygons Quadrilaterals Parallelograms Rhombuses Squares Parallelograms A Parallelogram is a Square. Sometimes Rectangles Rhombuses Squares January 30, 2017 13 Sometimes, Always, Never Polygons Quadrilaterals Parallelograms Rhombuses Squares Parallelograms Rectangles A Rhombus is a Rectangle. Sometimes – if it’s a square. Rhombuses Squares January 30, 2017 14 Sometimes, Always, Never Polygons Quadrilaterals Parallelograms Rhombuses Squares Parallelograms A Square is a Parallelogram. Always Rectangles Rhombuses Squares January 30, 2017 15 This chart should be memorized. Quadrilaterals Polygons Quadrilaterals Parallelograms Rhombuses Squares Parallelograms Rhombuses Rectangles Rectangles Squares January 30, 2017 16 Example • True or False? • The diagonals of a rectangle bisect each other. • What we know: the diagonals of a parallelogram bisect each other. • A rectangle is a parallelogram. • The statement is TRUE. January 30, 2017 17 Another Example • True or False? • Opposite sides of a square are parallel. • What we know: All squares are parallelograms. • Opposite sides of a parallelogram are parallel. • Therefore, opposite sides of a square are parallel. The statement is TRUE. January 30, 2017 18 Rhombus Properties: Th. 7.11: Rhombus Diagonals Theorem The Diagonals of a rhombus are perpendicular. The converse is also true: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. January 30, 2017 19 Theorem 7.12 • The diagonals of a rhombus bisect the angles. January 30, 2017 20 Finding Angle Measures in a Rhombus Find the measures of the numbered angles in rhombus ABCD. m1= 90 m2 = 61 m3 = 61 m1 + m3 + m4 = 180 90 + 61 + m4 = 180 151 + m4 = 180 m4 = 29 Theorem 7.13 • The diagonals of a rectangle are congruent. The converse is: if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. January 30, 2017 22 Do you know properties of a rectangle, rhombus, and square? • Which have congruent sides? • Rhombus & Square January 30, 2017 23 Do you know properties of a rectangle, rhombus, and square? • Congruent diagonals? • Rectangle & Square January 30, 2017 24 Do you know properties of a rectangle, rhombus, and square? • Which have four right angles? • Rectangle & Square January 30, 2017 25 Do you know properties of a rectangle, rhombus, and square? • Which have diagonals that bisect each other? • Rectangle & Rhombus & Square January 30, 2017 26 True or False? • The Diagonals of a square are perpendicular. • TRUE! • Reason: the diagonals of a rhombus are perpendicular, and a square is a rhombus. January 30, 2017 27 Diagonals Tell Us a Lot. • If the diagonals of a parallelogram are congruent, rectangle the parallelogram is a ______________. • If diagonals of a parallelogram are perpendicular, rhombus the parallelogram is a ___________. • If the diagonals of a parallelogram are perpendicular and congruent, the parallelogram is a square _____________. January 30, 2017 28 Figures inherit properties as we read down the chart. Polygons Quadrilaterals Parallelograms Parallelograms Rhombuses Diagonals Squares Rhombuses Rectangles Rectangles Diagonals Squares Diagonals and January 30, 2017 29 Coordinate Geometry • Given a parallelogram is formed by the points • P(-3, 1) • Q(1, 4) • R(1, -1) • S(-3, -4) • Graph these points. January 30, 2017 30 This is the graph: Q(1, 4) P(-3, 1) R(1, -1) S(-3, -4) January 30, 2017 31 What shape is it? Q(1, 4) 1. Show all sides congruent (we know P(-3, 1) R(1, -1) S(-3, -4) To prove it is a rhombus you could: opposite sides are parallel). 2. Or, show the diagonals are perpendicular. January 30, 2017 32 Find slopes of diagonals. Q(1, 4) mQS P(-3, 1) mPR R(1, -1) S(-3, -4) 4 (4) 8 2 1 (3) 4 1 (1) 2 1 3 1 4 2 January 30, 2017 33 Recall: lines are perpendicular if the product of slopes is -1. Q(1, 4) P(-3, 1) R(1, -1) S(-3, -4) 1 2 1 2 Diagonals are perpendicular so the parallelogram is a rhombus. January 30, 2017 34 Another view Quadrilaterals Parallelograms Rhombuses Rectangles Squares January 30, 2017 35
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