Properties of Special Parallelograms 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. February 15, 2019

Rhombus Parallelogram with four congruent sides. February 15, 2019

Rectangle A parallelogram with four right angles. February 15, 2019

Square A parallelogram with four congruent sides and four right angles. A rectangle with four congruent sides. A rhombus with four right angles. February 15, 2019

How these relate Polygons Quadrilaterals Parallelograms Squares Rectangles Rhombuses Squares February 15, 2019

Learning the relations Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Some Quadrilaterals are Parallelograms. Parallelograms All Parallelograms are quadrilaterals. Rhombuses February 15, 2019

Learning the relations Some Quadrilaterals are Rectangles. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares All Rectangles are Parallelograms… Parallelograms Rhombuses …and quadrilaterals, too. February 15, 2019

Learning the relations Some Quadrilaterals are Rhombuses. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Parallelograms All Rhombuses are Parallelograms… Rectangles …and quadrilaterals, too. February 15, 2019

Sometimes, Always, Never A Parallelogram is a Rhombus. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Sometimes Parallelograms Rhombuses Squares February 15, 2019

Sometimes, Always, Never A Square is a Rectangle. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Parallelograms Always Rhombuses Squares February 15, 2019

Sometimes, Always, Never A Parallelogram is a Square. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Sometimes Parallelograms Rhombuses Squares February 15, 2019

Sometimes, Always, Never A Rhombus is a Rectangle. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Sometimes – if it’s a square. Parallelograms Rhombuses Squares February 15, 2019

Sometimes, Always, Never Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares A Square is a Parallelogram. Always Parallelograms Rhombuses Squares February 15, 2019

This chart should be memorized. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Quadrilaterals Parallelograms Rhombuses Rectangles Squares February 15, 2019

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. February 15, 2019

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. February 15, 2019

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. February 15, 2019

Theorem 7.12 The diagonals of a rhombus bisect the angles. February 15, 2019

Finding Angle Measures in a Rhombus Find the measures of the numbered angles in rhombus ABCD. m1= 90 m2 = 61 m3 = 61 m1 + m3 + m4 = 180 90 + 61 + m4 = 180 151 + m4 = 180 m4 = 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. February 15, 2019

Do you know properties of a rectangle, rhombus, and square? Which have congruent sides? Rhombus & Square February 15, 2019

Do you know properties of a rectangle, rhombus, and square? Congruent diagonals? Rectangle & Square February 15, 2019

Do you know properties of a rectangle, rhombus, and square? Which have four right angles? Rectangle & Square February 15, 2019

Do you know properties of a rectangle, rhombus, and square? Which have diagonals that bisect each other? Rectangle & Rhombus & Square February 15, 2019

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. February 15, 2019

Diagonals Tell Us a Lot. rectangle rhombus square If the diagonals of a parallelogram are congruent, the parallelogram is a ______________. If diagonals of a parallelogram are perpendicular, the parallelogram is a ___________. If the diagonals of a parallelogram are perpendicular and congruent, the parallelogram is a _____________. rectangle rhombus square February 15, 2019

Figures inherit properties as we read down the chart. Polygons Quadrilaterals Parallelograms Rhombuses Rectangles Squares Parallelograms Rhombuses Rectangles Diagonals  Diagonals  Squares Diagonals  and  February 15, 2019

Coordinate Geometry Given a parallelogram is formed by the points Q(1, 4) R(1, -1) S(-3, -4) Graph these points. February 15, 2019

This is the graph: Q(1, 4) P(-3, 1) R(1, -1) S(-3, -4) February 15, 2019

What shape is it? To prove it is a rhombus you could: Q(1, 4) R(1, -1) S(-3, -4) To prove it is a rhombus you could: Show all sides congruent (we know opposite sides are parallel). Or, show the diagonals are perpendicular. February 15, 2019

Find slopes of diagonals. Q(1, 4) R(1, -1) S(-3, -4) February 15, 2019

Recall: lines are perpendicular if the product of slopes is -1. Q(1, 4) R(1, -1) S(-3, -4) Diagonals are perpendicular so the parallelogram is a rhombus. February 15, 2019

Another view Quadrilaterals Parallelograms Rhombuses Rectangles Squares February 15, 2019