2.3d:Quadrilaterals - Squares and Rhombi M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving polygons GSE’s CCSS:

Rhombi A parallelogram with 4 congruent sides (tilted square) Since the rhombus is a more specific parallelogram, all the properties of the parallelogram can be found in it.

POLYGONS Quadrilaterals Parallelograms RECTANGLES Has ALL the properties of the shapes above it RHOMBI 1 3) 4)

Rhombi Properties In addition to the parallelogram properties and the 4 congruent sides, its also has 3 other properties 1)The diagonals of a rhombus are perpendicular (Theorem 6-11) 2) If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus (Theorem 6-12) (the converse of 6-11 which proves a parallelogram is a rhombus) How could this theorem help us on the coordinate plane? If we have a parallelogram, we can use the slopes of the diagonals to determine they are perpendicular, therefore telling us it is a rhombus.

Rhombi Properties (Con’t) 3) Each diagonal of a rhombus bisects a pair of opposite angles. * And opposite angles are congruent since it has the properties of parallelograms

Square A Square is both a Rectangle and a Rhombus 1)Has 4 right angles like a rectangle 2) Has 4 congruent sides like a rhombus

POLYGONS Quadrilaterals Parallelograms RECTANGLES Has ALL the properties of the shapes above it RHOMBI Square 5)

Example PTRE is a square, solve for x 5x+10=45 5x = 35 x = 7

Example Determine whether the parallelogram ABDC is a rhombus, rectangle, or square if: A (-4,3) B (-2,3) C (-4, 1) D (-2,1) ANS: Square All work must be justified mathematically

Example 2 Determine whether parallelogram WXYZ is a rhombus, rectangle, or square. W (1,10) X (-4,0) Y (7,2) Z (12,12) All work must be justified mathematically ANS: Square

Kyle is building a barn for his horse. He measures the diagonals of the door opening to make sure that they bisect each other and they are congruent. How does he know that the corners are right angles? We know that. A parallelogram with congruent diagonals is a rectangle. Therefore, the corners are angles. Answer:

Example 5-2a Use rhombus LMNP to find the value of y if N EXAMPLE ANSWER: Why

Parallelograms Venn Diagram Place the following polygons according to the Venn Diagram: 1)Quadrilaterals 2)Parallelograms 3)Rhombi 4)Rectangles 5)Squares Quadrilaterals Parallelograms Rhombi Rectangles Squares