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2.3d:Quadrilaterals

2.3d:Quadrilaterals. - Squares and Rhombi. CCSS:. GSE’s. M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving polygons. Rhombi. A parallelogram with 4 congruent sides (tilted square).

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2.3d:Quadrilaterals

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  1. 2.3d:Quadrilaterals - Squares and Rhombi CCSS: GSE’s M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving polygons

  2. 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.

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

  4. Rhombi Properties In addition to the parallelogram properties and the 4 congruent sides, its also has 3 other properties • Thediagonals 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.

  5. 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

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

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

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

  9. 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) All work must be justified mathematically ANS: Square

  10. 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

  11. 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:

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

  13. Venn Diagram • Place the following polygons according to the Venn Diagram: • Quadrilaterals • Parallelograms • Rhombi • Rectangles • Squares Quadrilaterals Parallelograms Parallelograms Rectangles Rhombi Squares

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